On the double degenerate equation \(u_t - \text{div} a(x,t,u, \nabla u) = b(x,t,u,\nabla u)\)
The authors extend the local regularity theory for the saturation in the flow of two immiscible fluids in a porous medium presented in \textit{E.~DiBenedetto, U.~Gianazza} and \textit{V.~Vespri} [Indiana Univer. Math. J. 59, No.~6, 2041--2076 (2010; Zbl 1246.35110)] to a rather general double degenerate parabolic equation. More exactly, they prove the local continuity of the nonnegative solutions to NEWLINE\[NEWLINE u_t - \text{div }{\vec a}(x,t,\nabla u) = b(x,t,u,\nabla u), \quad \text{in } = \Omega \times (0,T], NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N \geq 1\), the diffusion field \({\vec a}\) and the forcing term \(b\) are real valued and measurable over \(\Omega_T\times \mathbb R\times \mathbb R^N\) and satisfy the structure conditions: NEWLINE\[NEWLINE \begin{aligned} & {\vec a}(x,t,u,\tilde u)\cdot\tilde u \geq \phi(|u|)|\tilde u|^2 - \varphi_0(x,t),\\ &|{\vec a}(x,t,u,\tilde u)| \leq \phi(|u|)|\tilde u| + \phi^{1/2}(|u|)\varphi_1(x,t),\\ & |b(x,t,u,\tilde u)|\leq \phi(|u|)|\tilde u| + \varphi_2(x,t), \end{aligned} NEWLINE\]NEWLINE being \(\phi\,:\, \mathbb R^+_0 \to \mathbb R^+_0\) a continuous function vanishing in \(u = 0\) and \(u = 1\). Here \(\varphi_i\) are non-negative functions satisfying NEWLINE\[NEWLINE \|\varphi_0 + \varphi_1^2 + \varphi_0^2\|_{\hat q,\hat r,\Omega_T} \leq C, NEWLINE\]NEWLINE where \(C\) is a given constant and \(\hat q\), \(\hat r\) are some positive linked numbers.NEWLINENEWLINEThe main result reads as follows: Every locally bounded weak solution \(u \in C(0,T; L^2_{loc}(\Omega))\cap L^2_{loc}(0,T; W^{1,2}_{loc}(\Omega))\) of the above-mentioned equation is locally continuous in \(\Omega_T\).
scientific article; zbMATH DE number 6030615
scientific article