engleski

Existence of strong traces for entropy solutions of degenerate parabolic equations

In this talk we study solutions to the degenerate parabolic equation $$ \partial_t u + \operatorname{; ; ; div}; ; ; _x f(u) = \operatorname{; ; ; div}; ; ; _x(a(u)\nabla u) \, , $$ subject to the initial condition $u(0, \cdot)=u_0$. Here the degeneracy appears as the matrix $a(\lambda)$ is only positive semi- definite, i.e.~it can be equal to zero in some directions. Moreover, the degeneracy can occur in any direction, which is the main novelty. Equations of this form often occure in modelling flows in porous media and sedimentation- consolidation processes. As a consequence of the degeneracy, solutions could be singular, so one needs to justify the meaning of the initial condition. A standard way is to show that $u_0$ is the strong trace of a solution $u$ at $t=0$. The notion of strong traces proved to be very useful in showing the uniqueness of the solution to scalar conservation laws with discontinuous flux. We prove existence of strong traces for entropy solutions to the equation above under the non- degeneracy conditions. The proof is based on the blow-up techniques, where a variant of microlocal defect functional is used and applied to the kinetic formulation of the equation above. This is joint work with Darko Mitrovi\'c.

degenerate parabolic equations ; strong traces ; defect measures

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