Continuous dependence of renormalized solution for convection-diffusion problems involving a nonlocal operator

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Résumé

In Ouédraogo A. et al (cf. [30]), it is provided existence and uniqueness results of L1-renormalized entropy solution for the Cauchy problem associated to the following vast class of nonlinear anisotropic degenerate parabolic-hyperbolic equations involving a nonlocal diffusion term:
\begin{equation*}\label{problem_(CP)}
\partial_{t}u+\nabla.F(u)-\displaystyle\sum_{i,j=1}^{N}\partial^{2}_{x_{i}x_{j}}A_{ij}(u)
-{\cal L}_{\mu}[u]= f(u) \hskip0.3 cm\hbox{in }\ \ Q=(0,T)\times \RR^N \ \text{with}\ T>0\ \text{and}\ N\geq 1.
\end{equation*}
Our goal is to complement this previous work with a continuous dependence result of the L1-solution with respect to the data set (F,a,μ,f, u0). The strategy is to follow the approach developed by Karlsen and Ulusoy in [28]. However, we must manage the difficulties due to the fact that we are working in the whole space RN with an only integrable initial datum u0 and the term source f depends on the unknown function u.

Mots-clés

continuous dependence on data, L1 solutions, advection-diffusion equation, nonlocal diffusion, anisotropic diffusion, Cauchy problem, whole space, nonlinear degenerate parabolic equations