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

2005/07/01 by Hans Lindblad · 329 citations
Mathematics · #Advanced Mathematical Physics Problems #Boundary (topology) #Boundary value problem #Classical mechanics #Compressibility #Euler equations #Euler's formula #Free boundary problem #Free surface #Geometry #Mathematical analysis #Mathematics #Mechanics #Motion (physics) #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Physics #Sobolev space #Surface (topology)

paper · pdf · doi:10.4007/annals.2005.162.109

published in Annals of Mathematics 162(1), 109-194 (Princeton University)

openalex publication_date 2005/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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