Séance Séminaire

Séminaire ACSIOM

Optimal continuous dependence estimates for fractional degenerate parabolic equations

This talk will be concerned with the Cauchy problem $$ u_t+\mbox{div} \, f(u)+(-\triangle)^{\frac{\alpha}{2}} \varphi(u)=0, \quad u(0)=u_0, $$ where $\alpha \in (0,2)$ and $\varphi$ is a nondecreasing nonlinearity. It will focus on continuous dependence estimates on the data; i.e. given another solution $$ v_t+\mbox{div} \, g(v)+(-\triangle)^{\frac{\beta}{2}} \psi(u)=0, \quad v(0)=v_0, $$ we will see how $u-v$ can be bounded by differences between $(f,\alpha,\varphi,u_0)$ and $(g,\beta,\psi,v_0)$. For instance, if $\alpha=2$ and if $u$ and $v$ have the same data, excepted $\psi \neq \varphi$, B. Cockburn and G. Gripenberg have shown in 1999 that $$ \|u(\cdot,t)-v(\cdot,t)\|_{L^1} = \mathcal{O} \left(\|\sqrt{\varphi'}-\sqrt{\psi'}\|_\infty \right). \quad \quad \quad \quad \quad \quad (1) $$ In the fractional case, we shall see that $$ \|u(\cdot,t)-v(\cdot,t)\|_{L^1} = \mathcal{O} \left( \|(\varphi')^{\frac{1}{\alpha}}-(\psi')^{\frac{1}{\alpha}}\|_\infty \right), \quad {\alpha>1,} $$ $$ \|u(\cdot,t)-v(\cdot,t)\|_{L^1} =\mathcal{O} \left( \|\varphi' \, \ln \varphi'-\psi' \, \ln \psi'\|_\infty \right), \quad {\alpha=1,} $$ $$ \|u(\cdot,t)-v(\cdot,t)\|_{L^1} =\mathcal{O} \left( \|\varphi'-\psi'\|_\infty\right), \quad {\alpha<1}, $$ giving in particular a new proof of Eq. (1) as $\alpha \to 2$. In the case where $\psi = \varphi$ but $\beta \neq \alpha$, we shall see that $$ \|u(\cdot,t)-v(\cdot,t)\|_{L^1} = \mathcal{O} (|\alpha-\beta|). $$ All these results are optimal. Joint work with Simone Cifani and Espen R. Jakobsen (Norwegian University of Science and Technology, Trondheim, Norway)