Preprint 2004-019

The Genuinely Nonlinear Non-Isotropic Ultra-Parabolic Equation

Sergei Sazhenkov

Abstract: The work is devoted to a study of the degenerate quasilinear parabolic-hyperbolic equation $$\partial_t u+\mbox{div}_x {\mathbf a}({\mathbf x},t,u) - \mbox{div}_x(A({\mathbf x},t)\nabla_x b(u)) =0$$ such that $b(u)$ is strictly increasing in $u$, the rank of the nonnegative diffusion $d\times d$-matrix $A$ may vary in ${\mathbf x}$ and $t$, the convection coefficients ${\mathbf a}=(a_1,\ldots,a_d)$ may be non-smooth in ${\mathbf x}$ and $t$, and the genuine nonlinearity condition holds The main results of the work consist in justification that any bounded in $L^\infty$ set of entropy solutions of the equation is relatively compact in $L1_{loc}$ and that the Cauchy problem with bounded initial data has an entropy solution. The proofs are based on the Chen--Perthame-type kinetic formulation of the equation and on Panov's theorem on a version of Tartar $H$-measures.

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Sergei Sazhenkov, <>
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