Archive of Issues
Russia Yekaterinburg
Section  Mathematics 
Title  Generalized solution for system of quasilinear equations 
Author(s)  Kolpakova E.A.^{a} 
Affiliations  Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{a} 
Abstract  We consider the Cauchy problem for the system of quasilinear first order equations of a special form. The system is symmetric, the state variable is $n$dimensional. The considered Cauchy problem is deduced from the Cauchy problem for the HamiltonJacobiBellman equation by means of the operation of differentiation of this equation and the boundary condition with respect to the variable $x_i$. It is assumed that the Hamiltonian and the initial condition are continuously differentiable functions. The Hamiltonian is convex with respect to the adjoint variable. The paper presents a new approach to the definition of the generalized solution of the system of quasilinear first order equations. The generalized solution belongs to the class of multivalued functions with convex compact values. We prove the existence, uniqueness and stability theorems. The semigroup property for the proposed generalized solution is obtained. It is shown that the potential for generalized solutions of quasilinear equations coincides with the unique minimax/viscosity solution of the corresponding Cauchy problem for the HamiltonJacobiBellman equation, and at the points of differentiability of the minimax solution its gradient coincides with the generalized solution of the Cauchy problem. Properties of the generalized solutions of the Cauchy problem for a system of quasilinear equations are obtained on the basis of this connection. In particular, it is shown that the introduced generalized solution coincides with the superdifferential of the minimax solution of the Cauchy problem and is singlevalued almost everywhere. The structure of the set of points at which the minimax solution is not differentiable is described by using the characteristics of the HamiltonJacobiBellman equation. It is shown that the property of the generalized solution of the quasilinear equation with a scalar state variable proposed by O.A. Oleinik, can be extended to the case of the system of quasilinear equations under consideration. 
Keywords  systems of quasilinear equations, HamiltonJacobiBellman equation, minimax/viscosity solution, method of characteristics 
UDC  517.956.3 
MSC  35L40, 35D35 
DOI  10.20537/vm140203 
Received  13 March 2014 
Language  Russian 
Citation  Kolpakova E.A. Generalized solution for system of quasilinear equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 4355. 
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