Evgeniy Panov

• Doctor of Sciences
• Professor (Full) at Yaroslav-the-Wise Novgorod State University

We study generalized solutions of an evolutionary equation related to some densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and suggest a criteria of uniqueness of this semigroup. We also find a stronger criteria of uniqueness of generalized sol...

In this paper, we study self-similar solutions of the multiphase Stefan problem for the heat equation on the half-line x > 0 with constant initial data and Dirichlet or Neumann boundary conditions. In the case of the Dirichlet boundary condition, we prove that the nonlinear algebraic system for determining the free boundaries is a gradient system,...

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coefficient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a finite number of variables. A similar r...

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case where the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem understood in the sense...

We find an explicit form of entropy solution to a Riemann problem for a degenerate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients. It is demonstrated that this solution corresponds to the minimum point of some strictly convex function of a finite number of variables. We also discuss the limit when piecewise...

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar resul...

We study the multi-phase Stefan problem with increasing Riemann initial data and generally negative latent specific heats for phase transitions. We propose a variational formulation of self-similar solutions, which allows us to find precise conditions for the existence and uniqueness of a solution.

We study multi-phase Stefan problem with increasing Riemann initial data and with generally negative latent specific heats for the phase transitions. We propose the variational formulation of self-similar solutions, which allows to find precise conditions for existence and uniqueness of the solution.

We introduce the notion of entropy solutions (e.s.) to a conservation law with an arbitrary jump continuous flux vector and prove the existence of the largest and the smallest e.s. to the Cauchy problem. The monotonicity and stability properties of these solutions are also established. In the case of a periodic initial function, we derive the uniqu...

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense...

We find an explicit form of entropy solution to a Riemann problem for a degenerate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients. It is demonstrated that this solution corresponds to the minimum point of some strictly convex function of a finite number of variables. MSC Classification: 35K55 , 35K65 , 35L6...

Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity in the appropriate sense.

We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to the minimum point of some strictly convex function of a finite number of variables. In the limit as number of...

We consider a second-order nonlinear degenerate parabolic equation in the case, where the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration a...

We introduce the notion of entropy solutions (e.s.) to a conservation law with an arbitrary jump continuous flux vector and prove existence of the largest and the smallest e.s. to the Cauchy problem. The monotonicity and stability properties of these solutions are also established. In the case of a periodic initial function we derive the uniqueness...

Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).

A first-order quasilinear equation with an odd flux function that has a single point of inflexion at zero is studied. A method for constructing sign-alternating discontinuous entropy solutions of this equation, based on the Legendre transform, is proposed. Bibliography: 18 titles.

Изучается квазилинейное уравнение первого порядка с нечетной функцией потока, имеющей в нуле единственную точку перегиба. Предложен способ построения разрывных знакопеременных энтропийных решений этого уравнения, основанный на преобразовании Лежандра. Библиография: 18 названий.

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an...

We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of...

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.

We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of...

We prove existence of the largest entropy sub‐solution and the smallest entropy super‐solution to the Cauchy problem for a nonlinear degenerate parabolic equation with only continuous flux and diffusion functions. Applying this result, we establish the uniqueness of entropy solution with periodic initial data. The more general comparison principle...

We prove existence of the largest entropy sub-solution and the smallest entropy super-solution to the Cauchy problem for a nonlinear degenerate parabolic equation with only continuous flux and diffusion functions. Applying this result, we establish the uniqueness of entropy solution with periodic initial data. The more general comparison principle...

We prove the asymptotic convergence of a space-periodic entropy solution of a one-dimensional degenerate parabolic equation to a traveling wave. It is also shown that on a segment containing the essential range of the limit profile the flux function is linear (with the slope equaled to the speed of the traveling wave) and the diffusion function is...

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux.

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.

It is shown that bounded sequences satisfying nonlinear differential constraints, strongly precompact under an exact condition of nondegeneration of these conditions. The proof is based on new localization principles for ultraparabolic H-measures with continuous indices.

We found the precise condition for the decay as \(t\rightarrow \infty \) of Besicovitch almost periodic entropy solutions of multidimensional scalar conservation laws. Moreover, in the case of one space variable we establish asymptotic convergence of the entropy solution to a traveling wave (in the Besicovitch norm). Besides, the flux function turn...

We prove the asymptotic convergence of a space-periodic entropy solution of a one-dimensional degenerate parabolic equation to a traveling wave. It is also shown that on a segment containing the essential range of the limit profile the flux function is linear (with the slope equaled to the speed of the traveling wave) and the diffusion function is...

This paper concerns the trace problem for quasi-solutions of scalar conservation laws defined in Ω ⊂ ℝⁿ⁺¹ with roughly nonautonomous flux functions f ∈ L¹loc(Ω;C(ℝ)ⁿ⁺¹). Under a nondegeneracy condition of f at the boundary of Ω, we show a strong trace on it.

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a fa...

The existence and uniqueness of a generalized entropy solution in the class of Besicovitch almost periodic functions is proved for the Cauchy problem for a multidimensional inhomogeneous quasilinear equation of the first order. © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

Установлено существование и единственность обобщенного энтропийного решения задачи Коши для многомерного неоднородного квазилинейного уравнения первого порядка в классе почти периодических функций Безиковича. Библиография: 12 названий.

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost p...

We found the precise condition for the decay as $t\to\infty$ of Besicovitch almost periodic entropy solutions of multidimensional scalar conservation laws. Moreover, in the case of one space variable we establish asymptotic convergence of the entropy solution to a traveling wave (in the Besicovitch norm). Besides, the flux function turns out to be...

Доказано, что периодическое обобщенное энтропийное решение одномерного закона сохранения сходится с ростом времени к бегущей волне. При этом функция потока является линейной на минимальном отрезке, содержащем существенный образ профиля бегущей волны, а скорость волны совпадает с угловым коэффициентом функции потока, ограниченной на этот отрезок. Би...

We prove that the periodic generalized entropy solution of a one-dimensional conservation law converges in time to a traveling wave. In this case, the flow function is linear on the minimal interval containing the essential image of the traveling wave profile and the wave velocity coincides with the angular coefficient of the flow function bounded...

In this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisf...

We establish a necessary and sufficient condition for decay of periodic renormalized solutions to a multidimensional conservation law with merely continuous flux vector.

We propose a procedure of reducing the notions of weak and strong two-scale convergence to the notions of weak and strong convergence in the Hilbert space L 2 respectively.

We show that, under the linear nondegeneracy condition, any generalized entropy solution, periodic in spatial variables, to the inhomogeneous quasilinear first order equation converges to a constant as the time tends to infinity. Bibliography: 16 titles.

We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.

We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients $a(x)$. It is shown that any generalized solution satisfies the renormalization property if and only if the operator $a\cdot\nabla u$, $u\in C_0^1(\mathbb{R}^n)$ in the Hilbert space $L^2(\mathbb{R}^n)$ is an essentially sk...

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We propose also the necessary and sufficient condition for the decay of entropy solutions as time $t\to+\infty$.

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We propose also the necessary and sufficient condition for the decay of almost periodic entropy solutions as time $t\t...

We introduce new variant of $H$-measures defined on spectra of general algebra of test symbols and derive the localization properties of such $H$-measures. Applications for the compensated compactness theory are given. In particular, we present new compensated compactness results for quadratic functionals in the case of general pseudo-differential...

We establish that λ- and μ-holomorphic functions can coincide on the boundary of a domain only if the complex numbers λ and μ lie in the same half-plane Im z > 0 or Im z Document Type: Research Article DOI: http://dx.doi.org/10.1007/s10958-014-1677-6 Affiliations: 1: Novgorod State University, 41, Bol’shaya St.-Peterburgskaya ul., Velikiy Novgorod,...

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector. Résumé Nous considérons les lois de conservation [hyperboliques] en plusieurs dimensions dʼespace avec la fonction de flux seulement continue. Nous établissons une condition nécessaire...

The concept of a renormalized entropy solution of the Cauchy problem for an inhomogeneous quasilinear equation of the first order is introduced. Existence and uniqueness theorems are proved, together with a comparison principle. Connections with generalized entropy solutions are investigated.

We study special hyperbolic systems of conservation laws, which can be written as single conservation laws on matrix algebras and include, in particular, the known systems of Keyfitz-Kranzer type. The theory of strong generalized entropy solutions of the Cauchy problem is developed.

We prove that weak limits of approximate entropy solutions to a one-dimensional degenerate parabolic equation are entropy solutions as well.

We introduce the notion of a renormalized entropy solution to the Cauchy problem for a quasilinear first order equation with an arbitrary measurable initial function. The existence and uniqueness theorems are proved. We also prove the comparison principle and study relations between renormalized and generalized entropy solutions. We also establish...

We consider a conservation law in the domain Ω ⊂ ℝn+1 with C1 boundary ∂Ω. For a wide class of functions including generalized entropy sub- and super-solutions, we prove the existence of strong traces for normal components of the entropy fluxes on ∂Ω. Non-degeneracy conditions on the flux are not required.

In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.

We study the Cauchy problem for a conservation law with space discontinuous flux of generalized Audusse–Perthame form. It is shown that, after a change of unknown function, entropy solutions in the sense of Audusse–Perthame correspond to Kruzhkov's generalized entropy solutions for the transformed equation. This observation allows to use the Kruzhk...

We introduce the notion of a periodic in spatial variables renormalized entropy solution to the Cauchy problem for a first order quasilinear conservation law. We prove the existence and uniqueness theorems and comparison principle. We also clarify relations with generalized entropy solutions. Bibliography: 19 titles. Illustrations: 1 figure.

We study the Dirichlet problem for a first order quasilinear equa- tion on a smooth manifold with boundary. Existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coecients. It is shown that generally the uniqueness fails. The non-uniqueness occurs becau...

We indicate conditions for the well-posedness of the Cauchy problem for a scalar quasilinear conservation law in the class of locally bounded functions. We construct examples showing that if these conditions are violated, then the Cauchy problem may fail to have a generalized entropy solution.

We present a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints. The main tool is the localization properties for ultra-parabolic H -measures corresponding to weakly convergent sequences. Résumé Nous pré...

For a one-dimensional conservation law with convex flux function we prove the uniqueness of a locally bounded generalized entropy solution to the Cauchy problem with an arbitrary bounded measurable initial function. Bibliography: 12 titles. Illustrations: 2 figures.

Sequences of entropy solutions of a non-degenerate first-order quasilinear equation are shown to be strongly pre-compact in the general case of a Caratheodory flux vector. Existence of the weak and entropy solution to the Cauchy problem for such an evolutionary equation is also established. The proofs are based on the general localization principle...

Under some non-degeneracy condition we show that sequences of entropy solutions of a semi-linear elliptic equation are strongly pre-compact in the general case of a Carathéodory flux vector. The proofs are based on localization principles for H-measures corresponding to sequences of measure-valued functions.

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. T...

A new symmetrizability criterion for linear matrix spaces is proposed, with applications to the theory of first order conservation laws.

Under some nondegeneracy condition, we show that sequences of entropy and approximate solutions of a semilinear ultra-parabolic equation are strongly precompact in the general case of a Caratheodory flux vector and a diffusion matrix. The proofs are based on localization principles for the parabolic H-measures corresponding to sequences of measure-...

For infinite-dimensional generalizations of the Keyfitz-Kranzer system of conservation laws in which the unknown vector ranges in an arbitrary Banach space, we single out the class of strong generalized entropy solutions of the Cauchy problem. Existence and uniqueness theorems are proved in this class.

We prove that weak limits of entropy solutions to a one-dimensional scalar conservation law with only a continuous flux function are entropy solutions as well.

We find a representation of prolonged systems in the form of an equation on a commutative matrix algebra. This representation is used to obtain a complete description of the entropies of the prolonged systems. In particular, we show that, for an essentially nonlinear equation, all such entropies are obtained by formal differentiation of “scalar” en...

We study scalar conservation laws with power-growth restriction on the flux vector. For such equations, we find correctness classes for the Cauchy problem among locally bounded generalized entropy solutions. These classes are determined by some exponents of admissible growth with respect to space variables. We give examples showing that increasing...

The well-posedness theory of the Cauchy problem for linear transport equations with only bounded measurable coefficients is presented. In the case of one spatial variable, the existence and uniqueness of generalized and renormalized solutions are established, the notion of generalized characteristics is introduced. This theory is also applied to pr...

The existence of the maximum and minimum generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation is proved in the general case of a flux vector that is merely continuous, when the uniqueness property of a generalized entropy solution does not necessarily hold. Some useful applications are presented. In particular,...

In this article it is proved that bounded sequences of measure-valued solutions of a non-degenerate first order quasilinear equation are precompact in the topology of strong convergence. The general case of flow functions which are merely continuous is considered.

Sequences of measure-valued solutions of a non-degenerate quasilinear equation of the first order are shown to be strongly precompact in the general case, when the flow functions contain independent variables and are merely continuous.

Hyperbolic systems of conservation laws with a functional-calculus operator on the right-hand side are considered in the space of second-order symmetric matrices. The entropies of such systems are described. The concept of a generalized entropy solution (g.e.s.) of the corresponding Cauchy problem is introduced, the properties of g.e.s.'s are analy...

We construct the theory of locally int egrable generalized entropy solutions (g.e.s.) of the Cauchy problem for a first order nonhomogeneous quasilinear equat ion in t he case when the flux is only cont inuous and satisfies the linear growth condit ion. We prove existence of maximal and minimal g.e.s., deduce sufficient conditions for uniqueness of...

Measure-valued solutions of the Cauchy problem are considered for a first-order quasilinear equation with only continuous flow functions. A measure-valued analogue of the maximum principle (in Lebesgue spaces) is proved. Conditions are found under which a measure-valued solution is an ordinary function. Uniqueness questions are studied. The class o...

We consider a hyperbolic system of conservation laws on the space of symmetric second-order matrices. The right-hand side of this system contains the functional calculus operator generated in the general case only by a continuous scalar function . For these systems we define and describe the set of singular entropies, introduce the concept of gener...

This paper studies isentropic solutions of quasilinear first-order equations with two independent variables and a flux function that is only continuous. The isentropic solutions are characterized by the requirement that the S. N. Kruzhkov entropy conditions hold for these solution with the equality sign. It turns out that the existence of a noncons...

We find some necessary and sufficient conditions for a plane curve to be the gradient range of a C 1-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function ϕ under which the differential equation \frac¶v¶t = j( \frac¶v¶x )\frac{{\partial v}}{{\partial t}} = \varphi \le...

The necessary and sufficient conditions for a curve to be an image of the gradient of a C1 function are identified. The graph a continuous mapping of a C1 function can be used such that the variation of the derivative of that continuous function is infinite on any interval from the real set. It was assumed that there is a measure-zero set that is c...

Conditions for the existence of non-trivial isentropic solutions of quasilinear conservation laws are found. Applications to the problem of the functional dependence between partial derivatives of a smooth function of two variables are presented. In particular, necessary conditions on a function \varphi for the equation \dfrac{\partial v}{\partial...

A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called δ(n)-shock wave, where δ(n) is nth derivative of the Dirac delta function (n=1,2,…). In this paper the case n=1 is studied in details. We introduce a definition of δ′-shock wave type solution for the system Within the framework of this defin...

A concept of a new type of singular solutions to hyperbolic systems of conservation laws is introduced. It is so-called - (n) -shock wave, where - (n) is n-th derivative of the delta function. We introduce a definition of - 0 -shock wave type solution for the system ut + ¡ f(u) ¢ x = 0; vt + ¡ f 0 (u)v ¢ x = 0; wt + ¡ f 00 (u)v 2 + f 0 (u)w ¢ x = 0...

The definition of a delta-shock wave type solution to hyperbolic systems of conservation laws is introduced. This solution is a new type of singular solutions. The Rankine-Hugoniot conditions for delta-shock are derived

It is shown that symmetrizability problems for hyperbolic equation and for the corresponding first-order system differ. The simple algebraic criterion is suggested for symmetrizability of general matrix families M: family M is reduced to symmetrical (Hermitian) form if and only if all matrices from Lie L(M) algebra have prime imaginary spectrum.