Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

By Hans Lindblad. 2004

367 citations recorded by the source corpus.

We study the motion of a compressible 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51213d451e85399ace14201942cabadaca0f8f23

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum (2012, background): “We refer the reader to [1,9,19,21,32,34,41,42,45], and [46] for a partial list of papers on this topic.”
Geometry and a priori estimates for free boundary problems of the Euler's equation (2006, background): “H. Lindblad [ LI05 ] proved existence of solutions for the general problem.”
Finite time singularities for the free boundary incompressible Euler equations (2011, neutral): “See Christodoulou-Lindblad [12], Lindblad [23], Coutand-Shkoller [16], Shatah-Zeng [28] and Zhang-Zhang [35].”
Two Dimensional Water Waves in Holomorphic Coordinates (2014, neutral): “M. Ifrim was supported by the National Science Foundation under Grant No. 0932078 000, while she was in residence at the Mathematical Science Research Institute in Berkeley, California, during the Fall semester 2013.”
Large time existence for 3D water-waves and asymptotics (2007, background): “The condition infR2 a(ζ 0 p, ψ 0 p) > 0 is the classical Taylor sign condition proper to the water-wave equations ([48,49,29,34,35,3,12,45], among others).”
A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations (2007, methodology): “The implementation of a Nash-Moser iterative scheme is however very technical, and is only used as a last recourse to solve nonlinear evolution equations, though some recent works show that it is a useful tool (e.g. [16, 17, 14, 13, 10, 8]).”
Global wellposedness of the 3-D full water wave problem (2011, background): “We mention the following recent work on local wellposedness [1, 5, 6, 15, 18, 19, 21, 24, 34].”
On the Cauchy problem for gravity water waves (2012, background): “In this direction, the beautiful work by Christodoulou– Lindblad [20], gives a priori bounds as long as the second fundamental form of the free surface is bounded, and the first-order derivatives of the velocity are bounded.”
Well‐posedness in smooth function spaces for moving‐boundary 1‐D compressible euler equations in physical vacuum (2009, background): “ with derivative loss in linearized equations when arbitrary domains are considered, or complex analysis tools for the irrotational problem with inﬁnite depth. We refer the reader to [2], [10], [20], [22], [33], [35], [41], [42], [43], and [46] for a partial list of papers on this topic. 6 D. COUTAN”
Local well-posedness of the viscous surface wave problem without surface tension (2013, methodology): “Local well-posedness without the irrotationality assumption was proved with a modified surface formulation by Zhang-Zhang [36] and with the original formulation by Christodoulou-Lindblad [10], Lindblad [23], Coutand-Shkoller”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, methodology): “[12] D. Christodoulou and H. Lindblad.”
Well‐posedness for compressible Euler equations with physical vacuum singularity (2008, methodology): “On the other hand, the free surface boundary problem was studied in [6] in the motion of a compressible liquid with vacuum by using Nash-Moser iteration; the physical boundary was treated in a sense that the pressure vanishes on the boundary and the pressure gradient is bounded away from zero, but t”
Almost global wellposedness of the 2-D full water wave problem (2009, background): “There has been much work recently concerning water waves with additional effects such as a nonzero surface tension, a bottom, and a bounded nonzero vorticity in the fluid region [1, 6, 7, 14, 24, 26, 28, 31, 41].”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, neutral): “Without being exhaustive, we list some of the more recent references [1, 2, 4, 5, 7, 8, 15, 17, 20, 25].”