Large time decay estimates for the Muskat equation

Abstract

Abstract. We prove time decay of solutions to the Muskat equation in 2D and in 3D. In [11] and [12], the authors introduce the normsin order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data f 0 ∈ H l (R 2 ) for some l ≥ 3 such that f 0 1 < k 0 for a constant k 0 ≈ 1/5, we prove uniform in time bounds of f s(t) for −d < s < l − 1 and assuming f 0 ν < ∞ we prove time decay estimates of the form f s(t) (1 + t) −s+ν for 0 ≤ s ≤ l − 1 and −d ≤ ν < 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.

On the Muskat problem with viscosity jump: Global in time results

“…Furthermore, we show uniform bounds of the interface in L ∞ and L 2 norms with analytic weights. Then, we show optimal decay rates for the analyticity of the critical solutions, improving the results in [PS17].…”

“…where k 0,d is an explicit constant, k 0,3 > 1/5 in 3D and k 0,2 > 1/3 in 2D. In [PS17], the optimal time decay of those solutions are proven, for initial data additionally bounded in subcritical Sobolev norms. We also point out work [Cam17], where the Lipschitz solutions given in [CCFG13] are shown to become smooth by using a conditional regularity result given in [CGSV17] together with an instant generation of a modulus of continuity.…”

Global existence and decay to equilibrium for some crystal surface models

“…Our approach is very adaptable and can lead to advances in other systems of PDE. For instance, it has been used Bruell & Granero-Belinchón to study the evolution of thin films in Darcy and Stokes flows [3] by Córdoba and Gancedo [6], Constantin, Córdoba, Gancedo, Rodriguez-Piazza, & Strain [5] for the Muskat problem (see also [7] and [17]), by Burczak & Granero-Belinchón [4] to analyze the Keller-Segel system of PDE with diffusion given by a nonlocal operator and by Bae, Granero-Belinchón & Lazar [2] to prove several global existence results (with infinite L p energy) for nonlocal transport equations.…”

Well-Posedness and Stability Results for Some Periodic Muskat Problems

“…However, many of the mathematical studies on this topic are quite recent and they cover various physical scenarios and mathematical aspects related to the original model proposed in [52], cf. [6,7,9,[11][12][13][14][15][16][17][18][21][22][23][24][25]27,30,32,36,[38][39][40][41][42][43]48,49,49,53,54,58,[60][61][62] (see also [55,56] for some recent research on the compressible analogue of the Muskat problem, the so-called Verigin problem).…”