A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations

A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in [4], and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the out...

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγ ρ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of sy...

A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary

vj = −∂jp− ∂jφ in Ωt, where ∂i = ∂ ∂x and v = δvj and where φ is the Newtonian gravity-potential defined by (1.2) φ(t, x) = −χΩt ∗ Φ(x) on Ωt, where χΩt is a function which takes the value 1 on Ωt and the value 0 on the complement of Ωt and where Φ is the fundamental solution to the Laplacean. Thus φ satisfies ∆φ = −1 on Ωt. We can impose the condition that the fluid be incompressible by requir...

The 2D Incompressible Boussinesq Equations

A Fourth Order Scheme for Incompressible Boussinesq Equations

1 Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742. 2 Institute for Scientific Computing and Applied Mathematics and Department of Mathematics, Indiana University, Bloomington, Indiana 47405. E-mail: [email protected] 3 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109. Received February ...