We prove time decay of solutions to the Muskat equation in 2D and in 3D. In \cite{ JEMS} and \cite{ CCGRPS}, the authors introduce the norms $\|f\|_{ s}(t)= \int_{ \mathbb{ R}^{ 2}} |\xi|^{ s}|\hat{ f}(\xi)| \ d\xi$ in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data $f_{ 0}\in H^{ l}(\mathbb{ R}^{ 2})$ for some $l\geq 3$ such that $\|f_{ 0}\|_{ 1} < k_{ 0}$ for a constant $k_{ 0} \approx 1/5$, we prove uniform in time bounds of $\|f\|_{ s}(t)$ for $-d < s < l-1$ and assuming $\|f_{ 0}\|_{ \nu} < \infty$ we prove time decay estimates of the form $\|f\|_{ s}(t) \lesssim (1+t)^{ -s+\nu}$ for $0 \leq s \leq l-1$ and $-d \leq \nu < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.
Categories: math.AP