Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form $$u_t + (-\Delta)^s u = f$$ where the nonlocal operator is the fractional Laplacian) in the setting of Sobolev spaces (as opposed to Hoelder spaces, which I hope to avoid completely if possible)? I am only interested in the well-posedness properties. Ideally, I would like an analogy of the heat equation with well-posedness in space $H^1(0,T;H^{-1}) \subset L^2(0,T;H^1)$ for the nonlocal case with a fractional diffusion. Thanks.
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
0
Basically the well-posedness is very much the same. I recommend the book by Majda and Bertozzi "Vorticity and incompressible flow". For nonlinear equations in different contexts you may check http://arxiv.org/pdf/1306.6197, http://arxiv.org/pdf/1211.5392, http://arxiv.org/pdf/0806.1180, http://arxiv.org/abs/math/0103040, http://arxiv.org/pdf/math/0608447v1.pdf, http://www.uam.es/personal_pdi/ciencias/acordoba/documentos/articulos/QuasiGeostrophicEquations.pdf and the references therein.
Well, let's assume that the domain is $\mathbb{T}^d$ and the initial data is $L^2$. Then, multiplying by $u$ and integrating by parts (using Plancherel Theorem and the multiplier for the fractional laplacian) you get the $L^2_tH^{s}_x$ bound. Now you regularize using mollifiers
$$ u_t^\epsilon=-\rho_\epsilon*((-\Delta)^s[\rho_\epsilon*u^\epsilon])=f, $$ with a mollified initial data.
Now you get the same uniform bounds. Moreover, $u_t^\epsilon$ verifies a uniform bound in $H^{-s}$. Thus you can pass to the limit by Aubin-Lions. Moreover, the solution will be $C([0,T],L^2)$, so the initial data has a meaning.
