| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief communications On the interaction of shock waves in two-dimensional isobaric media Yu. G. Rykov Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
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    This note discusses possible types of interaction of strong singularities in a two- dimensional medium in the absence of its own pressure drop. The immediate model of such media is the equations of gas dynamics in which the pressure $P$ is formally assumed to be zero. The resulting system of equations is a quasi-linear system with an incomplete set of eigenvalues and vectors. Therefore, its generalized solutions should be understood in the sense of Radon measures [1], and the corresponding shock waves are evolving strong singularities containing the delta function in the density $\varrho$ on manifolds of various dimensions. The system of equations under consideration models the processes of matter concentration and is used in describing a number of phenomena in dispersed and multiphase media (see, for example, [2]) and in astrophysical applications related to the large-scale structure of the Universe (see, for example, [3]). There are also parallels with the formalism of quantum mechanics (Schrödinger’s equation) that arise in considerations of self-gravitating media (see, for example, [4] and [5]). Note that non-classical shock waves of another type that satisfy evolutionary conditions different from the known Lax conditions also arise in applications: see, for example, [6]. Let $\mathbf{x}\equiv(x,y)$, $(t,\mathbf{x})\in\mathbb{R}_{+}\times \mathbb{R}^{2}$, and $\mathbf{\nabla}=(\partial/\partial x,\partial/\partial y)$; then the two-dimensional system of equations of pressureless gas dynamics can be written as follows:
$$
\begin{equation}
\frac{\partial\varrho}{\partial t}+\mathbf{\nabla}\cdot \varrho\mathbf{u}=0,\qquad \frac{\partial(\varrho\mathbf{u})}{\partial t}+\mathbf{\nabla}\cdot (\varrho\mathbf{u}\otimes\mathbf{u})=0,
\end{equation}
\tag{1}
$$
where $\varrho\geqslant 0$ is the medium density, $\mathbf{u}\equiv (u,v)$ is the velocity vector, and $\otimes$ denotes the tensor product. Let the initial Cauchy data also be given: $\varrho(0,\mathbf{x})=\varrho_0(\mathbf{x})\in L^{\infty}(\mathbb{R}^2)$ and $\mathbf{u}(0,\mathbf{x})=\mathbf{u}_{0}(\mathbf{x})\in L^{\infty}(\mathbb{R}^{2})$. It is natural for generalized solutions of system (1) that there arise curves in the $\mathbf{x}$-space or, equivalently, surfaces in the $(t,\mathbf{x})$-space on which mass and momentum concentrate with densities $M(t,l)$ and $\mathbf{I}(t,l)$. Let such surfaces be defined parametrically by $\mathbf{X}\equiv\bigl(\chi(t,l),\gamma(t,l)\bigr)$. Then for generalized solutions of (1) the Rankine–Hugoniot relations are fulfilled (see [7] and [8]):
$$
\begin{equation}
\begin{aligned} \, \frac{\partial M}{\partial t}&=\frac{\partial\chi}{\partial l} \{V[\varrho]-[\varrho v]\}-\frac{\partial\gamma}{\partial l} \{U[\varrho]-[\varrho u]\}, \\ \frac{\partial \mathbf{I}}{\partial t}&=\frac{\partial\chi}{\partial l} \{V[\varrho\mathbf{u}]-[\varrho v\mathbf{u}]\}- \frac{\partial\gamma}{\partial l}\{U[\varrho\mathbf{u}]- [\varrho u\mathbf{u}]\}, \end{aligned}
\end{equation}
\tag{2}
$$
where $\mathbf{I}=M\mathbf{U}$, $\mathbf{U}\equiv(U,V)\equiv (\partial\chi/\partial t,\partial\gamma/\partial t)$ and $[f]\equiv f(t,\mathbf{X}+0)-f(t,\mathbf{X}-0)$.
It was also shown in [7] that, in a collision, shock waves (2) form, generally speaking, a shock wave of the same type but with increased density. However, it turns out that there is also another type of interaction. Theorem 1. There exist initial data $\varrho_{0}$ and $\mathbf{u}_{0}$ such that in the generalized solution of system (1) there are two shock waves of type (2) which, when interacting, generate the delta function at a point in the measures of mass and momentum in the $\mathbf{x}$-space. The statement of Theorem 1 means that in the Cauchy problem for (1) it is possible to form an evolving hierarchy of shock waves on manifolds of different dimensions; for the modified Rankine-Hugoniot relation in this case, see [9]. Such a hierarchy will have a much richer structure when one goes over to the multidimensional case. Also, Theorem 1 implements an abstract construction in [10] and allows us to give the following variational interpretation of the generalized solution in terms of a Lagrangian mapping ${\mathcal L}_{t}\colon\mathbf{a}\to\mathbf{x}$, where $\mathbf{a}\equiv(a,b)$ are the coordinates on the initial (Lagrangian) plane, in the case of an interaction leading to the emergence of a delta function at a point. Fix $t>0$. Let $A_{1}$ be a set of disjoint domains in $\mathbb{R}^{2}$, and let $A_{2}$ be a set of domains obtained as follows. Consider a set of disjoint regions $G_{\alpha}\subset \mathbb{R}^{2}\setminus A_{1}$ of the form $G_{\alpha}=\{\mathbf{a}_{\alpha}(s,l),\, s\in [s_{\alpha}^{1},s_{\alpha}^{2}],\,l\in[l_{\alpha}^{1},l_{\alpha}^{2}]\}$. Then for each $\alpha$, $A_{2}$ includes all domains described by the parametrization $\mathbf{a}_{\alpha}$ but also satisfying the conditions $l\in[l^{1},l^{2}]\subset[l_{\alpha}^{1},l_{\alpha}^{2}]$ and $0<|l^{1}-l^{2}|\leqslant|l_{\alpha}^{1}-l_{\alpha}^{2}|$. Also let $A_{3}$ be the set of all domains contained in $\mathbb{R}^{2}\setminus(A_{1}\cup A_{2})$, and let $B=A_{1}\cup A_{2}\cup A_{3}$. Set
$$
\begin{equation}
\mathbf{F}(t,\mathbf{x};G)=\iint_{G}\biggl[\mathbf{u}_{0}(\mathbf{a})- \frac{\mathbf{x}-\mathbf{a}}{t}\biggr] \varrho_{0}(\mathbf{a})\,d\mathbf{a},\qquad G\in B.
\end{equation}
\tag{3}
$$
By the derivative of $\mathbf{F}$ over the area containing a point $\mathbf{a}$ we mean the quantity
$$
\begin{equation}
\frac{\delta\mathbf{F}}{\delta\mathbf{a}}=\lim_{|G|\downarrow\min} \frac{\mathbf{F}(t,\mathbf{x};G)}{|G|}\,,\qquad \mathbf{a}\in G\in B
\end{equation}
\tag{4}
$$
(the notation $|G|\downarrow\min$ means that the area of $G$ tends to a minimum under the condition $G\in B$).
Note that, generally speaking, instead of the measure $\varrho_{0}\,d\mathbf{a}$ an arbitrary non- negative Radon measure $M_{0}(d\mathbf{a})$ can be used in (3). However, in this case formula (4) requires some additional care. Theorem 2. Let there exists a generalized solution of system (1). Consider an arbitrary $t>0$. Then for any $\mathbf{a}\in\mathbb{R}^{2}$, except for a set of Lebesgue measure zero, there exists $\mathbf{x}$ such that $\delta\mathbf{F}(t,\mathbf{x};G)/\delta\mathbf{a}=0$. Theorem 2 describes the properties of the Lagrangian mapping ${\mathcal L}_{t}$, which can be constructed with the help of the function $\mathbf{F}$, and is an implementation of a more abstract form of the variational principle presented in [11]. 
Citation in format AMSBIB
\Bibitem{Ryk23} |
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