Naslov
Existence of strong traces of degenerate parabolic equations via velocity averaging

Autori
Erceg, Marko ; Mitrović, Darko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, neobjavljeni rad, znanstveni

Skup
Croatian German meeting on analysis and Mathematical physics

Mjesto i datum
Virtualni događaj, 22.03.2021. - 25.03.2021

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
degenerate parabolic equations ; strong traces ; kinetic formulation

Sažetak
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 directions can depend on $\lambda$. %, which is the main novelty. Equations of this form often occur 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. No non-degeneracy conditions on $f$ and $a$ are required. The proof is based on the blow-up techniques and the velocity averaging result for ultra-parabolic equations, applied to the kinetic formulation. In some special situations, we can allow even for heterogeneous fluxes and diffusion matrices, i.e.~$f$ and $a$ dependent of $(t, x)$ as well.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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