Russian Mathematical Surveys, 2023, Volume 78, Issue 5, Pages 971–981 DOI: https://doi.org/10.4213/rm10148e (Mi rm10148)

DOI: https://doi.org/10.4213/rm10148e

Bibliographic databases:

Zoom inZoom out

Vsevolod Alekseevich Solonnikov was born on 8 June 1933 in Leningrad, in a family of military officers. His grandfather Sergei Ivanovich Solonnikov was a lieutenant colonel in the Russian imperial army. His sons, Sergei and Orest, also took a military career. In the difficult 1930s Orest became the chief of staff of the Soviet Pacific Fleet. In spite of being a Cavallier of the Order of Lenin and other awards, he was arrested in 1937 and executed in 1938. In the 1950s Vsevolod’s uncle was fully rehabilitated. His named is still remembered with deep respect in the Navy.

The Leningrad apartment of uncle Orest, where little Vsevolod lived with his mother Galina Sergeevna, had a large library. Before WW2 he loved to tinker with books, browse a big geographic atlas, or leaf through an encyclopedia.

There was a nurse in the family, Ul’yana Antonovna Tumanova, who helped in raising the boy. She saved his life by managing to evacuate him from Leningrad in September 1941. His mother Galina stayed in Leningrad during the siege and died there in January 1942. On returning from evacuation, the nurse had to place Vsevolod in a foster house.

A few years later Vsevolod was admitted to cello classes in a ten-year musical boarding school at Leningrad Conservatory. In 1951 he passed exams and enrolled in the Faculty of Music Theory and Composition of Leningrad Conservatory. However, after the first year there he understood that physics and mathematics had more attraction for him. The lessons of Yulii Aronovich Mirskii, his wonderful physics teacher at the musical school, had played a great role in that.

In the summer of 1952 Solonnikov enrolled in the Faculty of Physics of Leningrad State University, and all of his further life was connected with mathematical physics. However, he retained his love for music during his whole life. As a student of the musical school and conservatory, he found plenty of friends and contacts in the musical world. He also met there his future wife Tat’yana Fedorovna Gamova. They lived a happy life together and had a son and grandchildren.

In his senior years at the university Solonnikov became a student of Ol’ga Aleksandrovna Ladyzhenskaya. His diploma thesis, devoted to the behaviour as $\varepsilon \to 0$ of solutions of elliptic equations with small parameter $\varepsilon$ multiplying higher derivatives, became a basis for his first paper [1].

In 1957, on graduating from the university with honours, Solonnikov began to work in Ladyzhenskaya’s group in the Leningrad Department of the Steklov Mathematical Institute (LDMI). He has been working in this group (which developed subsequently into the Laboratory of Mathematical Physics) for two thirds of a century.

Solonnikov made an enormous contribution to the development of mathematical physics and the theory of partial differential equations. For reasons of volume we can only briefly describe his main achievements, many of which are now in the ‘Golden Fund’ of mathematics.

I. Elliptic and parabolic equations and systems

The 1950s were the time when the contemporary understanding of the solvability of boundary-value problems for partial differential equations was formed. Ladyzhenskaya played one of the key roles in the solution of this fundamental problem, and her students were also active there. In particular, Solonnikov proved sharp (coercive) a priori estimates for solutions of the main boundary-value problems for second-order equations of elliptic and parabolic types, as well as for Stokes’ stationary system [2]–[4] in Sobolev spaces using the method of potentials. These works were the basis of his Ph.D. thesis One class of function spaces and a priori estimates for solutions of some boundary-value problems in mathematical physics, which he defended in 1961.

Next Solonnikov turned to linear and parabolic equations and systems of general form. The coercive estimates for solutions of boundary-value problems for elliptic equations, which he obtained simultaneously with and independently of S. Agmon, A. Douglis, and L. Nirenberg, were published slightly later, in [5] and [6]. At the same period he developed a solvability theory for initial-boundary value problems for parabolic systems in Sobolev and Hölder spaces [7] and thus completed the general theory of such problems going back to I. G. Petrovskii. In 1965 Solonnikov defended his D.Sc. thesis Boundary-value problems for linear parabolic systems of differential equations of general form. Some of these results were included in Chaps. 4 and 7 of the monograph Linear and quasi-linear equations of parabolic type [8], which he wrote jointly with Ladyzhenskaya and N. N. Uraltseva. In particular, they presented there the famous method for constructing a regularizer that had been introduced in [3]. This method enables one to find the unique solution of a boundary-value problem from explicit solutions of model problems in the whole space and a half-space. The monograph [8] became a desk book for several generations of experts in partial differential equations. It was translated into English and received international recognition.

Subsequently, in the 1980s, in the series of papers [9]–[11] (written, in particular, with his student E. V. Frolova and the Polish expert W. Zajączkowski) Solonnikov developed the solvability theory of boundary-value problems for elliptic and parabolic equations in domains with corners and edges on the boundary.

II. The theory of functions

The importance of the theory of function space for partial differential equations is well known. Solonnikov is the author of many results in this area, related to interpolation and multiplicative inequalities, and also of direct and converse trace theorems (including anisotropic ones). In this direction he often collaborated with V. P. Il’in, who can be viewed as his mentor in the theory of functions; he also worked with the young talented researcher K. K. Golovkin from LDMI (who sadly passed away prematurely). For instance, in [12] Solonnikov and Il’in established embedding and extension theorems for Sobolev–Slobodetskii, Besov, and Hölder spaces in domains of general form. In [13] and [14] the authors found conditions ensuring that operators in certain important classes (in particular, convolution operators) are bounded in spaces with fractional smoothness exponents. Note that in [14] these conditions were stated in terms of the operator symbol.

III. Mathematical problems in hydrodynamics

Investigations of mathematical models of hydrodynamics were among the main directions of the work of Ladyzhenskaya’s laboratory. Here we note Solonnikov’s papers [15]–[17] (one of which was written with Golovkin) on the creation of the theory of hydrodynamic potentials for three-dimensional linear problems. This theory enables one to study solvability ‘in the small’ for a general nonlinear problem, while keeping track on the improvement of smoothness of its solutions as the regularity of the data increases. At that time these techniques were quite refined.

Jointly with Shchadilov [18] Solonnikov proved that a generalized solution exists in the mixed boundary value problem for the stationary Stokes system and showed that the flow velocity belongs to $W_2^2(\Omega)$. In contrast to previous works, their proof did not use potential theory, but used instead an orthogonal decomposition of the space of vectors in $W_2^1(\Omega)$ with homogeneous boundary conditions. The velocity of the fluid was found by projecting Stokes’ system onto the subspace of divergence-free vectors, after which the pressure could be recovered separately. In addition, in [18] they gave an elementary proof of Korn’s inequality, which is central for problems in hydrodynamics.

In [19] Ladyzhenskaya and Solonnikov established a number of crucial results for vector-valued functions with prescribed divergence. Close results were independently obtained by J. Nečas, I. Babuška, F. Brezzi, and other authors. They are of importance for analytic and numerical investigation of problems with incompressibility condition, and are often stated in the contemporary literature as the so-called Ladyzhenskaya–Babuška–Brezzi condition (LBB condition).

Another interesting class consists of problems in domains with non-compact boundaries. Solonnikov began to engage in these problems in conjunction with Ladyzhenskaya. In [20] they distinguished the main function spaces where most precise results can be obtained and proved that such problems for the Stokes and Navier–Stokes systems with finite energy are solvable in domains with quite wide exits to infinity. Next (for instance, see [21]) they also investigated problems with arbitrary exits to infinity, but without the condition that the Dirichlet integral is finite. Solonnikov continued his investigations of this subject in [22], written together with his student Pileckas. A survey of results on problems with non-compact boundaries is available in [23].

IV. Problems in magnetic hydrodynamics

Ladyzhenskaya proposed to Solonnikov that they should generalize her results on the solvability of initial- boundary values problems for the Navier–Stokes system to the equations of magnetic hydrodynamics. In their first papers on this subject they established the unique solvability of non-stationary problems: local solvability for the three- dimensional problem and global solvability for the two-dimensional one [24]. They also showed that stationary boundary-value problems are solvable [25]. They returned to this topic later, and examined the stability and instability of stationary and forced periodic solutions [26].

Also subsequently Solonnikov was interested in various problems in magnetic hydrodynamics (including free boundary problems). In this area he worked with the Italian mathematicians G. Mulone, M. Padula, and S. Mosconi and also with his former postgraduate students Sh. Sahaev (from Kazakhstan) and Frolova. In particular, it was proved in [27] that the free boundary problem for the equations of magnetic hydrodynamics has a global solution for small initial data. In addition, in several papers by Solonnikov (for instance, see [28]) $L_p$-theory was developed for such problems.

V. Free boundary problems

In the late 1970s Solonnikov became engaged with free boundary problems. First he considered stationary problems of hydrodynamics, for example, the problem of a heavy viscous incompressible capillary fluid filling a part of a vessel. Not only the velocity and pressure in the fluid are unknown, but also the domain it occupies because the upper surface of the vessel is free: it is neither confined by a rigid wall nor under the action of external forces, apart from the surface tension of the fluid. Solonnikov proved that such problems for the Navier–Stokes systems are solvable both in the two-dimensional [29] and three-dimensional [30] cases. An additional complication in such problems is that the free surface is in contact with the rigid wall. This required of the author to find weighted Sobolev and Hölder spaces for domains with boundary edges in which coercive estimates in terms of the data of the problem holds for the solution.

In the 1990s Solonnikov considered free boundary problems for second-order parabolic equations. Jointly with some students he investigated so-called non- coersive problems with dynamical boundary conditions and obtained extremely precise estimate for these problems in various function spaces. For instance, in a series of papers with G. I. Bizhanova ([31] and some others) they proved theorems on the local solvability in time of the Stefan and Verigin problems in weighted Hölder spaces with minimal order of compatibility between the initial data and boundary conditions. A similar result in Sobolev spaces for the one-phase Stefan problem was obtained jointly with Frolova [32]. They returned to this topic several years later and considered the problem with dynamical boundary condition for the parabolic equation with small coefficient $\varepsilon$ multiplying the time derivative in the equation. Their uniform Hölder estimates for the solution of this problem as $\varepsilon\to 0$ allowed them to justify the quasistationary approximation to the Stefan problem [33].

Solonnikov is the author of pioneering works on non-stationary free boundary problems for a viscous fluid. He developed methods for the investigation of such problems which gave impetus to rapid development in this area, where he is one of the recognized leaders.

His first paper on this subject was [34] (1977). There, for a fluid without surface tension he showed that a local solution exists in Hölder spaces. He used a transition to Lagrangian variables, so that one can go over to a problem in a fixed domain, at the price of a significant complication of the coefficients of the system of equations. On the other hand, in [35] Solonnikov established the solvability of the problem for a capillary fluid, for all positive values of time, under the assumption that the initial data are close to equilibrium ones: the initial velocity is small, and the shape of the initial domain is close to a ball. Some details of the proof, related to the local unique solvability of a nonlinear problem in anisotropic Sobolev–Slobodetskii spaces, were presented in the subsequent papers of that series.

The problem turned out to be more difficult than the author believed originally. On the one hand the presence of surface tension stabilizes the solution, but on the other it makes the problem non-coercive, so that an integral term appears in the boundary conditions which cannot be treated as weak relative to the other terms in these conditions. For this reason one could not apply known methods to this problem and use the estimates established for the solution of a model problem in a half-space. So first of all, one had to find the solution explicitly in terms of its Fourier–Laplace image, and then estimate it using the Calderón–Zygmund theorem on estimates for singular integrals. To prove that the linear problem in a bounded domain is solvable they used the method mentioned above for the construction of a regularizer. The full proof of the local solvability of the problem of motion of a viscous drop was only complete in the early 1990s.

After that, jointly with his student I. S. Mogilevskii, Solonnikov obtained a similar result in Hölder spaces [36], [37]. In this case there arises an additional problem with estimating the explicit solution of the model problem in a half space: the absence of a result similar to the Calderón–Zygmund theorem in Hölder spaces. The authors used a fairly intricate method of Fourier multipliers for Hölder spaces, which is based on [14]. Subsequently, Solonnikov replaced it by a simpler method, which only works in Hölder spaces with reduced smoothness exponent with respect to time [38]. By developing these simpler techniques he (jointly with his student I. V. Denisova) could also prove that a similar problem for compressible bounded fluid mass is solvable, this time even without loss of smoothness with respect to time [39], [40]. Also note Solonnikov’s papers on the solvability of problems for a compressible fluid in Sobolev–Slobodetskii spaces and, in particular, [41], written with his Japanese colleague A. Tani.

VI. Stability of equilibrium shapes of an incompressible fluid

This classical problem, with contributions made historically by Newton, Maclaurin, Jacobi, Poincaré, Lyapunov, and many other authors, fascinated Solonnikov too. First he considered the motion of a drop close to a ball for small initial velocities. In several papers he proved that the problem of the motion of a finite volume of self-gravitating capillary fluid (compressible and incompressible alike) has a global solution and showed that in a coordinate system attached to the centre of mass of the drop the solution converges for large times to the rest state, and the volume occupied by the fluid tends to a ball. For an incompressible two-layer fluid he also obtained similar results jointly with Denisova [42]. In that paper they used the Hanzawa transformation, rather than Lagrangian coordinates. In this way they could relax assumptions on the smoothness of the initial surface, obtain sharp estimates, and extend the solution to an infinite interval of time. Moreover, to prove solvability they used the so-called method of generalized energy, developed by Solonnikov and his Italian colleague Padula [43], [44]. Using this method one can prove the existence of a global solution of the problem and its exponential decay in time. Solonnikov explained the main steps of the proof in his lectures at the Madeira Summer School [45].

In treating the more complicated problem of the stability of equilibrium shapes of isolated masses of rotating fluid Solonnikov turned to Lyapunov’s idea, who had proposed to examine the stability of equilibrium shapes using analytic methods. Lyapunov considered the second variation of the energy functional with respect to small perturbations of the boundary of the shape. Solonnikov developed this idea further and extended it to capillary fluids. In a large series of works he investigated the stability of axially symmetric and asymmetric equilibrium shapes. He showed [46] that for sufficiently small initial data (namely, the angular rotation velocity, the distribution of velocities, and the deviation of the initial shape of the drop from an equilibrium one) and for a positive second variation of the energy functional the perturbation of the shape tends to zero as $t\to\infty$. In addition, the motion of the drop transforms into the rotation of the fluid mass as a rigid body. On the other hand, symmetric equilibrium shapes of a rotating incompressible fluid are guaranteed instable in the case when the second variation of the energy functional can take negative values. A survey of results on this subject was presented in [47]. Summing up the above we can conclude that, thanks to Solonnikov’s papers, the theory developed before him acquired a finished look from the mathematical standpoint. Solonnikov himself believes that his achievements in this area are his most important results in mathematical hydrodynamics.

In recent years Solonnikov continued to develop techniques required for the investigation of free boundary problems for the Navier–Stokes equations. He generalized his methods to various cases of two-phase drops. With Denisova he wrote large surveys on these subjects, devoted to incompressible [48] and compressible fluid [49], and also the monograph Motion of a drop in an incompressible fluid, published in Russian and English [50], and devoted to the solvability of problems describing the motion of a two-layer fluid with unknown interface. On the basis of [51] they were able to find conditions for the stability of axially symmetric two-phase equilibrium shapes [52].

In addition, Solonnikov considered the case of two heterogeneous media, a compressible and an incompressible one. He established the global solvability of the problem of the motion of a two-phase drop in the $L_2$-setup [53]. In this way Solonnikov’s theory is repeatedly getting new development and finds numerous applications.

Apart from his research, Solonnikov has strongly been engaged in teaching activities. For 15 years he taught at the Department of Mathematical Physics in the Faculty of Mathematics and Mechanics of Leningrad State University: he read special courses, gave classes, supervised students’ research. Students distinguished the permanently high level of his lectures, exceptional thoroughness and pedagogical mastery in the presentation of his topics. In 1979 he was awarded the title of Professor. In tandem with Ural’tseva, he wrote a textbook on embedding theorems for Sobolev spaces [54]. Solonnikov had many diploma students in the 1980s. Subsequently, nine students defended their Ph.D. theses under his supervision, five of them later became Doctors of Sciences, and K. Pileckas was elected a member of the Lithuanian Academy of Sciences. Solonnikov’s students work in many universities in Russia, Lithuania, Kazakhstan, Armenia, Italy, and he is still in constant touch with them and follows their work closely. In this connection we can mention the paper [55], concerned with the flow of a viscous incompressible fluid with free boundary along a surface close to an inclined plane.

In the 1980s Solonnikov began travelling abroad on scientific trips. Subsequently, he worked in Germany, Italy, Portugal, Japan, and other countries. He has many co-authors in various countries and speaks five languages fluently.

Solonnikov is the author of more than 250 research papers, including two monographs. His research works are well known in Russia and other countries; they have significantly influenced the development of the contemporary theory of partial differential equations.

He was an invited speaker at many international conferences and scientific schools, including at the International Congress of Mathematicians in Berkeley (1986). A number of mathematical conferences in Russia, Portugal, Italy, Poland, and Lithuania were organized in his honour.

For many years Solonnikov was a member of the editorial boards of several Russian and International scientific publications. Also now he is a member of the editorial boards of the journals Interfaces and Free Boundaries and Journal of Mathematical Fluid Mechanics, which are among the most cited mathematical journals.

Solonnikov’s research achievements, his creative energy, and devotion to science are highly appreciated by the mathematical community in Russia and other countries. In 2003 we was awarded the prize of the Alexander von Humboldt Foundation (Germany). In hhe same year he got the title of Professor of Ferrara University (Italy). In 2009. together with V. V. Pukhnachev, he was awarded the M. A. Lavrentiev Prize of the Russian Academy of Sciences for the cycle of papers “Free boundary problems for the Navier–Stokes equations”. In 2013 we was awarded the P. L. Chebyshëv Prize of the Government of Saint Petersburg for Outstanding Results in Science and Technology in the nomination “Mathematics and Mechanics”. In 2015 he was awarded the title of a Honoured Scientist of the Russian Federation, and in 2017 he was elected a foreign member of the Lissabon Academy of Sciences. In 1992–1996 he was a member of the Executive Board of the European Mathematical Society.

Solonnikov’s fascination with mathematics, his broad vision, kindness, and open communication style gained him love and respect of many mathematicians all over the world. On behalf of his students and colleagues we congratulate Vsevolod Solonnikov on his 90th birthday and wish him good health, many years of creative work, and new interesting research results.

Bibliography
1.	V. A. Solonnikov, “Linear differential equations with a small parameter in the highest derivative term”, Dokl. Akad. Nauk SSSR, 119:3 (1958), 454–457 (Russian)
2.	V. A. Solonnikov, “A priori estimates for certain boundary value problems”, Soviet Math. Dokl., 2 (1961), 723–727
3.	V. A. Solonnikov, “A priori estimates for second-order parabolic equations”, Amer. Math. Soc. Transl. Ser. 2, 65, Amer. Math. Soc., Providence, RI, 1967, 51–137
4.	V. Solonnikov, “On estimates of Green's tensors for certain boundary problems”, Soviet Math. Dokl., 1 (1960), 128–131
5.	V. A. Solonnikov, “On general boundary problems for systems which are elliptic in the sense of A. Douglis and L. Nirenberg. I”, Amer. Math. Soc. Transl. Ser. 2, 56, Amer. Math. Soc., Providence, RI, 1964, 193–232
6.	V. A. Solonnikov, “General boundary value problems for Douglis–Nirenberg elliptic systems. II”, Proc. Steklov Inst. Math., 92 (1968), 269–339
7.	V. A. Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general form”, Proc. Steklov Inst. Math., 83 (1965), 1–184
8.	O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva (Uraltseva), Linear and quasi-linear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968, xi+648 pp.
9.	V. Zaionchkovskii (Zaja̧czkowski) and V. A. Solonnikov, “Neumann problem for second-order elliptic equations in domains with edges on the boundary”, J. Soviet Math., 27 (1984), 2561–2586
10.	V. A. Solonnikov, “Solvability of the classical initial-boundary-value problems for the heat-conduction equation in a dihedral angle”, J. Soviet Math., 32 (1986), 526–546
11.	V. A. Solonnikov and E. V. Frolova, “On a problem with the third boundary condition for the Laplace equation in a plane angle, and its applications to parabolic problems”, Leningrad Math. J., 2:4 (1991), 891–916
12.	V. P. Il'in and V. A. Solonnikov, “On some properties of differentiable functions of several variables”, Amer. Math. Soc. Transl. Ser. 2, 81, Amer. Math. Soc., Providence, RI, 1969, 67–90
13.	K. K. Golovkin and V. A. Solonnikov, “Estimates for integral operators in translation-invariant norms”, Amer. Math. Soc. Transl. Ser. 2, 61, Amer. Math. Soc., Providence, RI, 1967, 97–112        ; II, Proc. Steklov Inst. Math., 92 (1968), 3–32
14.	K. K. Golovkin and V. A. Solonnikov, “Estimates of convolution operators”, Semin. Math., 7, V. A. Steklov Math. Inst., Leningrad, 1968, 1–36
15.	K. K. Golovkin and V. A. Solonnikov, “On the first boundary problem for the nonstationary Navier–Stokes equations”, Soviet Math. Dokl., 2 (1961), 1188–1193
16.	V. A. Solonnikov, “Estimates of the solutions of a nonstationary linearized system of Navier–Stokes equations”, Amer. Math. Soc. Transl. Ser. 2, 75, Amer. Math. Soc., Providence, RI, 1968, 1–116
17.	V. A. Solonnikov, “On the differentiability properties of the solution of the first boundary-value problem for a non-stationary system of Navier–Stokes equations”, Boundary-value problems of mathematical physics. 2, Tr. Mat. Inst. Steklov., 73, Nauka, Moscow–Leningrad, 1964, 221–291 (Russian)
18.	V. A. Solonnikov and V. E. Shchadilov, “A certain boundary value problem for the stationary system of Navier–Stokes equations”, Proc. Steklov Inst. Math., 125 (1973), 186–199
19.	O. A. Ladyzhenskaya and V. A. Solonnikov, “Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations”, J. Soviet Math., 10:2 (1978), 257–286
20.	O. A. Ladyzhenskaya and V. A. Solonnikov, “On the solvability of boundary and initial-boundary value problems for Navier–Stokes equations in domains with non-compact boundaries”, Vestn. Lenongr. Univ., 13:3 (1977), 39–47 (Russian)
21.	O. A. Ladyzhenskaya and V. A. Solonnikov, “Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral”, J. Soviet Math., 21:5 (1983), 728–761
22.	V. A. Solonnikov and K. I. Piletskas (Pileckas), “Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier–Stokes system of equations in domains with noncompact boundaries”, J. Soviet Math., 34:6 (1986), 2101–2111
23.	V. A. Solonnikov, “On problems of the hydrodynamics of a viscous incompressible fluid in domains with noncompact boundaries”, St. Petersburg Math. J., 4:6 (1993), 1081–1102
24.	O. A. Ladyzhenskaya and V. A. Solonnikov, “Solution of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid”, Mathematica problems in hydrodynamics and magnetohydrodynamics for a viscous incompressible fluid, Tr. Mat. Inst. Steklov., 59, Publishing house of the USSR Academy of Sciences, Moscow–Leningrad, 1960, 115–173 (Russian)
25.	V. A. Solonnikov, “Some stationary boundary-value problems in magnetohydrodynamics”, Mathematica problems in hydrodynamics and magnetohydrodynamics for a viscous incompressible fluid, Tr. Mat. Inst. Steklov., 59, Publishing house of the USSR Academy of Sciences, Moscow–Leningrad, 1960, 174–187 (Russian)
26.	O. A. Ladyzhenskaya and V. A. Solonnikov, “The linearization principle and invariant manifolds for problems of magnetohydrodynamics”, J. Soviet Math., 8 (1977), 384–422
27.	V. A. Solonnikov and E. V. Frolova, “Solvability of a free boundary problem of magnetohydrodynamics in an infinite time interval”, J. Math. Sci. (N. Y.), 195:1 (2013), 76–97
28.	V. A. Solonnikov, “$L_p$-theory of free boundary problems of magnetohydrodynamics in multi-connected domains”, Ann. Univ. Ferrara Sez. VII Sci. Mat., 60:1 (2014), 263–288
29.	V. A. Solonnikov, “Solvability of a problem on the plane motion of a heavy viscous incompressible capillary liquid partially filling a container”, Math. USSR-Izv., 14:1 (1980), 193–221
30.	V. A. Solonnikov, “Solvability of three-dimentional problem with a free boundary for a stationary system of Navier–Stokes equations”, J. Soviet Math., 21:3 (1983), 427–450
31.	G. I. Bizhanova and V. A. Solonnikov, “Free boundary problems for second order parabolic equations”, St. Petersburg Math. J., 12:6 (2001), 949–981
32.	V. A. Solonnikov and E. V. Frolova, “$L_p$-theory for the Stefan problem”, J. Math. Sci. (N. Y.), 99:1 (2000), 989–1006
33.	V. A. Solonnikov and E. V. Frolova, “Justification of a quasistationary approximation for the Stefan problem”, J. Math. Sci. (N. Y.), 152:5 (2008), 741–768
34.	V. A. Solonnikov, “Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface”, Math. USSR-Izv., 11:6 (1977), 1323–1358
35.	V. A. Solonnikov, “Unsteady motion of a finite mass of fluid, bounded by a free surface”, J. Soviet Math., 40:5 (1988), 672–686
36.	I. S. Mogilevskii and V. A. Solonnikov, “Solvability of a noncoercive initial-boundary value problem for a Stokes system in Holder classes of functions (half-space case)”, Z. Anal. Anwendungen, 8:4 (1989), 329–347 (Russian)
37.	I. S. Mogilevskii and V. A. Solonnikov, “On the solvability of an evolution free boundary problem for the Navier–Stokes equations in Hölder spaces of functions”, Mathematical problems relating to Navier–Stokes equations, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992, 105–181
38.	V. A. Solonnikov, “Estimates of solutions of the second initial boundary-value problem for the Stokes system in the spaces of functions having Hölder continuous derivatives with respect to spatial variables”, J. Math. Sci. (N. Y.), 109:5 (2002), 1997–2017
39.	I. V. Denisova and V. A. Solonnikov, “Classical solvability of a model problem in a half-space, related to the motion of an isolated mass of a compressible fluid”, J. Math. Sci. (N. Y.), 115:6 (2003), 2753–2765
40.	I. V. Denisova and V. A. Solonnikov, “Classical solvability of a problem on the motion of an isolated mass of compressible fluid”, St. Petersburg Math. J., 14:1 (2002), 53–74
41.	V. A. Solonnikov and A. Tani, “Free boundary problem for a viscous compressible flow with a surface tension”, Constantin Carathéodory: an international tribute, World Sci. Publ., Teaneck, NJ, 1991, 1270–1303
42.	I. V. Denisova and V. A. Solonnikov, “$L_2$-theory for two incompressible fluids separated by a free interface”, Topol. Methods Nonlinear Anal., 52:1 (2018), 213–238
43.	M. Padula and V. A. Solonnikov, “On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation”, Topics in mathematical fluid mechanics, Quad. Mat., 10, Dept. Math., Seconda Univ. Napoli, Caserta, 2002, 185–218
44.	V. A. Solonnikov, “Estimate of the generalized energy in a free-boundary problem for a viscous incompressible fluid”, J. Math. Sci. (N. Y.), 120:5 (2004), 1766–1783
45.	V. A. Solonnikov, “Lectures on evolution free boundary problems: classical solutions”, Mathematical aspects of evolving interfaces (Funchal 2000), Lecture Notes in Math., 1812, Springer-Verlag, Berlin, 2003, 123–175
46.	V. A. Solonnikov, “On the stability of axially symmetric equilibrium figures of a rotating viscous incompressible fluid”, St. Petersburg Math. J., 16:2 (2005), 377–400
47.	V. Solonnikov, “On problem of stability of equilibrium figures of uniformly rotating viscous incompressible liquid”, Instability in models connected with fluid flows. II, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008, 189–254
48.	V. A. Solonnikov and I. V. Denisova, “Classical well-posedness of free boundary problems in viscous incompressible fluid mechanics”, Handbook of mathematical analysis in mechanics of viscous fluids, Springer, Cham, 2018, 1135–1220
49.	I. V. Denisova and V. A. Solonnikov, “Local and global solvability of free boundary problems for the compressible Navier–Stokes equations near equilibria”, Handbook of mathematical analysis in mechanics of viscous fluids, Springer, Cham, 2018, 1947–2035
50.	I. V. Denisova and V. A. Solonnikov, Motion of a drop in an incompressible fluid, Adv. Math. Fluid Mech., Lect. Notes Math. Fluid Mech., Birkhäuser/Springer, Cham, 2021, vii+316 pp.
51.	V. A. Solonnikov, “On the problem of non-stationary motion of two viscous incompressible liquids”, J. Math. Sci. (N. Y.), 142:1 (2007), 1844–1866
52.	I. V. Denisova and V. A. Solonnikov, “Rotation problem for a two-phase drop”, J. Math. Fluid Mech., 24 (2022), 40, 26 pp.
53.	V. A. Solonnikov, “$L_2$-theory for two viscous fluids of different types: compressible and incompressible”, St. Petersburg Math. J., 32:1 (2021), 91–137
54.	V. A. Solonnikov and N. N. Uraltseva, “Sobolev spaces”, Publishing house of Lenngrad State University, Leningrad, 1981, 129–199 (Russian)
55.	K. Pileckas and V. A. Solonnikov, “Viscous incompressible free-surface flow down an inclined perturbed plane”, Ann. Univ. Ferrara Sez. VII Sci. Mat., 60:1 (2014), 225–244

Citation in format AMSBIB

\Bibitem{BizDenNaz23}
\by G.~I.~Bizhanova, I.~V.~Denisova, A.~I.~Nazarov, K.~I.~Pileckas, V.~V.~Pukhnachev, S.~I.~Repin, J.-F.~Rodrigues, G.~A.~Seregin, N.~N.~Uraltseva, E.~V.~Frolova
\paper On the 90th birthday of Vsevolod Alekseevich Solonnikov
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 5
\pages 971--981
\mathnet{http://mi.mathnet.ru//eng/rm10148}
\crossref{https://doi.org/10.4213/rm10148e}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4723257}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023RuMaS..78..971B}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001184355800009}