A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

Zhang

2020

Nonlinearity

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.

Section: History and Backgroundsupporting

confidence: 84%

Section: Mhd System In Lagrangian Coordinates and Main Resultsmentioning

confidence: 99%

A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary

Zhang

2020

Nonlinearity

“…It is worth mentioning here that when the fluid domain is unbounded and the velocity u 0 is irrotational (i.e., curl u 0 = 0, a condition that preserved by the evolution), this problem is called the (incompressible) water wave problem, which has received a great deal of attention. The local wellposedness (LWP) for the free-boundary incompressible Euler equations in either bounded or unbounded domains have been studied in [1,4,5,6,9,10,13,24,32,34,36,39,43,44,45,46,47,52,54,55,56,59,60]. In addition, the long time well-posedness for the water wave problem with small initial data is available in [2,16,23,25,27,53,57,58,61], and there are recent results concerning the life-span for the water wave problem with vorticity [17,26,48].…”

Section: History and Backgroundmentioning

confidence: 99%

On the Motion of a Compressible Gravity Water Wave with Vorticity

2018

Ann. PDE

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in [20] to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed κ. As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of [20] to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

“…This is known to be the reference domain. Using a partition of unity, e.g., [7,26], a general domain can also be treated with the same tools we shall present. However, choosing Ω as above allows us to focus on the real issues of the problem without being distracted by the cumbersomeness of the partition of unity.…”

Section: Reformulation In Lagrangian Coordinatesmentioning

confidence: 99%

“…Later, Coutand-Shkoller [5] avoided the Nash-Moser iteration and extended the result to the case σ > 0. See also [45,31,32,33,6,7] and references therein for the related works.…”

Section: Introductionmentioning

confidence: 99%

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

Zhang

2021

Preprint

We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh-Taylor sign condition holds true initially. This result is a continuation of the authors' previous works [13,27,12]. Our proof is based on the combination of the techniques developed in our previous works [13,27,12], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.