Strong traces to degenerate parabolic equations (CROSBI ID 698805)
Prilog sa skupa u zborniku | sažetak izlaganja sa skupa | međunarodna recenzija
Podaci o odgovornosti
Erceg, Marko ; Mitrović, Darko
engleski
Strong traces to 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. 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 quasi- 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. This is joint work with Darko Mitrovi\'c.
degenerate parabolic equations ; strong traces ; defect measures
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Podaci o prilogu
1-1.
2019.
objavljeno
Podaci o matičnoj publikaciji
The Sixth Najman Conference on Spectral Theory and Differential Equations-Conference: schedule & abstracts
Podaci o skupu
6th Najman Conference on Spectral Theory and Differential Equations
predavanje
08.09.2019-13.09.2019
Sveti Martin na Muri, Hrvatska