Convexity and the Hele–Shaw Equation

Abstract

Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the Dirichlet to Neumann operator, normal forms transformations. In this paper, we introduce several identities for the Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we study convex changes of unknowns and obtain a large class of strong Lyapunov functions; in addition to be non-increasing, these Lyapunov functions are convex functions of time. The analysis relies on a compact elliptic formulation of the Hele-Shaw equation, which is of independent interest. Then we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh-Taylor coefficient a (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function ϕ(x) = x log x and find that ϕ(1/ √ a) is a sub-solution of a well-posed equation.

Global existence in the Lipschitz class for the N-Peskin problem

“…As noted before [19,22], the Peskin problem has certain similarities with the Muskat problem (see [1, 2, 5-7, 10, 13-15, 18, 20, 21, 23, 27] and the references therein) 1) in Ω ± (t ) (1.4a)…”

Self-similar solutions for the Muskat equation

“…If the initial data is sufficiently small in 9 H 3 2 , then the slope can be arbitrarily large [27] and even unbounded [5,7]. The result [5] also shows local existence and uniqueness in H 3 2 . This is currently the best (lowest) regularity result in terms of the space of the initial data, which is a problem that has garnered a lot of attention recently (e.g.…”

“…Medium-size initial data in critical spaces but with uniformly continuous slope guarantees global wellposedness [20]. If the initial data is sufficiently small in 9 H 3 2 , then the slope can be arbitrarily large [27] and even unbounded [5,7]. The result [5] also shows local existence and uniqueness in H 3 2 .…”

“…We must note however that those works correspond to a horizontal Hele-Shaw cell (hence without gravity) and some include fluid injection. Moreover, some of these works are in a one-phase setting, which even for Muskat significantly changes the possible behaviors [2,4,15,29,34].…”

Well-posedness of an asymptotic model for free boundary Darcy flow in porous media in the critical Sobolev space

“…The next result we need is a particular commutation property which is specific to the nonlinearity of the equation (1.4) Lemma 3.2. Let s, σ ≥ 0, α ∈ [0, σ] and φ ∈ Ḣ s+ 1 4 +α , ψ ∈ Ḣ σ+ 1 4 −α then we have that Using the monotonicity property (3.2) we obtain that…”