Generalized solution for system of quasi-linear equations

E. A. Kolpakova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2014), pp. 43-55

Résumé

We consider the Cauchy problem for the system of quasi-linear 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 Hamilton–Jacobi–Bellman 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 quasi-linear 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 quasi-linear equations coincides with the unique minimax/viscosity solution of the corresponding Cauchy problem for the Hamilton–Jacobi–Bellman 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 quasi-linear 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 Hamilton–Jacobi–Bellman 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 quasi-linear equations under consideration.

Détail

Keywords: systems of quasilinear equations, Hamilton–Jacobi–Bellman equation, minimax/viscosity solution, method of characteristics.

Bibliographie

[1] Evans L. C., Partial differential equations, American Mathematical Society, 1997, 662 pp. | MR

[2] Rozhdestvenskii B. L., Yanenko N. N., System of quasi-linear equations and their application to gas dynamics, Nauka, M., 1968, 687 pp. | MR

[3] Glimm J., “Solutions in the large for nonlinear hyperbolic systems of equations”, Comm. Pure Appl. Math., 18 (1965), 697–715 | DOI | MR | Zbl

[4] Danilov V. G., Shelkovich V. M., “Delta-shock wave type solution of hyperbolic systems of conservation laws”, Quarterly of Applied Mathematics, 63:3 (2005), 401–427 | MR

[5] Dafermos C. M., Hyperbolic conservation law in continuum physics, Springer, 2005, 708 pp. | MR | Zbl

[6] Bressan A., Colombo R. M., “Unique solutions of $2\times 2$ conservation laws with large data”, Indiana U. Math. J., 44 (1995), 677–725 | DOI | MR | Zbl

[7] Subbotin A. I., Generalized solutions of first order PDE's. The dynamical optimization perspective, Birkhauser, Boston, 1995, 312 pp. | MR | Zbl

[8] Subbotina N., Kolpakova E., “On structure of the value function”, Proceedings of the 18th IFAC World Congress, Part I, v. 18, 2011, 8040–8045 | DOI

[9] Subbotina N. N., “The method of characteristics for Hamilton–Jacobi equations and applications to dynamical optimization”, Journal of Mathematical Sciences, 135:3 (2006), 2955–3091 | DOI | MR

[10] Crandall M. G., Lions P.-L., “Viscosity solutions of Hamilton–Jacobi equations”, Trans. Amer. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[11] Oleinik O. A., “About the Cauchy problem for nonlinear equations in the class of discontinuous functions”, Doklady Akademii Nauk SSSR, 95:3 (1954), 451–454 (in Russian) | MR