TAILIEUCHUNG - Đề tài " Well-posedness for the motion of an incompressible liquid with free surface boundary "

Annals of Mathematics 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. . | Annals of Mathematics Well-posedness for the motion of an incompressible liquid with free surface boundary By Hans Lindblad Annals of Mathematics 162 2005 109 194 Well-posedness for the motion of an incompressible liquid with free surface boundary By Hans Lindblad 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. 1. Introduction We consider Euler s equations describing the motion of a perfect incompressible fluid in vacuum dt Vk dk Vj dj p 0 j 1 . n in D div V dk Vk 0 in D where di d dx and D u 0 t T t X Dt Dt c Rra Here Vk ỗkiVi vk and we use the convention that repeated upper and lower indices are summed over. V is the velocity vector field of the fluid p is the pressure and Dt is the domain the fluid occupies at time t. We also require boundary conditions on the free boundary dD u 0 t T t X dDt p 0 on dD dt Vkdk U e T dD . Condition says that the pressure p vanishes outside the domain and condition says that the boundary moves with the velocity V of the fluid particles at the boundary. Given a domain D0 c Rra that is homeomorphic to the unit ball and initial data Vo satisfying the constraint we want to find a set D The author was supported in part by the National Science Foundation. 110 HANS LINDBLAD u 0 t T t X Dt Dt c Rra and a vector field v solving - with initial conditions x 0 x G D Do and v v0 on 0 X Do- Let N be the exterior unit normal to the free surface dDt. Christodoulou C2 conjectured that the initial value problem - is well-posed in Sobolev

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