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On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium Yu. G. Rykov Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 2, DOI: https://doi.org/10.4213/im9507 
Bibliographic databases:
    § 1. Introduction1.1. Origin of interest in the system of equations of isobaric mediaThe study of the motion of media in which one can neglect media’s own pressure drop at a given time (briefly, pressureless media) is of both mathematical and applied interest. The immediate model of such environments is equations of gas dynamics in which the pressure $P$ is formally assumed to be zero. From the point of view of applications, pressureless media arise when describing various physical phenomena, such as the evolution of multiphase flows, the movement of dispersed media, in particular, dust particles or droplets, the phenomenon of cumulation, the interaction of hypersonic flows in some extreme cases, the movement of granular media, etc. Examples of various gas-dynamic problems involving pressureless media can be found, for example, in the classical monographs [1]–[3]. Pressureless media are also used in astrophysics to approximate large-scale distribution of matter in the Universe (see, for example, the fundamental paper [4] and the survey [5]). Mathematically, the model of isobaric media also provides a wide range of different problems. For example, an extensive program was carried out to study classical solutions of the corresponding system of equations up to the moment of occurrence of singularities based on the technique of group-theoretic analysis (see, among others, [6]–[8]). In particular, in these publications it was found that singularities in solutions can arise on manifolds of different dimensions. However, A. N. Kraiko, even in the late 1970s , when considering the above-mentioned gas-dynamic problems, showed at the physical level of rigor that solutions of equations for pressureless media can make sense even after the appearance of singularities, while a new type of discontinuous solutions arises in which strong density singularities are formed on hypersurfaces of different codimension. In particular, in [9], also, at the physical level of rigor, the laws of evolution of such hypersurfaces were obtained. Additionally, we note that there is extensive literature of an applied nature devoted to the description and calculation of two-phase flows in a three-dimensional formulation, where one phase is an ordinary gas and the other is a medium of solid particles with constant pressure (see, for example, [10], [11]). Here, along with gas-dynamic shock waves, there are also specific singularities associated with a medium without pressure, which are known as “films or sheets” and “filaments or strings” in order to distinguish them from gas-dynamic shock waves. Additionally, we note the recent paper [12], where the model of the medium consisted of finely dispersed particles is considered with zero pressure, but only until a certain density threshold is reached. This leads to the disappearance of strong singularities, but the emergence of non-standard solutions to the Riemann problem, on which the corresponding numerical algorithm is based. In the following sections, we will be interested in the mathematical aspect of the evolution of specific strong singularities associated with pressureless medium. Therefore, we will approach this issue from the standpoint of the general theory of conservation laws and call all the singularities arising from the intersection of any fields of characteristics by “shock waves”, without distinguishing their special types. Consequently, from this point of view, both “films” and “filaments” are shock waves, and the only type of shock waves, since gas-dust media will not be considered in this publication. In addition, unlike the above studies in the field of applied gas dynamics, where the three-dimensional case was studied, we will consider only the case of two spatial variables. In this formulation, there is only one type of singularity with the presence of density delta function on spatial hypersurfaces — curves in the space $\mathbb{R}^2$, which can be identified as a “filament”. In this paper, it is rigorously shown that the interaction of such “filaments” (shock waves) can lead to the occurrence of singularities of another type with delta density function at a point in space $\mathbb{R}^2$, which from the point of view of the theory of conservation laws is also a “shock wave”. Previously, the hypothetical possibility of the existence of this type of interaction of shock waves for pressureless media was theoretically indicated, and then obtained, but only in numerical calculations, by the author and his co-authors [13]–[15]. 1.2. Model description and historical digressionLet us proceed with the description of the model of a pressureless medium under consideration. Let $\mathbf{x}\equiv(x,y)$, $(t,\mathbf{x})\in \mathbb{R}_{+}\times\mathbb{R}^2$, $\mathbf{\nabla}=(\partial/\partial x,\partial/\partial y)$, then the two-dimensional system of equations of pressureless gas dynamics is as follows
$$
\begin{equation}
\frac{\partial\varrho}{\partial t} +\mathbf{\nabla}\cdot\varrho\mathbf{u}=0,\qquad \frac{\partial (\varrho\mathbf{u})}{\partial t} +\mathbf{\nabla}\cdot (\varrho\mathbf{u}\otimes\mathbf{u})=0,
\end{equation}
\tag{1.1}
$$
where $\varrho>0$ is the density of matter, $\mathbf{u}\equiv (u,v)$ is the velocity vector, and $\otimes$ denotes the tensor product. The traditional system of equations of gas dynamics (when dealing with non- isentropic flows) also includes the equation of conservation of total energy; therefore the following equation should be added to system (1.1), putting the pressure equal to zero in the traditional conservation law,
$$
\begin{equation}
\frac{\partial E}{\partial t}+\mathbf{\nabla}\cdot(\mathbf{u}E)=0,
\end{equation}
\tag{1.2}
$$
where $E\equiv\varrho(e+|\mathbf{u}|^2/2)$, and $e$ is the specific internal energy. However, equation (1.2) within the framework of the model of pressureless media is independent of system (1.1) in the sense that the evolution of both smooth parts of the solution and shock waves is determined only by (1.1), but (1.2) defines the law of change of an additional quantity (the specific internal energy $e$) in accordance with a known solution (possibly with shock waves) containing density and velocity field. Because of this fact, the proposed paper considers only the Cauchy problem for system (1.1) with initial data
$$
\begin{equation}
\varrho(0,\mathbf{x}) = \varrho_0(\mathbf{x})\geqslant 0,\qquad \mathbf{u}(0,\mathbf{x}) = \mathbf{u}_0(\mathbf{x}),
\end{equation}
\tag{1.3}
$$
where the functions $\varrho_0$, $\mathbf{u}_0$, generally speaking, are arbitrary bounded measurable functions. Despite the external simplicity, the emergence of strong singularities brings to problem (1.1), (1.3) additional difficulties and increases its attractiveness for mathematical research. The literature in this direction is quite extensive, therefore, without claiming to be complete, we will cite a number of characteristic publications that, from the author’s point of view, give a sufficient idea of the state of research in this area. The paper [16] marks the beginning of an intensive study of system (1.1) in the one-dimensional setting. In [17], [18], with the help of various approaches, a theorem on existence of generalized solutions in the space of measures was proved. Note that in what follows, the generalized solutions of system (1.1) will, be generally interpreted in the sense of Radon measures, and the initial data will also be understood accordingly. Namely, the initial Cauchy data will be specified as Radon measures $P_0(dx,dy)$, $I_0(dx,dy)$, $J_0(dx,dy)$, defined on Borel subsets of $\mathbb{R}^2$. Here, $P_0\geqslant 0$, and $dI_0/dP_0=u_0(x,y)$ and $dJ_0/dP_0=v_0(x,y)$ exist in the Radon–Nikodým sense. In the case of initial conditions (1.3), the corresponding measures $P_0$, $I_0$, $J_0$ will be absolutely continuous with respect to the standard Lebesgue measure. In [19], a detailed description of a variational principle for a one-dimensional version of system (1.1) was given capable of constructively describing the structure of generalized solutions. An existence theorem on the basis of this principle was proved. The essence of this variational principle is as follows. To obtain a generalized solution $u(t,x)$ to problem (1.1), (1.3) in the one-dimensional setting, we first must find the global minimum $a(t,x)$ with respect to $a$ in the function
$$
\begin{equation}
F(t,x;a)=\int_{0-0}^{a-0}(u_0(\sigma)t+\sigma-x)\, P_0(d\sigma).
\end{equation}
\tag{1.4}
$$
If such a minimum is unique, then $u(t,x) = (x-a(t,x))/t$ and is continuous, and the measure $P_t(dx;x)$ is absolutely continuous with respect to the Lebesgue measure and has density $\varrho$ such that $\varrho(t,x)\, dx = P_0(da)$. If $F$ has several minima, then there is a discontinuity in the velocity $u$, and $P_t(dx;x)=\int_{a_{\min}(t,x)}^{a_{\max}(t,x)}P_0(da)$. For completeness, we note that non-classical solutions of a one-dimensional system of equations of pressureless gas dynamics not satisfying the variational principle described above were considered in [20] and [21]. Such generalized solutions are not entropic, but can describe a number of physical processes, for example, deposition or decomposition of matter. Problems of both existence and uniqueness of generalized solutions to problem (1.1), (1.3) in the one-dimensional setting, in which the initial data can be measures, were considered, for example, in [22], [23]. The system of equations of pressureless gas dynamics is a quasi-linear system of equations having a single eigenvalue and an incomplete set of eigenvectors. Therefore, strictly speaking, it is not hyperbolic. However, the behaviour of solutions of this system, including generalized ones, can be described in terms characteristic of the theory of hyperbolic systems of conservation laws. Therefore, here, we will adhere to the terminology of this theory, and characterize the system of equations of pressureless gas dynamics as a degenerate non-strictly hyperbolic system of equations. Due to the degeneracy, the uniqueness conditions for the system of equations under consideration can take various forms, but are generally conditioned by, firstly, the requirement of the necessary number of characteristics coming to the singularity manifold (in accordance with the general theory of conservation laws). For example, in the above studies, the uniqueness condition included a condition of type $E$, was proposed by O. A. Oleinik. Second, one more condition is necessary, containing some form of prohibition of the decomposition of the substance. Finally, we note the recent papers [24], [25], where in a one-dimensional formulation the variational approach is generalized to the case of presence of external forces. The case of many spatial variables is much less studied. In the book [26] for the case of two spatial variables, Rankine–Hugoniot type relations were obtained in differential form for the case of strong singularities on surfaces in the $(t,\mathbf{x})$-space and solutions of the two-dimensional Riemann problem that involving singularities were studied. This was done as part of the study of solutions to the two-dimensional Riemann problem for a system of equations of traditional gas dynamics. The Rankine–Hugoniot relations obtained in [26] in the case of pressureless media represent an evolutionary system of partial differential equations, which includes, in addition to the dynamics of the singularity surface, also the dynamics of the density of the concentration of matter. Therefore, generally speaking, they are more complex than the Rankine–Hugoniot relations for ordinary gas dynamics. Nevertheless, these more involved relations are natural for a system of equations of pressureless gas dynamics from the point of view of the general theory of conservation laws. Therefore, in what follows, any relations arising on special surfaces or curves in the $(t,\mathbf{x})$-space, will simply be called Rankine–Hugoniot relations. Rankine–Hugoniot relations in differential form, which are similar to those of [26], were obtained on the basis of Colombeau’s theory of new generalized functions (see, for example, [27]) in [28] (for a more detailed account, see [29]), where their description was also given in the integral form. The integral description of the Rankine–Hugoniot relations actually suggests the possibility of evolving singularities on manifolds of different dimensions, which was noted in [29]. Thus, a hierarchy of shock waves arises. Later in [30], [31], this assumption was given a more constructive form based on a variational approach. We also note that in [32], the results of [26] in relation to the Riemann problem for a two-dimensional system of equations of pressureless gas dynamics were transferred to the complete system (1.1), (1.2), which also contains the energy equation. The concentration of matter in the multidimensional case on surfaces of codimension one was studied, for example, in [33]–[35]. In addition, the involved issues of propagation of the velocity vector field to the points $(t,\mathbf{x})$ located inside the singularities of different dimensions are considered in [36], [37] in the case when system (1.1) can be represented as the Hamilton–Jacobi equation; a generalization to an arbitrary convex Hamiltonian can also be made. In this case, a variational interpretation of the “viscous solutions” of this equation is used. The results obtained can be useful, in particular, in astrophysical applications (see, for example, [5]). The paper is organized as follows. In § 2, we give necessary concepts and definitions including those of a generalized solution, the Rankine–Hugoniot relations. We also give a brief description of the adhesion model and a self-similar form of solutions for a two-dimensional pressureless gas dynamics system. The main results are formulated in § 3. Theorem 1 establishes the existence of an interaction of shock waves for which a delta singularity is formed at a point on the plane. In Theorem 2, the complete Rankine–Hugoniot relations are formulated in the case of such singularities at a point. These results are proved in § 4. In this section, we also establish some additional results, which we formulate Theorems 3 and 4, which establish some qualitative properties of shock waves whose interaction leads to the appearance of a delta singularity at a point on the plane. § 2. Basic concepts and preliminary considerations2.1. The concept of a generalized solution and its main elementsIn this section, we introduce the necessary notation and definitions, and formulate a number of well-known results characterizing the structure of generalized solutions of system (1.1). We set
$$
\begin{equation*}
d\mathbf{x}\equiv(dx,dy),\quad \mathbf{u}(t,\mathbf{x}) \equiv(u(t,\mathbf{x}), v(t,\mathbf{x})), \quad \mathbf{I}_t(d\mathbf{x})\equiv (I_t(dx,dy),J_t(dx,dy)).
\end{equation*}
\notag
$$
By a generalized solution of problem (1.1), (1.3) we will understand a Radon family of measures $P_t(d\mathbf{x})$, $\mathbf{I}_t(d\mathbf{x})$, where the index $t$ is the time variable. The initial data are understood accordingly. Namely, in the case of initial data (1.3), appropriate measures $P_0(d\mathbf{x})$, $\mathbf{I}_0(d\mathbf{x})$ are absolutely continuous with respect to the standard Lebesgue measure with densities $\varrho_0(\mathbf{x})$, $\varrho_0(\mathbf{x})\mathbf{u}_0(\mathbf{x})$. Definition 1. Let $\mathfrak{P}_t(d\mathbf{x})\equiv(P_t(d\mathbf{x}), \mathbf{I}_t(d\mathbf{x}))$ be families of Radon measure defined on Borel subsets of $\mathbb{R}^2$, weakly continuous with respect to $t$. Let $P_t\geqslant 0$, and let the measure $\mathbf{I}_t$ be absolutely continuous with respect to $P_t$ for almost all $t>0$. We define the vector function $\mathbf{u}(t,\mathbf{x})$ as the Radon–Nikodým derivative $\mathbf{u}(t,\mathbf{x})=d\mathbf{I}_t/dP_t$. The measures $\mathfrak{P}_t$ are called a generalized solution of the Cauchy problem (1.1), (1.3) if: 1) for any vector function $\mathbf{f}\colon\mathbb{R}^2\to\mathbb{R}^3$, $\mathbf{f}\in C_0^1(\mathbb{R}^2)$ and any $0<t_1<t_2<+\infty$,
$$
\begin{equation}
\iint \mathbf{f}(\mathbf{x})\odot\mathfrak{P}_{t_2}\, (d\mathbf{x})- \iint \mathbf{f}(\mathbf{x})\odot\mathfrak{P}_{t_1}\, (d\mathbf{x})= \int_{t_1}^{t_2}\iint(\mathbf{\nabla} \otimes\mathbf{f})^{\top}\cdot\mathbf{u} \odot \mathfrak{P}_{\tau}\, (d\mathbf{x})\, d\tau,
\end{equation}
\tag{2.1}
$$
where $\odot$ is the componentwise multiplication (the Hadamard product), the index $\top$ denotes transposition, and $\iint$ is the integration over $\mathbb{R}^2$; 2) in the weak sense as $t\to +0$ $P_t\to P_0$, $\mathbf{I}_t\to \mathbf{I}_0$. If we consider the classical solutions of system (1.1) and perform the corresponding product differentiations, then (1.1) can be written in the matrix form, $\mathcal{U}\equiv(\varrho,u,v)$,
$$
\begin{equation}
A_0\, \frac{\partial\mathcal{U}}{\partial t} + A_1\, \frac{\partial\mathcal{U}}{\partial x} + A_2\, \frac{\partial\mathcal{U}}{\partial y}=0,
\end{equation}
\tag{2.2}
$$
where $A_i$, $i=0,1,2$, are appropriate matrices of size $3\times 3$. According to the general theory, the characteristic surfaces $\varphi(t,x,y)=0$ for (2.2) are given by the characteristic equation
$$
\begin{equation}
\biggl|\frac{\partial\varphi}{\partial t}\, A_0 + \frac{\partial\varphi}{\partial x}\, A_1 + \frac{\partial\varphi}{\partial y}\, A_2\biggr|=0,
\end{equation}
\tag{2.3}
$$
where $|\,{\cdot}\,|$ denotes the determinant. The next result can be easily obtained by direct calculations. Proposition 1. Equation (2.3) has a triple root $D\equiv\partial\varphi/\partial t+\mathbf{u}\cdot\mathbf{\nabla}\varphi=0$ and corresponding incomplete sets of left $L_1\equiv(u,-1,0)$, $L_2\equiv(v,0,-1)$ and right $R_1\equiv(1,0,0)$, $R_2\equiv(1,\partial\varphi/\partial y,-\partial\varphi/\partial x)$ eigenvectors. From Proposition 1 it follows that the bicharacteristics of (2.2) are the lines $\dot{\mathbf{x}}=\mathbf{u}$. Also, the left multiplication of (2.2) by left eigenvectors leads to characteristic relations for (2.2), which turn out to be an inviscid two-dimensional Burgers equation
$$
\begin{equation}
\frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\mathbf{\nabla})\mathbf{u}=0.
\end{equation}
\tag{2.4}
$$
It immediately follows from (2.4) that the bicharacteristics are straight lines, and the characteristic surfaces $\varphi(t,\mathbf{x})=0$ correspond to such functions $\varphi$ that are constant along the bicharacteristics. The described results make it possible to immediately obtain a classical solution (up to the time it exists) of problem (1.1), (1.3). Given $\mathbf{a}\equiv (a,b)$, let the mapping $\mathcal{L}_{\tau}\colon\mathbf{a}\to\mathbf{x}$ be defined by ($0\leqslant\tau\leqslant t$)
$$
\begin{equation}
\mathbf{x}=\mathbf{x}^*(\tau,\mathbf{a})\equiv\mathbf{a}+\tau\mathbf{u}_0(\mathbf{a}).
\end{equation}
\tag{2.5}
$$
We now arrive at the following result, which is obtained by using equation (2.4) and the continuity equation. Proposition 2. Let $\mathbf{a}=(\mathbf{x}^*)^{-1}(\mathbf{x})$, then as long as $|\partial\mathbf{x}^*(\tau,\mathbf{a})/\partial\mathbf{a}|\ne 0$, the solution to problem (1.1), (1.3) is classical and is given by If the initial data is such that the determinant $|\partial\mathbf{x}^*(\tau,\mathbf{a})/\partial\mathbf{a}|$ vanishes at some time, then the solution ceases to be classical, and shock waves occur, which, due to the degeneracy of (1.1), have a density delta function defined, in general, on manifolds of different dimensions in the $\mathbf{x}$-space. In the two-dimensional case, known singularities of this kind are evolving curves in the $\mathbf{x}$-space or, equivalently, surfaces in the $(t,\mathbf{x})$-space. In order for a solution with delta density function on a curve to be generalized (in the sense of Definition 1) solution of system (1.1) it is necessary, in analogy with ordinary gas dynamics, to fulfill certain relations. These relations will also be called Rankine–Hugoniot relations, even though they are substantially different from those in gas dynamics. This will be discussed in more details below. We describe such relations in the considered case of singularities along curves in the $\mathbf{x}$-space. To do this, consider a surface $\Gamma(t,\mathbf{x})=0$, $\Gamma\in C^1([t_1,t_2]\times\mathbb{R}^2)$ , in the $(t,\mathbf{x})$-space on the time interval $t\in [t_1,t_2]$. Let the surface $\Gamma$ be given parametrically by
$$
\begin{equation*}
\mathbf{x}=\mathbf{X}_i(t,l)\in C^1([t_1,t_2]\times R),\qquad \mathbf{X}\equiv(\chi,\gamma).
\end{equation*}
\notag
$$
The orientation of $\Gamma$ coincides with that is of $(t,\mathbf{x})$. In accordance with the direction of the positive normal, we define the positive “$+$” and negative “$-$” sides of $\Gamma$. Accordingly, the values related to the two sides of the surface will be provided with the same indices. Consider a family of measures of the form
$$
\begin{equation}
\begin{aligned} \, P_t &=P^-+(P^+-P^-)H(\Gamma)+\widetilde{P}(t,l)\delta(\Gamma), \\ \mathbf{I}_t &=\mathbf{I}^-+(\mathbf{I}^+-\mathbf{I}^-)H(\Gamma) +\widetilde{\mathbf{I}}(t,l)\delta(\Gamma), \end{aligned}
\end{equation}
\tag{2.7}
$$
where $\Gamma$ is the above surface; $H$ is the Heaviside function; $P^{\pm}$, $\mathbf{I}^{\pm}$ are measures that are absolutely continuous with respect to the Lebesgue measure, with densities $\varrho^{\pm}$, $\varrho^{\pm}\mathbf{u}^{\pm}$, respectively; $\delta$ is the Dirac measure on a curve. The functions $\varrho^{\pm}$, $\varrho^{\pm}\mathbf{u}^{\pm}$, $\widetilde{P}$, $\widetilde{\mathbf{I}}$ are assumed to be piecewise continuously differentiable on their domains of definition. Proposition 3 (see, for example, [26], and also [28]). Let the family of measures $\mathfrak{P}_t(d\mathbf{x}) \equiv(P_t(d\mathbf{x}), \mathbf{I}_t(d\mathbf{x}))$, as defined by (2.7), form a generalized solution to problem (1.1), (1.3) in the sense of Definition 1. Then the functions $(\varrho^{\pm},\mathbf{u}^{\pm})$ satisfy (1.1) in the classical sense, and, along the surface $\Gamma$ Formulae (2.8) are the Rankine–Hugoniot relations, which are via the general theory of conservation laws. If we put $\widetilde{P}=\widetilde{\mathbf{I}}=0$, then (2.8) will go into the Rankine–Hugoniot relations characteristic of the conservative notation of the system of conservation laws (1.1). However, the presence of delta singularities leads naturally to the appearance of an additional evolutionary (containing the time derivative) term in (2.8), but, by their nature, these more involved relations remain Rankine–Hugoniot relations, which, however, are simply obtained for a more involved kind of generalized solutions. That is why here we will avoid terms like “generalized Rankine–Hugoniot relations” or “extension of generalized Rankine–Hugoniot relations”. To implement solutions of the form (2.7), it is necessary that the bicharacteristics, which we will be simply called characteristics, come to $\Gamma$ from both sides of this surface. It is also necessary that the projection to the positive normal of the velocity $(U,V)$, as defined on $\Gamma$, lies between the corresponding projections of velocities from the “$+$” and “$-$” sides. The positive normal is defined as $\mathbf{N}\equiv(-\partial\gamma/\partial l,\partial\chi/\partial l)$. In fact, relations (2.9) are stability conditions of P. Lax type. In [26] and many subsequent studies, the conditions (2.9) are used to introduce the concept of entropy solutions in the study of the two-dimensional Riemann problem for (1.1). Proposition 4. Let $\varrho^\pm$, $\mathbf{u}^\pm$ be constant. Then system (2.8) has the first integral where $|\mathbf{p}\ \mathbf{q}|$ denotes the determinant of the matrix constructed on vectors $\mathbf{p}$ and $\mathbf{q}$. Proof. We write (2.8) as a quasilinear system of equations. Namely, consider the vector of variables $\mathbf{W}\equiv(\mathbf{x},\widetilde{P},\widetilde{\mathbf{I}})$. Now (2.8) can be written as
$$
\begin{equation}
\begin{gathered} \, \frac{\partial}{\partial t}\mathbf{W} +\mathcal{A}\, \frac{\partial}{\partial l}\mathbf{W}= \begin{pmatrix} U\\V\\0\\0\\0 \end{pmatrix}, \\ \mathcal{A} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ [\varrho v]-V[\varrho] & U[\varrho]-[\varrho u] & 0 & 0 & 0 \\ [\varrho uv]-V[\varrho u] & U[\varrho u]-[\varrho u^2] & 0 & 0 & 0 \\ [\varrho v^2]-V[\varrho v] & U[\varrho v]-[\varrho uv] & 0 & 0 & 0 \end{pmatrix}. \end{gathered}
\end{equation}
\tag{2.11}
$$
The matrix $\mathcal{A}$ has a five-fold null eigenvalue, while the left eigenvectors with coordinates $l_i$, $i=1,\dots,5$, satisfy the relations
$$
\begin{equation}
\begin{aligned} \, l_3([\varrho v]-V[\varrho])+ l_4([\varrho uv]-V[\varrho u])+ l_5([\varrho v^2]-V[\varrho v]) &=0, \\ l_3(U[\varrho]-[\varrho u])+ l_4(U[\varrho u]-[\varrho u^2])+ l_5(U[\varrho v]-[\varrho uv]) &=0. \end{aligned}
\end{equation}
\tag{2.12}
$$
From (2.12) it follows that the vector with coordinates $(l_3,l_4,l_5)$ is parallel to the vector $(|\mathbf{u}^+\mathbf{u}^-|,[v],-[u])$. Left-multiplying (2.11) by the left eigenvector, we find that
$$
\begin{equation*}
\frac{\partial}{\partial t}\bigl(l_3\widetilde{P} + l_4\widetilde{I}+l_5\widetilde{J}\,\bigr)=0,
\end{equation*}
\notag
$$
from which Proposition 4 follows. 2.2. On the construction of a generalized solution by the dynamics of adhesionIn [19], it was shown that, in the case of one spatial variable, generalized solutions of problem (1.1), (1.3) can be constructed using the so-called sticking particles dynamics. In other words, it was proposed to consider the evolution a system of particles distributed in accordance with the initial data (1.3), which move by inertia, and are subject to absolutely inelastic collisions in accordance with the laws of conservation of mass and momentum. It was shown that, by increasing the number of particles, it is possible, after a passage to the limit, to obtain generalized solutions to problem (1.1), (1.3). The justification of the limiting transition was based on the variational representation of solutions (1.4), which is also valid for the case of a set of particles, that is, in the case when the measure $P_0(d\sigma)$ is discrete. If only one spatial variable $x$ is considered, then the particles flying towards each other will necessarily collide and will continue to move as a whole. Consequently, at each time $t$, it is possible to define the mapping $x=\varphi_t(a)$, where $a$ is the Lagrangian coordinate of the particle. A mapping of this kind will be called a Lagrangian in what follows. A Lagrangian mapping, which in general is not one-one, shows which particles with the initial coordinate $a$ will be at the point $x$ at time $t$. Note that according to [19], the construction of Lagrangian maps is closely related to the existence and structure of generalized solutions of a system of equations of pressureless gas dynamics. For system (1.1), in the case of one spatial variable for small values of time $t$ and smooth initial data, the mapping $\varphi_t(a)$ has the form $\varphi_t(a)=a+tu_0(a)$. With increasing $t$, this mapping may cease to be one-one, so that one value of $x$ will correspond to an entire segment $[a_1,a_2]$ of values of $a$. Such a loss of mutual unambiguity means the presence of a process of concentration of the substance. The form a Lagrangian mapping in the one-dimensional case is determined by the variational principle (1.4). For the multidimensional case, an attempt to construct an analog of the mapping $x=\varphi_t(a)$ under very general assumptions was made in [38], however, as noted in [39], the corresponding proofs are incomplete. Apparently, the corresponding difficulties can be explained by a quite involved structure of solutions, even in the two-dimensional case, not to mention the multidimensional one, to obtain results without studying a number of specific examples. For example, in the two-dimensional case, as we will see below, the preimage of a point $\mathbf{x}$ with a Lagrangian mapping can be both a point and a curve, and a domain in the $\mathbf{a}$-space. In the generalized solution, accordingly, a hierarchy of shock waves arises. Also, in the multidimensional case, particles flying towards each other (in some sense) may not collide: the corresponding trajectories will be skew lines. For example, in the two-dimensional case, the skewness situation is typical for most trajectories for a complete set, with respect to a certain measure, of initial particle distributions (see [40]). If there is an infinite initial set of particles in a bounded set on the plane, then non-existence and non-uniqueness of a generalized solution are possible, see [39]. Taking into account the above difficulties, in [30], [31] and in the earlier publication [41] a principle was proposed for construction of Lagrangian maps that does not rely on the dynamics of adhesion in a system of moving particles, but uses a variational representation for generalized solutions. In these studies it was shown, among other things, that, in the two-dimensional case, it is necessary to use as a natural generalization of (1.4) the vector functional
$$
\begin{equation}
\mathbf{F}(t,\mathbf{x};\mathbf{a}) =\iint_{G} (\mathbf{u}_0(\mathbf{a})t+\mathbf{a}-\mathbf{x})\, P_0(d\mathbf{a}),
\end{equation}
\tag{2.13}
$$
where $G\in \mathbb{R}^2$ is some domain. It is possible to define the derivative of $\mathbf{F}$ over $G$ (see [42] and [43]), and build a Lagrangian mapping on this basis. But, in order to be constructive, this approach proposes to consider specific solutions that contain a hierarchy of shock waves. The present paper is devoted to a detailed description of such a solution. Nevertheless, at least in the two-dimensional case, the approximate adhesion dynamics can be used in the construction of numerical methods (see [44]). In a later paper [15], it was shown that even the direct numerical implementation of the adhesion dynamics is capable of providing sufficiently accurate information on generalized solutions of system (1.1). 2.3. Self-similar form of Rankine–Hugoniot relationsIn this section, we will mainly follow [26], [32], but the corresponding formulations will be given in a more convenient form for us. In the theory of systems of quasilinear conservation laws, self-similar solutions play an important role. Next, we will be interested in special solutions of the Riemann problem for (1.1), so we will consider self-similar solutions for system (2.8). In what follows, we will agree to drop the “tilde” over the densities $\widetilde{P}$, $\widetilde{\mathbf{I}}$. Let $\overline{l}\equiv l/t\geqslant 0$, let the “prime” denote differentiation with respect to $\overline{l}$, and let $\mathbf{U}\equiv(U,V)$, $\mathcal{X}\equiv(X,Y)$. We will search for a solution of system (2.8) in the form $\mathbf{x}=t\mathcal{X}(\overline{l})$, $(U,V)=\mathbf{U}(\overline{l})$, $P=tm(\overline{l})$, and, for any $f$, we keep the notation $[f]\equiv f^+-f^-$. Differentiating, we get from (2.8) that
$$
\begin{equation}
\begin{cases} X-\overline{l}X'=U, \\ Y-\overline{l}Y'=V, \\ m-\overline{l}m'=X'(V[\varrho]-[\varrho v])-Y'(U[\varrho]-[\varrho u]), \\ I-\overline{l}I'=X'(V[\varrho u]-[\varrho uv])-Y'(U[\varrho u]-[\varrho u^2]), \\ J-\overline{l}J'=X'(V[\varrho v]-[\varrho v^2])-Y'(U[\varrho v]-[\varrho uv]), \end{cases}
\end{equation}
\tag{2.14}
$$
where $I=mU$, $J=mV$. The stability conditions (2.9) can now be written as
$$
\begin{equation}
\begin{aligned} \, d^+ &\equiv|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}-\mathbf{u}^+)| = |(\mathcal{X}-\mathbf{u}^+)\ (\mathbf{U}-\mathbf{u}^+)|>0, \\ d^- &\equiv|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}-\mathbf{u}^-)| = |(\mathcal{X}-\mathbf{u}^-)\ (\mathbf{U}-\mathbf{u}^-)|<0. \end{aligned}
\end{equation}
\tag{2.15}
$$
Substituting $\mathbf{I}=m\mathbf{U}$ into (2.14) and differentiating the product, we obtain the system of first-order ordinary differential equations
$$
\begin{equation}
\begin{aligned} \, \begin{aligned} \, X'&=\dfrac{X-U}{\overline{l}}, \\ Y'&=\dfrac{Y-V}{\overline{l}}, \\ m'&=\dfrac{m}{\overline{l}}-\dfrac{1}{\overline{l}^{\,2}}\{\varrho^+d^+-\varrho^-d^-\}, \\ U'&=\dfrac{1}{m\overline{l}^{\,2}}\{\varrho^+(U-u^+)d^+ -\varrho^-(U-u^-)d^-\}, \\ V'&=\dfrac{1}{m\overline{l}^{\,2}}\{\varrho^+(V-v^+)d^+ -\varrho^-(V-v^-)d^-\}. \end{aligned} \end{aligned}
\end{equation}
\tag{2.16}
$$
We note that system (2.16) also has the first integral, as in (2.10). In this case, the first integral can be written as
$$
\begin{equation}
m\cdot|(\mathbf{u}^+-\mathbf{u}^-)\ (\mathbf{U}-\mathbf{u}^-)|=C\overline{l},
\end{equation}
\tag{2.17}
$$
where $C$ is some constant. Below, for system (2.16), the Cauchy problem will be considered for $0\leqslant\overline{l}\leqslant\overline{l}_0$ for some $\overline{l}_0>0$:
$$
\begin{equation}
X(\overline{l}_0)=X_0,\quad Y(\overline{l}_0)=Y_0,\quad m(\overline{l}_0)=m_0>0,\quad U(\overline{l}_0)=U_0,\quad V(\overline{l}_0)=V_0.
\end{equation}
\tag{2.18}
$$
In [26], [32], the Riemann problem for (1.1) was studied in detail. Its extensions due to the energy equation (1.2) were also considered. Definition 3. The Riemann problem for a system (1.1) is a Cauchy problem in which the initial data (1.3) are constant in each quadrant of $\mathbb{R}^2$, that is, where $i=1,\dots,4$ is the natural numbering of the quadrants of $\mathbb{R}^2$. As in the case of traditional gas dynamics, the solution of the Riemann problem was sought in a self-similar form, and, so, was based mainly on the study of solutions to problem (2.16), (2.18), which determine the shock wave with concentration of matter on a curve in $\mathbb{R}^2$. In particular, the solution of the Riemann problem depended on the fact that if such shock waves collide, then a shock wave of the same type, but with a larger mass, is formed. In the following sections of the present paper, we will show that there is another type of interaction in which a shock wave is formed with concentration of substance at the point of $\mathbb{R}^2$. In this case, a hierarchy of shock waves arises. § 3. The main results3.1. The existence of interaction of shock waves leading to concentration of matter at a pointLet $u>0$, $v>0$, $\varrho>0$, $R>0$ be constant values, and let $\varrho<R$. Consider the following initial data (2.19):
$$
\begin{equation}
\begin{gathered} \, u_1=u_4=-u,\qquad u_2=u_3=u,\qquad v_1=v_2=-v,\qquad v_3=v_4=v, \\ \varrho_i=\varrho,\quad i\ne 4, \qquad \varrho_4=R. \end{gathered}
\end{equation}
\tag{3.1}
$$
In the initial data (3.1), the vector field $\mathbf{u}_0$ is directed to the origin in all quadrants and has a sufficient degree of symmetry. Theorem 1. For a generalized solution of problem (1.1), (3.1) there is a straight line $\mathbf{x}=t\mathcal{X}^*$, $\mathcal{X}^*\equiv(X^*,Y^*)$, such that, for any $t$, the measure $P_t$ has a delta singularity at the point $t\mathcal{X}^*$. In this case, the point $\mathcal{X}^*$ is the intersection point of shock waves described by system (2.16). The meaning of Theorem 1 is that shock waves (2.16) represent some curved lines that may not intersect if the stability conditions (2.15) are met. If they intersect, then, for a delta singularity to occur, the preimage of the intersection point under the Lagrangian mapping must have a positive measure. The proof of these properties is far from being trivial. After the proof of Theorem 1, we will establish a number of qualitative properties of the behaviour of shock waves of type (2.16), which interact to form a substance concentration at a point of $\mathbb{R}^2$. These qualitative properties will be presented as Theorems 3 and 4. It is also worth pointing out that the solution of problem (1.1), (3.1) containing the concentration of substance at a point in the $\mathbf{x}$-plane was obtained numerically in [15]. This result was announced in [42]. 3.2. Rankine–Hugoniot relations when concentration of substance occurs at a pointTo formulate the following theorem, it is necessary to carry out additional constructions. Let, within this section, the index $i$ takes the values $i=1,\dots,Q$. Consider in the $(t,\mathbf{x})$-space, on the time interval $t\in [t_1,t_2]$, $Q$ surfaces $\Gamma_i(t,\mathbf{x})=0$, $\Gamma_i\in C^1([t_1,t_2]\times\mathbb{R}^2)$. Suppose that all surfaces intersect along some curve $L$, where $\Gamma_i$ and $L$ are given parametrically by
$$
\begin{equation}
\begin{aligned} \, \mathbf{x} &=\mathbf{X}_i(t,l)\in C^1([t_1,t_2]\times\mathbb{R}),\qquad \mathbf{X}\equiv(\chi,\gamma), \\ \mathbf{x} &=\mathbf{S}(t)\in C^1([t_1,t_2]),\qquad \mathbf{S}\equiv(s^{x},s^{y}). \end{aligned}
\end{equation}
\tag{3.2}
$$
Also let, for some continuously differentiable $l_i(t)$, the equality $\mathbf{X}(t,l_i(t))=\mathbf{S}(t)$, $t\in [t_1,t_2]$, hold. We will consider the surfaces $\Gamma_i$ not for all values of $l$ (namely, for $\Gamma_i$, we assume that $l\leqslant l_i(t)$). For the resulting parts of the surfaces, we keep the notation $\Gamma_i$. Let $\Gamma_i$ be oriented in the same way as to $(t,\mathbf{x})$, and, from the direction of the positive normal, we define the positive “$+$” and negative “$-$” sides of $\Gamma_i$. They are defined by the inequalities $\Gamma_i(t,\mathbf{x})>0$, $\Gamma_i(t,\mathbf{x})<0$. Accordingly, the entities related to the two sides of the surfaces will be provided with the same indices. For each $t$, consider the domains $G_i(t)\subset \mathbb{R}^2$ between these surfaces, that is, $G_i(t)=\{\mathbf{x}\colon\Gamma_i(t,\mathbf{x})>0,\,\Gamma_{i+1}(t,\mathbf{x})<0\}$, $i=1,\dots,Q$. In addition, $\Gamma_{Q+1}\equiv\Gamma_1$. We will also assume that the corresponding characteristics (2.5) lying on both the positive and negative sides of $\Gamma_i $ intersect it at some time point, that is, the stability conditions (2.9) are met. Consider the family of measures
$$
\begin{equation}
\begin{aligned} \, P_t &=P_1^-+\sum_{i=1}^{Q}(P_i^+-P_i^-)H(\Gamma_i) + \sum_{i=1}^{Q}P_i(t,l)\delta(\Gamma_i) +M(t)\delta(\mathbf{x}-\mathbf{S}(t)), \\ \mathbf{I}_t &=\mathbf{I}_1^-+\sum_{i=1}^{Q}(\mathbf{I}_i^+-\mathbf{I}_i^-)H(\Gamma_i) + \sum_{i=1}^{Q}\mathbf{I}_i(t,l)\delta(\Gamma_i) +\boldsymbol{\Pi}(t)\delta(\mathbf{x}-\mathbf{S}(t)), \end{aligned}
\end{equation}
\tag{3.3}
$$
where $H$ is the Heaviside function; $\delta$ is the Dirac measure on curves and at a point; $P_i^{\pm}$, $\mathbf{I}_i^{\pm}$ are are absolutely continuous measures with respect to the Lebesgue measure and which have densities $\varrho_i^{\pm}$, $\varrho_i^{\pm}\mathbf{u}_i^{\pm}$, respectively. In this case, the measures $P_i^+$, $\mathbf{I}_i^+$ are defined only in the domains $G_i$, and, for any $f$, we define $f_i^+=f_{i+1}^-$, and the number $Q+1$ is replaced by $1$. The functions $\varrho_i^{\pm}$, $\varrho_i^{\pm}\mathbf{u}_i^{\pm}$, $P_i$, $\mathbf{I}_i$, $M$, $\boldsymbol{\Pi}$ are assumed to be piecewise continuously differentiable on their domains of definition. Theorem 2. Let the family of measures $\mathfrak{P}_t(d\mathbf{x})\equiv(P_t(d\mathbf{x}), \mathbf{I}_t(d\mathbf{x}))$, as defined by (3.3), form a generalized solution of problem (1.1), (1.3) in the sense of Definition 1. Then 1) there exists a Lagrangian mapping $\mathcal{L}_t$; 2) the functions $(\varrho_i^{\pm},\mathbf{u}_i^{\pm})$, $i=1,\dots,Q$, satisfy (1.1) in the classical sense; 3) along the surfaces of $\Gamma_i$, relations (2.8) are fulfilled for each $i$, where, in accordance with the above convention, the “tilde” sign is dropped; 4) along the curve $L$, where $(M(t),\boldsymbol{\Pi}(t))=\mathfrak{P}_0(\mathcal{L}_t^{-1}(\mathbf{S}(t)))$.Theorem 2 describes a situation where several colliding shock waves form a delta singularity at a point on the plane. A concrete example of this behaviour for $Q=2$ is presented in Theorem 1. At the same time, a sufficient condition for a family of measures of type (3.3) to form a generalized solution of (1.1) is that, in addition to (2.8), relation (3.4) should also hold. That is, in this configuration, the Rankine–Hugoniot relations will consist of a pair of equations (2.8) and (3.4). The first relation is an evolutionary system of partial differential equations, and the second one is a system of ordinary differential equations. From the point of view of the general theory of systems of conservation laws, both relations describe shock waves, which nevertheless are of a different character, since they focus on manifolds of different dimensions in the $\mathbf{x}$-space. Note that a preliminary version of these results was given in [14]. In addition, Theorem 2 emphasizes the usefulness of the notion of a Lagrangian mapping $\mathcal{L}_t$, in particular, for determining the type of interaction of shock waves. So, if the preimage of a point in the $\mathbf{x}$-space under a Lagrangian map $\mathcal{L}_t$ is empty, then there is a vacuum zone; if the preimage is of zero dimension, then the solution is continuous at a given point; if the preimage is a curve, then there is a shock wave in the form of a concentration of matter on the curve; and if the preimage is of positive measure, then there is a concentration of matter at the point. Further, if two shock waves along a curve collide, then they simply merge if the preimage of the collision point is also a curve, and a point singularity is formed if the specified preimage is of positive measure. § 4. Proofs of the main theorems and additional results4.1. Proof of Theorem 1As shown in [26], the solution of system (1.1) with initial data of type (3.1) consists, in general, of vacuum zones, constants, and shock waves along curves whose parameters obey equations (2.16), which follow from (2.8) under the solution self-similarity assumption. With this in mind, we will build a generalized solution to problem (1.1), (3.1) as a set of constant vectors $\mathcal{U}$, separated by evolving shock waves along curves in the space $\mathbf{x}=(x,y)$. We set
$$
\begin{equation*}
\lambda\equiv\frac{\sqrt{R}-\sqrt{\varrho}}{\sqrt{R}+\sqrt{\varrho}},\qquad \theta\equiv\frac{2\sqrt{\varrho}}{\sqrt{R}+\sqrt{\varrho}},\qquad \Theta\equiv\frac{2\sqrt{R}}{\sqrt{R}+\sqrt{\varrho}}.
\end{equation*}
\notag
$$
According to the evolution of characteristics (2.5), we can immediately conclude about the formation of four shock waves. In view of (2.16) and the initial data (3.1), these waves have the form described below. In what follows, we will also construct the corresponding Lagrangian mapping, and so, for convenience, we will write the following formulae in the $\mathbf{x}$-variable. If necessary, they can easily be rewritten in the $\mathbf{x}/t$-variables, depending on $\overline{l}\equiv l/t$. In addition, we will further agree to drop (if appropriate) the index $t$ in the notation for measures. Shock wave I:
$$
\begin{equation}
\begin{gathered} \, x_1=l-ut\geqslant 0,\qquad y_1=\lambda vt, \\ P=2vt\sqrt{\varrho R}\, dl,\qquad l\geqslant l_1(t)\equiv ut. \end{gathered}
\end{equation}
\tag{4.1}
$$
Shock wave II:
$$
\begin{equation}
\begin{gathered} \, x_2=0,\qquad y_2=l-vt\geqslant \lambda vt, \\ P=2ut\varrho dl,\qquad l\geqslant l_2(t)\equiv (\lambda+1)vt. \end{gathered}
\end{equation}
\tag{4.2}
$$
We reserve number III for the shock wave that arises due to the interaction of waves I and II. Shock wave IV:
$$
\begin{equation}
\begin{gathered} \, x_4=-l+ut\leqslant -\lambda ut,\qquad y_4=0, \\ P=2vt\varrho dl,\qquad l\geqslant l_4(t)\equiv (\lambda+1)ut. \end{gathered}
\end{equation}
\tag{4.3}
$$
Shock wave V:
$$
\begin{equation}
\begin{gathered} \, x_5=-\lambda ut,\qquad y_5=-l+vt\leqslant 0, \\ P=2ut\sqrt{\varrho R}\,dl,\qquad l\geqslant l_5(t)\equiv vt. \end{gathered}
\end{equation}
\tag{4.4}
$$
Finally, the result of the interaction of waves IV and V will be denoted as shock wave VI. For convenience, we also denote the combination of shock waves III and VI as wave VII. Waves III and VI will be obtained as a result of solution of system (2.16) with the corresponding initial data $(\mathcal{X}_{0,3}, m_{0,3},\mathbf{U}_{0,3})$, $(\mathcal{X}_{0,6}, m_{0,6},\mathbf{U}_{0,6})$ and will be described later in the direct construction. We will also show that waves III and VI intersect at some point $\overline{\mathbf{x}}\equiv (\overline{x},\overline{y})=t\cdot(X^*,Y^*)$, in which a density delta singularity with total mass $M(t)$ will be formed. Waves III and VI will be considered accordingly only up to the intersection point of $\overline{\mathbf{x}}$. Let wave VII (without taking into account the delta singularity) for each $t$ be given by some equation $y=F(t,x)$. Generally speaking, the existence of such a representation follows from the results of [26], but in this paper we will use it only for convenience of description of the structure of the generalized solution. Thus, the generalized solution of problem (1.1), (3.1) is as follows (first we mark the zones of constant values $\varrho$, $\mathbf{u}$, and then the location of shock waves):
$$
\begin{equation}
\begin{aligned} \, &P=\varrho\, d\mathbf{x},\ \mathbf{I}=(-uP,-vP) &&\text{for }x>0,\ y>\lambda vt; \\ &P=\varrho \, d\mathbf{x},\ \mathbf{I}=(uP,-vP) &&\text{for } -\lambda ut<x<0,\ y>F(t,x) \\ &&&\qquad\text{and}\ x<-\lambda ut,\ y>0; \\ &P=\varrho \, d\mathbf{x}, \ \mathbf{I}=(uP,vP) &&\text{for } x<-\lambda ut,\ y<0; \\ &P=R d\, \mathbf{x}, \ \mathbf{I}=(-uP,vP) &&\text{for } -\lambda ut<x<0,\ y<F(t,x) \\ &&&\qquad\text{and}\ x>0,\ y<\lambda vt; \\ &P=2vt\sqrt{\varrho R}\, dl,\ \mathbf{I}=\frac{\partial\mathbf{x}}{\partial t}\cdot P &&\text{on the shock wave I, see (4.1)}; \\ &P=2\varrho utdl,\ \mathbf{I}=\frac{\partial\mathbf{x}}{\partial t}\cdot P &&\text{on the shock wave II, see (4.2)}; \\ &P,\ \mathbf{I},\ \mathbf{x}(t,l) &&\text{on the shock wave III are defined} \\ &&&\text{from (2.16), as mentioned above}; \\ &P=2\varrho vtdl, \ \mathbf{I}=\frac{\partial\mathbf{x}}{\partial t}\cdot P &&\text{on the shock wave IV, see (4.3)}; \\ &P=2ut\sqrt{\varrho R}\, dl,\ \mathbf{I}=\frac{\partial\mathbf{x}}{\partial t}\cdot P &&\text{on the shock wave V, see (4.4)}; \\ &P,\ \mathbf{I},\ \mathbf{x}(t,l) &&\text{on the shock wave VI are defined} \\ &&&\text{from (2.16), as mentioned above}; \\ &P = M(t)\delta(\mathbf{x}\,{-}\,\overline{\mathbf{x}}),\ \mathbf{I} = (X^*,Y^*)\cdot P &&\text{as } \mathbf{x}=\overline{\mathbf{x}}. \end{aligned}
\end{equation}
\tag{4.5}
$$
Now let construct waves III and VI in succession. Wave III, with due account of self-similarity, is given by $\mathbf{x}=t\mathcal{X}(\overline{l})$, $P=tm(\overline{l})$, where $\overline{l}\equiv l/t$, $\mathcal{X}\equiv(X,Y)$, and the values $\mathcal{X}$, $\mathbf{U}\equiv(U,V)$, $m$ are determined by the following system of equations arising from (2.16) after substitution of the corresponding “$+$” and “$-$” values
$$
\begin{equation}
\begin{cases} \mathcal{X}'=\dfrac{\mathcal{X}-\mathbf{U}}{\overline{l}}, \\ m'=\dfrac{m}{\overline{l}}-\dfrac{1}{\overline{l}^{\,2}}\{\varrho d^+-R d^-\}, \\ \mathbf{U}'=\dfrac{1}{m\overline{l}^{\,2}} \{\varrho d^+(\mathbf{U}+(-u,v)) -Rd^-(\mathbf{U}+(u,-v))\}, \end{cases}
\end{equation}
\tag{4.6}
$$
where
$$
\begin{equation}
\begin{aligned} \, d^+ &=|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}+(-u,v))| = |(\mathcal{X}+(-u,v))\ (\mathbf{U}+(-u,v))|>0, \\ d^-&=|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}+(u,-v))| = |(\mathcal{X}+(u,-v))\ (\mathbf{U}+(u,-v))|<0. \end{aligned}
\end{equation}
\tag{4.7}
$$
Relations (4.7) follow from (2.15) after substituting the corresponding “$+$” and “$-$” values. For the system of equations (4.6), we will set the initial data $(\mathcal{X}_0\equiv\mathcal{X}(\overline{l}_0),\, m_0\equiv m(\overline{l}_0), \, \mathbf{U}_0\equiv\mathbf{U}(\overline{l}_0))$ for some $\overline{l}_0>0$ (the specific value of $\overline{l}_0$ will then turn out to be unimportant), and we will solve the Cauchy problem for (4.6) at $0<\overline{l}\leqslant\overline{l}_0$. The initial data will be formed from the condition of continuity of the corresponding measures at the point of interaction of shock waves I and II, namely ($l_0\equiv\overline{l}_0t$),
$$
\begin{equation}
\begin{aligned} \, \dot{l}_0\mathfrak{P}|_{l_0} &= 2vt\sqrt{\varrho R}\,{\cdot}\,\dot{l}_1\, {\cdot}\,(1,\dot{\mathbf{x}}_1)+ 2ut\varrho\dot{l}_2\cdot(1,\dot{\mathbf{x}}_2) \nonumber \\ &= 2uvt\sqrt{\varrho R}\cdot(2-\lambda,-u,v(2\lambda-1)). \end{aligned}
\end{equation}
\tag{4.8}
$$
In view of $\mathbf{U}=\mathbf{I}/P$, we have
$$
\begin{equation}
\mathcal{X}_0=(0,\lambda v),\qquad m_0=2uv\sqrt{\varrho R}\, \frac{2-\lambda}{\overline{l}_0},\qquad \mathbf{U}_0=\frac{(-u,v(2\lambda-1))}{2-\lambda}.
\end{equation}
\tag{4.9}
$$
For wave VI, the direction of motion along it changes with increasing $\overline{l}$, as compared to wave III, the sides “$+$” and “$-$” swap, and instead of system (4.6), from (2.16) we have the following system of equations:
$$
\begin{equation}
\begin{cases} \mathcal{X}'=\dfrac{\mathcal{X}-\mathbf{U}}{\overline{l}}, \\ m'=\dfrac{m}{\overline{l}}-\dfrac{1}{\overline{l}^{\,2}}\{Rd^+- \varrho d^-\}, \\ \mathbf{U}'=\dfrac{1}{m\overline{l}^{\,2}}\{Rd^+(\mathbf{U}+(u,-v)), -\varrho d^-(\mathbf{U}+(-u,v))\}, \end{cases}
\end{equation}
\tag{4.10}
$$
here, the expressions for the values $d^{\pm}$ are different from (4.7)
$$
\begin{equation}
\begin{aligned} \, d^+ &=|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}+(u,-v))| = |(\mathcal{X}+(u,-v))\ (\mathbf{U}+(u,-v))|>0, \\ d^- &=|(\mathcal{X}-\mathbf{U})\ (\mathbf{U}+(-u,v))| = |(\mathcal{X}+(-u,v))\ (\mathbf{U}+(-u,v))|<0. \end{aligned}
\end{equation}
\tag{4.11}
$$
The continuity condition of the corresponding measures at the point of interaction of shock waves IV and V takes the form (the value $l_0\equiv\overline{l}_0t$ can be taken the same as for wave III)
$$
\begin{equation}
\begin{aligned} \, \dot{l}_0\mathfrak{P}|_{l_0} &= 2vt\varrho\,{\cdot}\,\dot{l}_4\,{\cdot}\,(1,\dot{\mathbf{x}}_4) + 2ut\sqrt{\varrho R}\, \dot{l}_5\cdot(1,\dot{\mathbf{x}}_5) \nonumber \\ &= 2uvt\sqrt{\varrho R} \cdot(2-\lambda,u(1-2\lambda),v). \end{aligned}
\end{equation}
\tag{4.12}
$$
Another appeal to the relation $\mathbf{U}=\mathbf{I}/P$ shows that
$$
\begin{equation}
\mathcal{X}_0=(-\lambda u,0),\qquad m_0=2uv\sqrt{\varrho R}\, \frac{2-\lambda}{\overline{l}_0},\qquad \mathbf{U}_0=\frac{(u(1-2\lambda),v)}{2-\lambda}.
\end{equation}
\tag{4.13}
$$
Above, $\mathfrak{P}$, $\mathcal{X}$, $m$, $\mathbf{U}$, $\mathbf{I}$, $d^{\pm}$ for waves III and VI are not provided with indexes for ease of notation. Therefore, in what follows, the indices $1,\dots,6$ corresponding to shock waves $\mathrm{I},\dots,\mathrm{VI}$ will also be used only if ambiguity arises. Let $L_1$ be the straight line $uY+vX=0$ on the plane $\mathbf{x}/t$, and let $L_2$ be the the straight line $uY-vX=\lambda uv$ on the same plane. Existence of solutions to problems (4.6), (4.9) and (4.10), (4.13) is proved in [26]. This follows from the general theorems on existence of solutions for systems of ordinary differential equations and estimates from below for $m$. In the same paper, it was shown that inequalities (4.7), (4.11) are true as long as the corresponding curves intersect the line $L_1$. Consider the following transformation of unknown functions $\mathcal{X}$, $m$, $\mathbf{U}$ (the indexes denote quantities related to the corresponding shock wave)
$$
\begin{equation}
\begin{gathered} \, X_6=-\frac{u}{v}Y_3,\qquad Y_6=-\frac{v}{u}X_3,\qquad m_6=m_3, \\ U_6=-\frac{u}{v}V_3,\qquad V_6=-\frac{v}{u}U_3. \end{gathered}
\end{equation}
\tag{4.14}
$$
Transforming with respect to (4.14), we get $d_3^+=-d_6^-$ and $d_3^-=-d_6^+$. We prove the first equality; the second is treated similarly,
$$
\begin{equation*}
\begin{aligned} \, d_3^+ &=|(\mathcal{X}_3+(-u,v))\ (\mathbf{U}_3+(-u,v))| \\ &= \left|\begin{pmatrix} -\dfrac{u}{v}Y_6-u \\ -\dfrac{v}{u}X_6+v \end{pmatrix}\ \begin{pmatrix} -\dfrac{u}{v}V_6-u \\ -\dfrac{v}{u}U_6+v \end{pmatrix}\right| \\ &=\biggl(-\frac{u}{v}\biggr)\cdot\biggl(-\frac{v}{u}\biggr) \biggl|\begin{pmatrix} Y_6+v \\ X_6-u \end{pmatrix}\ \begin{pmatrix} V_6+v \\ U_6-u \end{pmatrix}\biggr|=-d_6^-. \end{aligned}
\end{equation*}
\notag
$$
A transformation (4.14) carries over the system of equations (4.6) to the system of equations (4.10), the initial data (4.9) go to (4.13), and the straight line $L_1$ remains in place. As shown in [26], shock waves of type III, VI will necessarily cross the line $L_1$, after which conditions (4.7), (4.11) may fail to hold. But then shock wave VI will intersect the line $L_1$ at the same point as the shock wave III, that is, on the line $L_1$ there is a point $\mathcal{X}^*$ at which shock waves III and VI intersect. Also, for these shock waves, the stability conditions (2.9) (that is, inequalities (4.7) for wave III and inequalities (4.11) for wave VI) are met. For further reference, it is worth pointing out that transformation (4.14) carries the straight line $L_2$ to itself. In order to understand what happens at the point $\mathbf{x}=t\mathcal{X}^*$, it is necessary to construct a Lagrangian mapping $\mathcal{L}_t\colon\mathbf{a}\to\mathbf{x}$, which, for each $t$, shows which points with coordinates $\mathbf{a}$ at the initial time will end up at a point with coordinates $\mathbf{x}$ at time $t$. Based on the type of solution (4.5), we can generally say that at first the motion of points occurs along the characteristics, and the Lagrangian mapping has the form (2.5). Then there is a sequential concentration of substance on shock waves I, II, IV, V, and then, on waves III, VI. The question is how does this happen and what arises at the point $\mathbf{x}=t\mathcal{X}^*$? Next, we describe the Lagrangian mapping sequentially for different regions of the $\mathbf{a}$-plane. Considering the results obtained in [26], we assume that waves III and VI can be described by the relations $Y=F_3(X)$, $X^*<X<0$; $Y=F_6(X)$, $-\lambda u<X<X^*$, respectively, while $F_3$, $F_6$ are continuously differentiable functions. Thus, wave VII (recall that this is how the union of waves III and VI is denoted) is given by the composite function $Y=F_7(X)$, $-\lambda u<X<0$. On the shock wave $Y=F_7(X)$, the characteristics come from the regions of the second and fourth quadrants of the $\mathbf{a}$-plane. Accordingly, on this plane in accordance with the transformation (2.5) there will be two preimages of the curve $Y=F_7(X)$: in the second quadrant of the $\mathbf{a}$-plane, $b=B^*(t,a)$, $-\Theta ut<a<-ut$, and in the fourth quadrant of the $\mathbf{a}$–plane, $b=B^{**}(t,a)$, $\theta ut<a<ut$. According to the structure of solution (4.5), for those domains where representation (2.5) remains valid, we have
$$
\begin{equation}
\begin{aligned} \, &\mathcal{L}_t\colon \{a>ut,\, b>\Theta vt\} \to \{x>0,\, y>\lambda vt\} \\ &\qquad\text{according to the formula } \mathbf{x}=\mathbf{a}+t\cdot(-u,-v); \\ &\mathcal{L}_t\colon \{a<-\Theta ut,\, b>vt\} \cup \{-\Theta ut<a<-ut,\, b>B^*(a,t)\} \\ &\qquad \to\{x<-\lambda ut,\, y>0\} \cup \biggl\{-\lambda ut<x<0,\, y>tF_7\biggl(\frac{x}{t}\biggr)\biggr\} \\ &\qquad \text{according to the formula }\mathbf{x}=\mathbf{a}+t\cdot(u,-v); \\ &\mathcal{L}_t\colon \{a<-\Theta ut,\, b<-vt\} \to \{x<-\lambda ut,\, y<0\} \\ &\qquad \text{according to the formula } \mathbf{x}=\mathbf{a}+t\cdot(u,v); \\ &\mathcal{L}_t\colon \{a>ut,\, b<-\theta vt\} \cup \{\theta ut<a<ut,\, b<B^{**}(a,t)\} \\ &\qquad \to\{x>0,\, y<\lambda vt\} \cup \biggl\{-\lambda ut<x<0,\, y<tF_7\biggr(\frac{x}{t}\biggr)\biggr\} \\ &\qquad\text{according to the formula }\mathbf{x}=\mathbf{a}+t\cdot(-u,v). \end{aligned}
\end{equation}
\tag{4.15}
$$
For shock waves I, II, IV and V, which are straight lines parallel to one of the coordinate axes on the $\mathbf{x}$-plane, the Lagrangian mapping is as follows: the preimage of each point of these waves is a segment on the $\mathbf{a}$-plane
$$
\begin{equation}
\begin{aligned} \, &\mathcal{L}_t\colon\{a=x+ut,\, -\theta vt<b<\Theta vt\} \to (t,\mathbf{x})&&\forall\, (t,\mathbf{x})\in\text{wave I}; \\ &\mathcal{L}_t\colon\{b=y+vt,\, -ut<a<ut\}\to (t,\mathbf{x}) &&\forall\, (t,\mathbf{x})\in\text{wave II}; \\ &\mathcal{L}_t\colon\{a=x-ut,\, -vt<b<vt\}\to (t,\mathbf{x}) &&\forall\, (t,\mathbf{x})\in\text{wave IV}; \\ &\mathcal{L}_t\colon\{b=y-vt,\, -\Theta ut<a<\theta ut\}\to (t,\mathbf{x}) &&\forall\, (t,\mathbf{x})\in\text{wave V}. \end{aligned}
\end{equation}
\tag{4.16}
$$
The scenario of concentration on waves III, VI is more complex, since a part of substance first concentrates on shock waves I, II, IV and V, and then, due to the interaction of waves, passes into waves III, VI. In parallel, the concentration of matter from the second and fourth quadrants also occurs, from where characteristics come to waves III, VI. Let us consider in more detail the picture of concentration of matter on wave III (for wave VI, the picture is similar). We fix the time $t$ and introduce the auxiliary quantities $s_1$, $s_2$, and a sufficiently small $\varepsilon>0$ such that $s_1<ut$, $s_2<\Theta vt$ and $\max(|s_1-ut|,\,|s_2-\Theta vt|)\leqslant\varepsilon$. Consider wave I first. If $s\geqslant ut$, then this wave is a set of trajectories
$$
\begin{equation}
x=s-u\tau,\qquad y=\lambda v\tau,\qquad 0\leqslant\tau\leqslant t,
\end{equation}
\tag{4.17}
$$
which by time $t$ accumulate points according to the Lagrangian mapping (4.16). For $s=ut$, the corresponding trajectory (4.17) hits the shock wave II, and the interaction of shock waves occurs. If $s<ut$, then such interaction will occur at an earlier time $s/u$. Therefore, if the trajectory on wave I starts at the point $(s_1,0)$, then it will interact with wave II at time $\tau_1=s_1/u$ and brings to the interaction point particles from the segment $\{a=s_1,\, -\theta v\tau_1\leqslant b\leqslant\Theta v\tau_1\}$ lying in the $\mathbf{a}$-plane. If $\tau>\tau_1$, then the trajectory now moves along wave III in accordance with the formulae
$$
\begin{equation}
\mathbf{x}=\tau \mathcal{X}_3\biggl(\frac{\overline{l}_0\tau_1}{\tau}\biggr),
\end{equation}
\tag{4.18}
$$
where $\mathcal{X}$ is determined from the solution of system (4.6), and $\overline{l}_0$ refers to the initial data (4.8). At $|s_1-ut|\leqslant\varepsilon$, our trajectory at $\tau=t$ will not reach the straight line $L_1$, that is, it will not reach the point of interaction with the shock wave VI. However, with a further decrease in $s_1$ there appears an $s_1^*$ such that
$$
\begin{equation}
\mathcal{X}^*=\mathcal{X}_3\biggl(\frac{\overline{l}_0\tau_1^*}{t}\biggr),\qquad s_1^*=u\tau_1^*,
\end{equation}
\tag{4.19}
$$
that is, the trajectory under consideration will fall into the point of interaction of waves III and VI. A similar analysis can be also carried out for wave II, and the corresponding values can be determined. Namely, the trajectory under consideration for wave II will start at the point $(0,s_2)$, interaction with wave I will occur at the time $\tau_2=s_2/(\Theta v)$, and wave II will bring to the point of interaction particles from the segment $\{- u\tau_2\leqslant a\leqslant u\tau_2,\, b=s_2\}$ (lying in the $\mathbf{a}$-plane). Since at $\tau>\tau_2$, the trajectory under consideration moves in the same way as that of wave I just considered, we can assume that $\tau_1=\tau_2\equiv\tau_3$,, and for $\tau>\tau_2$, a further evolution occurs according to formulae (4.18). Formulae (4.19) also hold for $s_2^*=\Theta v\tau_2^*$, where $\tau_2^*=\tau_1^*\equiv\tau_3^*$. So, consider, for example, some $s_1$ such that $s_1^*<s_1<ut$. This value $s_1$ will correspond to $s_2$ such that $s_2^*<s_2<\Theta vt$ and $s_1/u=s_2/(\Theta v)$. The trajectories from the points $(s_1,0)$ and $(0,s_2)$ lying on the $\mathbf{a}$-plane will lie on waves I and II accordingly, as described above. At $\tau>\tau_3$, both of these trajectories will move into the same trajectory on wave III, which is described by formulae (4.18) (with the equivalent replacement of $\tau_1$ by $\tau_3$). At time $t$, this trajectory will be at some point $\widehat{\mathbf{x}}$ lying on the wave III. Let $\Upsilon_3^2(s_1)$ be the projection of this trajectory onto the $\mathbf{a}$-plane via transformation (2.5) in the second quadrant of $\mathbf{a}$, and let $\Upsilon_3^4(s_1)$ in the fourth quadrant of $\mathbf{a}$. For wave III, the Lagrangian mapping looks is as follows (since $s_1^*<s_1<ut$ and since $s_1/u=\tau_1=\tau_2=s_2/(\Theta v)$:
$$
\begin{equation}
\begin{aligned} \, &\mathcal{L}_t\colon\{a=s_1,\, -\theta v\tau_1\leqslant b\leqslant\Theta v\tau_1\} \cup\Upsilon_3^2(s_1)\cup \{-u\tau_2\leqslant a\leqslant u\tau_2,\, b=s_2\}\cup\Upsilon_3^4(s_1) \nonumber \\ &\qquad\to(t,\widehat{\mathbf{x}})\quad \forall\, (t,\widehat{\mathbf{x}})\in\text{wave III}. \end{aligned}
\end{equation}
\tag{4.20}
$$
The picture of the concentration of matter on wave VI; the corresponding formulae can be written with $1$, $2$, $3$ replaced, if necessary, by $4$, $5$, $6$. Therefore, the Lagrangian mapping can be written as
$$
\begin{equation}
\begin{aligned} \, &\mathcal{L}_t\colon \{a=s_4,\, -v\tau_4\leqslant b\leqslant\tau_4\} \cup\Upsilon_6^2(s_4) \cup \{-\Theta u\tau_5\leqslant a\leqslant \theta u\tau_5,\, b=s_5\}\cup\Upsilon_6^4(s_4) \nonumber \\ &\qquad\to(t,\widehat{\mathbf{x}})\quad \forall\, (t,\widehat{\mathbf{x}})\in\text{wave VI} \end{aligned}
\end{equation}
\tag{4.21}
$$
(since $-\Theta ut<s_4< s_4^*$ and since $-s_4/(\Theta u)=\tau_4=\tau_5=-s_5/v$). Next, the constructed Lagrangian mapping (4.15), (4.16), (4.20), (4.21) covers all points of the $\mathbf{x}$-plane except for a single point $\overline{\mathbf{x}}=t\mathcal{X}^*$. At the same time, on the $\mathbf{a}$-plane, the preimage under the already constructed Lagrangian mapping does not contain the entire domain $\Omega$, for which the boundary $\partial\Omega$ is given by the following curves
$$
\begin{equation}
\begin{aligned} \, &\partial\Omega =\biggl\{a=s_1^*,-\frac{s_1^*\theta v}{u}\leqslant b\leqslant s_2^*\biggr\} \cup \biggl\{-\frac{us_2^*}{\Theta v}\leqslant a\leqslant s_1^*,\, b=s_2^*\biggr\} \cup \Upsilon_3^2(s_1^*) \cup \Upsilon_6^2(s_4^*) \nonumber \\ &\quad\cup \biggl\{a=s_4^*,\, s_5^*\leqslant b\leqslant -\frac{s_4^*v}{\Theta u}\biggr\} \cup\biggl\{s_4^*\leqslant a\leqslant -\frac{s_5^*\theta u}{v},\, b=s_5^*\biggr\} \cup \Upsilon_6^4(s_4^*) \cup\Upsilon_3^4(s_1^*). \end{aligned}
\end{equation}
\tag{4.22}
$$
That is, one point $\overline{\mathbf{x}}$ on the $\mathbf{x}$-plane corresponds to the whole area $\Omega$ on the $\mathbf{a}$-plane, and this means the concentration of matter at the point $\overline{x}$, that is, the occurrence of a delta singularity of the measure $P_t$. In addition, by construction, the point $\overline{x}$ for every $t$ is the intersection point of shock waves described by concretizations of system (2.16). This completes the proof of Theorem 1. 4.2. Proof of Theorem 2First, for convenience, we will give one technical lemma that was used in [29]. Lemma 1. Let on the plane $\mathbf{a}\equiv (a,b)$ there exist a family of oriented domains $G(t)$ with piecewise continuously differentiable boundaries $\partial G(t)$, $0<t_1\leqslant t\leqslant t_2$, such that $G(\tau_1)\subset G(\tau_2)$ for $t_1<\tau_1<\tau_2<t_2$. Let $\partial G(t)$ be a closed curve that can be set parametrically as $\mathbf{a}(t,l)$, where $l$ is the parameter along the curve. Then, for $t_1\leqslant t\leqslant t_2$, where $\varphi(t,\mathbf{a})\in C^1\bigl([t_1,t_2] \times\overline{\bigcup_{t\in [t_1,t_2]}G(t)}\,\bigr)$. Proof of Theorem 2. By $\{t\}$ we denote the plane $(t = \mathrm{const},\mathbf{x})$. It is enough to consider a function $\mathbf{f}$ from the Definition 1 such that $(\Gamma_i\cap\{t\})\cap \operatorname{supp}\mathbf{f}\ne\varnothing$, for $0<t_1<t<t_2$, for $1\leqslant i\leqslant Q$ and $\mathbf{S}(t)\in \operatorname{supp}\mathbf{f}$. For $0<t_1<t<t_2$ the curves $(\Gamma_i\cap\{t\})$ divide $\operatorname{supp}\mathbf{f}$ into $Q$ domains $G_i(t)$, that is,
Based on the geometric construction described in Theorem 2, we define the Lagrangian map $\mathcal{L}_t$ as follows. Consider an arbitrary point $\mathbf{a}$ and the characteristic emanating from it in accordance with formulae (2.5). By the assumptions on the structure of the geometric construction in question, there is a time $\tau_{\mathbf{a}}^1>0$, such that, for some $i_{\mathbf{a}}$ and $l_{\mathbf{a}}$, $\mathbf{x}^*(\tau_{\mathbf{a}}^1,\mathbf{a}) =\mathbf{X}_{i_{\mathbf{a}}}(\tau_{\mathbf{a}}^1,l_{\mathbf{a}})$. Further, for the same reasons, there is such $\tau_{\mathbf{a}}^2\geqslant\tau_{\mathbf{a}}^1$, that $\mathbf{X}_{i_{\mathbf{a}}}(\tau_{\mathbf{a}}^2,l_{\mathbf{a}}) =\mathbf{S}(\tau_{\mathbf{a}}^2)$. We put
$$
\begin{equation}
\mathcal{L}_t(\mathbf{a})=\begin{cases} \mathbf{x}^*(t,\mathbf{a})\text{ in accordance with (2.5) for } 0\leqslant t\leqslant\tau_{\mathbf{a}}^1; \\ \mathbf{X}_{i_{\mathbf{a}}}(t,l_{\mathbf{a}}) \text{ for } \tau_{\mathbf{a}}^1\leqslant t\leqslant\tau_{\mathbf{a}}^2; \\ \mathbf{S}(t)\text{ for } \tau_{\mathbf{a}}^2\leqslant t. \end{cases}
\end{equation}
\tag{4.25}
$$
Assertion 2) of Theorem 2 easily follows from (2.1) if we take trial functions $\mathbf{f}$ such that $\operatorname{supp}\mathbf{f}$ does not contain singularities. We set $G_i^*(t)\equiv \mathcal{L}_t^{-1}(G_i(t))$, $D_{\Gamma_i}(t)\equiv \mathcal{L}_t^{-1}(\Gamma\cap\{t\})$ and $D_{\mathbf{S}}(t)\equiv \mathcal{L}_t^{-1}(\mathbf{S}(t))$ according to (4.25). Next, substituting a family of measures of the form (3.3) into the right-hand side of the integral identity (2.1), we get, in view of decomposition (4.24) and formulae (2.5),
$$
\begin{equation*}
\begin{aligned} \, &\int_{t_1}^{t_2}d\tau\, \biggl\{\sum_{i=1}^{Q} \iint_{G_i(\tau)} (\mathbf{\nabla}\otimes\mathbf{f})^{\top} \cdot \mathbf{u}\odot\mathfrak{P}_{\tau}(d\mathbf{x}) \\ &\qquad+\sum_{i=1}^{Q}\int_{\Gamma_i\cap\{t\}}(\mathbf{\nabla}\otimes\mathbf{f})^{\top} \cdot\frac{\partial\mathbf{X}_i}{\partial\tau}(\tau,l)\odot\mathfrak{P}_{\tau,i}(dl)+ (\mathbf{\nabla}\otimes\mathbf{f})^{\top}\cdot \frac{d\mathbf{S}}{dt}\odot\mathfrak{P}_{\tau}(\mathbf{S}(\tau))\biggr\} \\ &=\int_{t_1}^{t_2}d\tau\, \biggl\{\sum_{i=1}^{Q} \iint_{G_i^*(\tau)}\frac{\partial\mathbf{f}(\mathbf{x}^*)}{\partial\tau} \odot\mathfrak{P}_0(d\mathbf{a}) \\ &\qquad+\sum_{i=1}^{Q}\int_{\Gamma_i\cap\{t\}} \frac{\partial\mathbf{f}(\mathbf{X}_i(\tau,l))}{\partial\tau} \odot\mathfrak{P}_{\tau,i}(dl)+\frac{d\mathbf{f}(\mathbf{S})}{d\tau} \odot\mathfrak{P}_{\tau}(\mathbf{S}(\tau))\biggr\}\equiv\mathfrak{I}. \end{aligned}
\end{equation*}
\notag
$$
Applying Lemma 1 to the first term in $\mathfrak{I}$, we have
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{I} &=\int_{t_1}^{t_2}d\tau\, \biggl\{\sum_{i=1}^{Q}\frac{d}{d\tau} \iint_{G_i^*(\tau)}\mathbf{f}(\mathbf{x}^*)\odot\mathfrak{P}_0(d\mathbf{a}) \\ &\qquad+ \sum_{i=1}^{Q}\frac{d}{d\tau}\, \int_{\Gamma_i\cap\{t\}} \mathbf{f}(\mathbf{X}_i(\tau,l)) \odot\mathfrak{P}_{\tau,i}(dl) + \frac{d}{d\tau}\{\mathbf{f}(\mathbf{S}) \odot\mathfrak{P}_{\tau}(\mathbf{S}(\tau))\}\biggr\} \\ &\qquad-\int_{t_1}^{t_2}d\tau\, \biggl\{\sum_{i=1}^{Q} \int_{\Gamma_i\cap\{t\}} \mathbf{f}(\mathbf{X}_i(\tau,l)) \odot\biggl(\frac{\partial\mathbf{a}_i^+}{\partial (\tau,l)} \odot\mathfrak{P}_0+ \frac{\partial\mathfrak{P}_{\tau,i}}{\partial\tau}\biggr)\, dl \\ &\qquad-\sum_{i=1}^{Q}\int_{\Gamma_{i+1}\cap\{t\}} \mathbf{f}(\mathbf{X}_{i+1}(\tau,l)) \odot\frac{\partial\mathbf{a}_{i+1}^-}{\partial(\tau,l)}\odot\mathfrak{P}_0\, dl \\ &\qquad+\mathbf{f}(\mathbf{S})\odot\biggl(\frac{d}{d\tau}\mathfrak{P}_{\tau} (\mathbf{S}(\tau)) + \sum_{i=1}^{Q}\mathfrak{P}_{\tau,i}(\tau,l_i(\tau))\dot{l_i}\biggr)\biggr\}, \end{aligned}
\end{equation*}
\notag
$$
where $Q+1$ is replaced by $1$, and $\mathbf{a}_i^{\pm}(\tau,l)$ refers to functions derived from (2.5) with $\mathbf{X}_i(\tau,l)$ substituted for $\mathbf{x}$; in this case, the “$\pm$” signs means the use of characteristics passing from the positive and negative sides of $\mathbf{X}_i(\tau,l)$, respectively.
After integration, the first time integral in $\mathfrak{I}$ represents the left-hand part in (2.1) of Definition 1. Therefore, the second time integral in $\mathfrak{I}$ should vanish. So, recalling the notation $[f]_i\equiv f_i^+-f_i^-$, and by arbitrariness of the function $\mathbf{f}$, we have, for each $i$, $i=1,\dots,Q$,
$$
\begin{equation}
\biggl[\frac{\partial\mathbf{a}}{\partial(\tau,l)}\odot\mathfrak{P}_0\biggr]_i + \frac{\partial\mathfrak{P}_{\tau,i}}{\partial\tau}=0,\qquad \frac{d}{d\tau}\mathfrak{P}_{\tau}(\mathbf{S}(\tau)) + \sum_{i=1}^{Q}\mathfrak{P}_{\tau,i}(\tau,l_i(\tau))\dot{l_i}=0.
\end{equation}
\tag{4.26}
$$
Recalling that $\mathfrak{P}_{\tau,i}=(P_i(\tau,l),\mathbf{I}_i(\tau,l))$ on $\Gamma_i$, from the first equation in (4.26) we obtain the relations on the surface of the concentration of the substance in integral form, which, as shown in [29], are equivalent to (2.8).
Further, in the second equation of (4.26), on the curve (in the $(t,\mathbf{x})$)-space of substance concentration, the measure $\mathfrak{P}_{\tau}$ is defined by $\mathfrak{P}_{\tau}(\mathbf{S}(\tau))=(M(\tau),\boldsymbol{\Pi}(\tau)) =\mathfrak{P}_0(D_{\mathbf{S}}(\tau))$ by construction. The curves $\mathbf{a}_i^+(\tau,l_i(t))$, $\mathbf{a}_{i+1}^-(\tau,l_{i+1}(t))$, $i=1,\dots,Q$, on the $\mathbf{a}$-plane consecutively define the boundary of the area $D_{\mathbf{S}}(\tau)$, which we will denote for brevity by $\mathbf{a}_{\mathbf{S}}(\tau,l(t))$. In this we mean that $l(t)$ assumes the value $l_i(t)$ of $\mathbf{a}_{\mathbf{S}}$ takes the values $\mathbf{a}_i^{\pm}$. Using Lemma 1, it can be shown that
$$
\begin{equation*}
\frac{d}{d\tau}\mathfrak{P}_0(D_{\mathbf{S}}(\tau))= \oint_{\partial D_{\mathbf{S}}(\tau)} \frac{\partial\mathbf{a}}{\partial (\theta,l)} \odot\mathfrak{P}_0 \dot{l}(\tau)\, d\theta,
\end{equation*}
\notag
$$
and this is equivalent to the second equality in (4.26).
Thus, if the measures (3.3) form a generalized solution of problem (1.1), (1.3), then (2.8), (3.4) necessarily hold. This completes the proof of Theorem 2. 4.3. Proof of the results associated with Theorem 1Now we will prove several properties of the family of measures $\mathfrak{P}_t$ of the form (3.3) constructed during the proof of Theorem 1. For convenience of the following formulations, we set
$$
\begin{equation}
\widetilde{\mathcal{X}}\equiv(\widetilde{X},\widetilde{Y}) = (uY+vX,uY-vX),\qquad \widetilde{\mathbf{U}} \equiv(\widetilde{U},\widetilde{V})= (uV+vU,uV-vU),
\end{equation}
\tag{4.27}
$$
where the corresponding values, if necessary, will be provided with the index $3$ if they relate to the solution of problem (4.6), (4.8) and the index $6$ if they relate to the solution of problem (4.10), (4.12). Theorem 3. Let the point $\overline{\mathbf{x}}$ corresponds to the domain $\Omega$ on the $\mathbf{a}$-plane, and the corresponding quantities $\mathfrak{P}_t$ at this point are evaluated as the mass and momentum at the initial time of the domain $\Omega$ in accordance with the initial data (3.1). Then if $\widetilde{X}_3=0$ implies the equality $\widetilde{V}_3=\widetilde{Y}_3$, then the equality $\mathbf{I}_t/P_t=\mathcal{X}^*$ is true. Remark 1. Theorem 3 means that measures (4.5) form a generalized solution of problem (1.1), (3.1), since the law of conservation of momentum is also fulfilled at the point $\overline{\mathbf{x}}$. Proof of Theorem 3. For a proof, it is necessary to calculate the total mass $M(t)$ and the total momentum $\mathbf{I}(t)$ of the domain $\Omega$. Given the initial distribution (3.1) of density and velocity vector, we obtain
$$
\begin{equation*}
\begin{aligned} \, M &=\varrho(S_{\mathrm{I}}+S_{\mathrm{II}}+S_{\mathrm{III}}) +RS_{\mathrm{IV}} =\varrho(S_{\mathrm{I}}+S_{\mathrm{III}}) +\varrho S_{\mathrm{II}}+RS_{\mathrm{IV}}, \\ \frac{I}{u} &=\varrho(-S_{\mathrm{I}}+S_{\mathrm{II}}+S_{\mathrm{III}})-RS_{\mathrm{IV}} =\varrho(S_{\mathrm{III}}-S_{\mathrm{I}})+\varrho S_{\mathrm{II}}-RS_{\mathrm{IV}}, \\ \frac{J}{v} &=\varrho(-S_{\mathrm{I}}-S_{\mathrm{II}}+S_{\mathrm{III}})+RS_{\mathrm{IV}} =\varrho(S_{\mathrm{III}}-S_{\mathrm{I}})-\varrho S_{\mathrm{II}}+RS_{\mathrm{IV}}, \end{aligned}
\end{equation*}
\notag
$$
where $S_{\mathrm{I}}$, $S_{\mathrm{II}}$, $S_{\mathrm{III}}$, $S_{\mathrm{IV}}$ are the intersection areas of $\Omega$ with the corresponding quadrant. To calculate these areas, we will use the classical formula for the calculation of the area of the domain if the parametrization of its boundary is known, that is, formulae (4.22). We have
$$
\begin{equation*}
\begin{aligned} \, S_{\mathrm{I}} &=s_1^*s_2^*,\qquad S_{\mathrm{III}} =s_5^*s_4^*, \\ S_{\mathrm{II}} &=\frac{1}{2\Theta}\biggl\{(s_2^*)^2\frac{u}{v} + (s_4^*)^2\frac{v}{u}\biggr\} +\frac{\overline{l}_0\tau_3^*}{2} \int_{\tau_3^*}^{t}\{Y_3X_3'-X_3Y_3'+uY_3'+vX_3'\} \biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau \\ &\qquad+\frac{\overline{l}_0\tau_6^*}{2} \int_t^{\tau_6^*} \{Y_6X_6'-X_6Y_6'+uY_6'+vX_6'\} \biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau, \\ S_{\mathrm{IV}} &=\frac{\theta}{2}\biggl\{(s_1^*)^2\frac{v}{u}+ (s_5^*)^2\frac{u}{v}\biggr\} +\frac{\overline{l}_0\tau_3^*}{2} \int_t^{\tau_3^*}\{Y_3X_3'-X_3Y_3'-uY_3'-vX_3'\} \biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau \\ &\qquad+\frac{\overline{l}_0\tau_6^*}{2} \int_{\tau_6^*}^{t}\{Y_6X_6'-X_6Y_6'-uY_6'-vX_6'\} \biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau. \end{aligned}
\end{equation*}
\notag
$$
Now we use the expressions obtained earlier for $s_i^*$, $i=1,\dots,4$; the first equation in (4.6), and expressions (4.7) for wave III; similar relations (4.10), (4.11) for wave VI. As a result, we get
$$
\begin{equation*}
\begin{aligned} \, S_{\mathrm{I}} &=(\tau_3^*)^2uv\Theta,\qquad S_{\mathrm{III}}=(\tau_6^*)^2uv\Theta, \\ S_{\mathrm{II}} &=\frac{uv\Theta}{2}\{(\tau_3^*)^2+(\tau_6^*)^2\} +\frac12 \int_{\tau_3^*}^{t}\tau d_3^+\biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau + \frac12\int_t^{\tau_6^*}\tau d_6^-\biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau, \\ S_{\mathrm{IV}} &=\frac{uv\theta}{2}\{(\tau_3^*)^2+(\tau_6^*)^2\} +\frac12 \int_t^{\tau_3^*}\tau d_3^-\biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau + \frac12\int_{\tau_6^*}^{t}\tau d_6^+\biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau. \end{aligned}
\end{equation*}
\notag
$$
Next, for $M$, $\mathbf{I}$, we have
$$
\begin{equation*}
\begin{aligned} \, M &=uv\Sigma\biggl\{\varrho\frac{3\Theta}{2}+R\frac{\theta}{2}\biggr\} +\frac12 \int_{\tau_3^*}^{t}\tau(\varrho d_3^+-Rd_3^-) \biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau \\ &\qquad+\frac12\int_{\tau_6^*}^{t}\tau(Rd_6^+-\varrho d_6^-) \biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau, \\ \frac{I}{u} &=\varrho uv\Theta\Delta +uv\Sigma\biggl\{\varrho\frac{\Theta}{2} -R\frac{\theta}{2}\biggr\}+ \frac12\int_{\tau_3^*}^{t}\tau(\varrho d_3^++Rd_3^-) \biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau \\ &\qquad-\frac12\int_{\tau_6^*}^{t}\tau(Rd_6^++\varrho d_6^-) \biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau, \\ \frac{J}{v} &=\varrho uv\Theta\Delta -uv\Sigma\biggl\{\varrho\frac{\Theta}{2} -R\frac{\theta}{2}\biggr\}- \frac12\int_{\tau_3^*}^{t}\tau(\varrho d_3^++Rd_3^-) \biggl(\frac{\overline{l}_0\tau_3^*}{\tau}\biggr)\, d\tau \\ &\qquad+\frac12\int_{\tau_6^*}^{t}\tau(Rd_6^++\varrho d_6^-) \biggl(\frac{\overline{l}_0\tau_6^*}{\tau}\biggr)\, d\tau, \end{aligned}
\end{equation*}
\notag
$$
here, $\Sigma\equiv (\tau_3^*)^2+(\tau_6^*)^2$ and $\Delta\equiv (\tau_6^*)^2-(\tau_3^*)^2$. Changing the variables $\overline{l}=\overline{l}_0\tau_3^*/\tau$ or $\overline{l}=\overline{l}_0\tau_6^*/\tau$ in integrals, we have
$$
\begin{equation}
\begin{aligned} \, M &=uv\Sigma\sqrt{\varrho R}(2-\lambda)-\frac12 \int_{\overline{l}_0}^{\overline{l}_0\tau_3^*/t} \frac{(\overline{l}_0\tau_3^*)^2}{\overline{l}^{\,3}} (\varrho d_3^+-Rd_3^-)\, d\overline{l} \\ &\qquad-\frac12\int_{\overline{l}_0}^{\overline{l}_0\tau_6^*/t} \frac{(\overline{l}_0\tau_3^*)^2}{\overline{l}^{\,3}}(Rd_6^+-\varrho d_6^-)\, d\overline{l}, \\ \frac{I}{u} &=\varrho uv\Theta\Delta-\lambda\sqrt{\varrho R}\, uv\Sigma - \frac12 \int_{\overline{l}_0}^{\overline{l}_0\tau_3^*/t} \frac{(\overline{l}_0\tau_3^*)^2}{\overline{l}^{\,3}} (\varrho d_3^++Rd_3^-)\, d\overline{l} \\ &\qquad+\frac12\int_{\overline{l}_0}^{\overline{l}_0\tau_6^*/t} \frac{(\overline{l}_0\tau_3^*)^2}{\overline{l}^{\,3}} (Rd_6^++\varrho d_6^-)\, d\overline{l},\qquad vI+uJ=2\varrho\Theta (uv)^2\Delta. \end{aligned}
\end{equation}
\tag{4.28}
$$
From equations (4.6), (4.10) and the initial conditions (4.8), (4.12), we have
$$
\begin{equation*}
\begin{gathered} \, \varrho d_3^+-Rd_3^-=\overline{l}(m_3-m_3'\overline{l}),\qquad (m_3U_3)'=\frac{m_3U_3}{\overline{l}}- \frac{u}{\overline{l}^{\,2}}(\varrho d_3^++Rd_3^-), \\ Rd_6^+-\varrho d_6^-=\overline{l}(m_6-m_6'\overline{l}),\qquad (m_6U_6)'=\frac{m_6U_6}{\overline{l}}+\frac{u}{\overline{l}^{\,2}}(Rd_6^++\varrho d_6^-), \\ m_{0,3}=m_{0,6}=2uv\sqrt{\varrho R}\,\frac{2-\lambda}{\overline{l}_0}. \end{gathered}
\end{equation*}
\notag
$$
Substituting these expressions into (4.28), we finally get
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, M &=uv\Sigma\cdot\sqrt{\varrho R}(2-\lambda)+ \sum_{k=3,\,6}m_k^*\biggl\{\cdot\frac{t\overline{l}_0\tau_k^*}{2} -(\tau_k^*)^2uv\sqrt{\varrho R}(2-\lambda)\biggr\} \\ &=\frac{t\overline{l}_0}{2}(m_3^*\tau_3^*+m_6\tau_6^*), \\ \frac{I}{u} &=\varrho uv\Theta\Delta\,{-}\,\lambda\sqrt{\varrho R}\, uv \cdot\Sigma\,{+} \!\sum_{k=3,\,6}m_k^*U_k^*\cdot\frac{t\overline{l}_0\tau_k^*}{2u} \,{+}\, uv\sqrt{\varrho R}\bigl((\tau_3^*)^2{-}\,(\tau_6^*)^2(1\,{-}\,2\lambda)\bigr), \end{aligned} \\ vI+uJ=2\varrho\Theta (uv)^2\Delta. \end{gathered}
\end{equation*}
\notag
$$
By the symmetry of the problem, we can assume that $\tau_3^*=\tau_6^*$. Hence $\Delta=0$, $\Sigma=2(\tau_3^*)^2$, and
$$
\begin{equation*}
M=\frac{t\overline{l}_0\tau_3^*}{2}(m_3^*+m_6^*),\qquad \frac{I}{u}=\frac{t\overline{l}_0\tau_3^*}{2u}(m_3^*U_3^*+m_6^*U_6^*),\qquad vI+uJ=0.
\end{equation*}
\notag
$$
Applying transformation (4.14), we have $m_3=m_6$, $U_6=-(u/v)V_3$. Using the assumptions of Theorem 3 and (4.27), we find that
$$
\begin{equation}
\frac{I}{M}=\frac{m_3^*U_3^*+m_6^*U_6^*}{m_3^*+m_6^*} = \frac12\biggl(U_3^*-\frac{u}{v}V_3^*\biggr) = \frac12\biggl(X_3^*-\frac{u}{v}Y_3^*\biggr).
\end{equation}
\tag{4.29}
$$
Since the equality $uY_3+vX_3=0$ is satisfied on the line $L_1$, from (4.29) we have that $I/M=X_3^*$, and the corresponding equality also holds for $J$.
This completes the proof of Theorem 3. Theorem 4. For waves III and VI, the estimate $\widetilde{Y}\geqslant\lambda uv$ holds. In addition, for wave III, Remark 2. Theorem 4, means, first, that shock waves III and VI on the $\mathbf{x}$-plane lie above the line $L_2$. Second, since $C<0$, the discriminant of the quadratic expression with respect to $\widetilde{U}$ in inequality (4.30) is positive, and one of the roots of the corresponding quadratic equation is negative. Therefore, (4.30) implies an upper bound for $\widetilde{U}$ (in addition to the lower bound, which will be obtained by proving the first part of Theorem 4). Proof of Theorem 4. Consider wave III; for wave VI similar results are secured by the symmetry conditions (4.14). For brevity, in the proof of this theorem we will drop the index $3$ in the corresponding quantities.
First, note that, for wave III, the first integral (2.17) takes the form according to the new variables (4.27). Using (4.27) again and excluding the equation for $m$ due to (4.31), we can write system (4.6) and the initial data (4.9) as
$$
\begin{equation}
\begin{gathered} \, \begin{cases} \widetilde{\mathcal{X}}' =\dfrac{\widetilde{\mathcal{X}}-\widetilde{\mathbf{U}}}{\overline{l}}, \\ \widetilde{U}'=\dfrac{2}{C \overline{l}^{\,3}}\widetilde{U}^2 (\varrho d^+-Rd^-), \\ \widetilde{V}'=\dfrac{2}{C \overline{l}^{\,3}}\widetilde{U} \{\widetilde{V}(\varrho d^+-Rd^-)+ 2uv(\varrho d^++Rd^-)\}, \end{cases} \\ \widetilde{\mathcal{X}}_0 =(\lambda uv,\lambda uv),\qquad (\widetilde{U}_0,\widetilde{V}_0) =2uv\frac{(-\theta,\lambda)}{2-\lambda}, \nonumber \end{gathered}
\end{equation}
\tag{4.32}
$$
where
$$
\begin{equation*}
\frac{d^+}{\overline{l}}=\frac{1}{2uv}(\widetilde{Y}\widetilde{X}' - \widetilde{X}\widetilde{Y}')+\widetilde{X}',\qquad \frac{d^-}{\overline{l}}=\frac{1}{2uv}(\widetilde{Y}\widetilde{X}' - \widetilde{X}\widetilde{Y}')-\widetilde{X}'.
\end{equation*}
\notag
$$
It is easy to see that $C<0$ due to the first integral (4.31) and initial data from (4.32). From inequalities (4.7) we have $\varrho d^+-Rd^- > 0$. Excluding this expression from the third equation in (4.32), and using the second equation in (4.32), we obtain
$$
\begin{equation*}
\widetilde{V}'=\frac{\widetilde{U}'}{\widetilde{U}} \{\widetilde{V}+2uv\vartheta(\mu)\}
\end{equation*}
\notag
$$
or, what is the same,
$$
\begin{equation}
\biggl(\frac{\widetilde{V}}{\widetilde{U}}\biggr)' =- \biggl(\frac{1}{\widetilde{U}}\biggr)'\cdot 2uv\cdot\vartheta(\mu),
\end{equation}
\tag{4.33}
$$
where
$$
\begin{equation*}
\mu\equiv -\frac{d^-}{d^+},\qquad\vartheta(\mu) \equiv \frac{\varrho/R-\mu}{\varrho/R+\mu}.
\end{equation*}
\notag
$$
According to [26], we have $\mu\geqslant\sqrt{\varrho/R}$ for $\overline{l}\leqslant\overline{l}_0$, which implies that $\vartheta (\mu)\leqslant\vartheta (\sqrt{\varrho/R})=-\lambda$. Now from(4.33) we obtain
$$
\begin{equation*}
\biggl(\frac{\widetilde{V}}{\widetilde{U}}\biggr)' \geqslant 2uv\lambda\biggl(\frac{1}{\widetilde{U}}\biggr)'
\end{equation*}
\notag
$$
since $\widetilde{U}'<0$ by the second equation in (4.32). Solving the corresponding differential inequality for $\overline{l}\leqslant\overline{l}_0$, we find that
$$
\begin{equation}
\widetilde{V}\geqslant\lambda\widetilde{U}+2uv\lambda.
\end{equation}
\tag{4.34}
$$
Further, taking into account the sign $\widetilde{U}'$, we have $\widetilde{U}\geqslant\widetilde{U}_0\geqslant -uv$ for $\overline{l}\leqslant\overline{l}_0$.
Now from (4.34) we have $\widetilde{V}\geqslant\lambda uv$. Next, from the first equation in (4.32), we get the differential inequality
$$
\begin{equation*}
\biggl(\frac{\widetilde{Y}}{\overline{l}}\biggr)' =- \frac{\widetilde{V}}{\overline{l}^{\,2}} \leqslant -\frac{uv\lambda}{\overline{l}^{\,2}},
\end{equation*}
\notag
$$
solving which we come to inequality $\widetilde{Y}\geqslant\lambda uv$.
Now let us take a closer look at the second equation in (4.32) and write $\overline{U}\equiv\widetilde{U}/\overline{l}^{\,2}$. As a result, we have
$$
\begin{equation*}
\overline{U}'=\frac{2}{C}\overline{U}^{\,2}\frac{\varrho d^+-Rd^-}{\overline{l}}- 2\overline{l}\overline{U}.
\end{equation*}
\notag
$$
Since $C<0$ and since $\widetilde{U}<0$, which follows from the first integral (4.30), we arrive at the inequality
$$
\begin{equation}
\overline{U}'\geqslant\frac{2R}{C}\overline{U}^{\,2}\cdot\frac{d^+-d^-}{\overline{l}}.
\end{equation}
\tag{4.35}
$$
Further, based on the general relations (2.15), (2.16), it is easy to check that
$$
\begin{equation*}
\begin{gathered} \, d^+-d^-=|(\mathbf{u}^+-\mathbf{u}^-)\ (\mathcal{X}-\mathbf{u}^-)|- |(\mathbf{u}^+-\mathbf{u}^-)\ (\mathbf{U}-\mathbf{u}^-)|, \\ \biggl(\frac{d^+-d^-}{\overline{l}}\biggr)' =-\frac{1}{\overline{l}} |(\mathbf{u}^+-\mathbf{u}^-)\ \mathbf{U}'|. \end{gathered}
\end{equation*}
\notag
$$
Differentiating in the second equality, we obtain, for wave III,
$$
\begin{equation}
\frac{d^+-d^-}{\overline{l}}=\bigl(2\widetilde{U}+d^+-d^-\bigr)'.
\end{equation}
\tag{4.36}
$$
Combining (4.35), (4.36), we come to the inequality
$$
\begin{equation}
\frac{\overline{U}'}{\overline{U}^{\,2}} \geqslant\frac{2R}{C} \bigl(2\widetilde{U}+d^+-d^-\bigr)'.
\end{equation}
\tag{4.37}
$$
Solving inequality (4.37), we get the estimate
$$
\begin{equation*}
1+\frac{2R}{C}\, \overline{U}\bigl(2\widetilde{U}+d^+-d^-\bigr)\leqslant \overline{U}\chi_0,
\end{equation*}
\notag
$$
from which (4.30) follows. This proves Theorem 4.
Citation in format AMSBIB
\Bibitem{Ryk24} |
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