2021/05/21 by Chao Wang, Zhifei Zhang, Weiren Zhao +1 · 9 citations
Mathematics · #Advanced Mathematical Physics Problems #Algorithm #Boundary (topology) #Euler equations #Geology #Geometry #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Sobolev space #Surface (topology) #Type (biology)
paper · pdf · doi:10.1090/memo/1318
published in Memoirs of the American Mathematical Society 270(1318) (American Mathematical Society)
openalex publication_date 2021/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript three halves plus epsilon"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^\frac 32+ε </mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.