Well-posedness for the motion of an incompressible liquid with free surface boundary

Abstract

We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

“…In this, the second part of the paper, we use our methodology to prove wellposedness of the free-surface Euler equations with σ = 0 and the Taylor sign condition (1.2) imposed, previously established by Lindblad in [13]. The main advantages of our method over the Nash-Moser approach of [13] are the significantly shorter proof and the fact that we provide directly the optimal space in which the problem is set, instead of having to separately perform an optimal energy study once a solution is known as in [6].…”

“…Although the Laplace-Young condition (1.1c) provides improved regularity for the boundary, the required nonlinear estimates are more difficult to close due to the complexity of the mean curvature operator and the need to study time-differentiated problems which do not arise in the σ = 0 case. It appears that the use of the timedifferentiated problem in Lindblad's paper [13] is due to the use of certain tangential projection operators, but this is not necessary. We note that our energy function is different from that in [13] and provides better control of the Lagrangian coordinate.…”

“…It appears that the use of the timedifferentiated problem in Lindblad's paper [13] is due to the use of certain tangential projection operators, but this is not necessary. We note that our energy function is different from that in [13] and provides better control of the Lagrangian coordinate.…”

“…Lannes [11] provided a proof for the finite depth case with varying bottom by implementing a Nash-Moser iteration. The first well-posedness result for the full Euler equations with zero surface tension, σ = 0, is due to Lindblad [13] with the additional "physical condition" that (1.2) ∇p · n < 0 on ∂Q,…”

“…In [12], Lindblad proved well-posedness of the linearized Euler equations, but the estimates were not sufficient for well-posedness of the nonlinear problem. The estimates were improved in [13], wherein Lindblad implemented a Nash-Moser iteration to deal with the manifest loss of regularity in his linearized model and thus established the well-posedness result in the case that (1.2) holds and σ = 0. Local existence for the case of positive surface tension, σ > 0, remained open.…”