[Paper Review] Decay rates for the Viscous Incompressible MHD with and without Surface Tension

Boling Guo, Lan Zeng+1 more|arXiv (Cornell University)|Jul 25, 2018

Navier-Stokes equation solutions3 citations

TL;DR

This paper establishes global well-posedness for a 3D viscous incompressible magnetohydrodynamic (MHD) system with a free surface under periodic boundary conditions, both with and without surface tension. It proves that solutions decay exponentially when surface tension is present, and almost exponentially when absent, using a flattening coordinate transformation and energy estimates in Sobolev spaces with careful control of nonlinear and boundary terms.

ABSTRACT

In this paper, we consider a layer of a viscous incompressible electrically conducting fluid interacting with the magnetic filed in a horizontally periodic setting. The upper boundary bounded by a free boundary and below bounded by a flat rigid interface. We prove the global well-posedness of the problem for both the case with and without surface tension. Moreover, we show that the global solution decays to the equilibrium exponentially in the case with surface tension, however the global solution decays to the equilibrium at an almost exponential rate in the case without surface tension.

Motivation & Objective

• To establish global well-posedness of a 3D viscous incompressible MHD system with a free upper boundary and rigid lower boundary under horizontal periodicity.
• To analyze the long-time behavior of solutions, particularly decay rates toward equilibrium.
• To compare the influence of surface tension on the decay rate of solutions to the MHD system.
• To develop a robust energy method in flattened coordinates to handle the moving free boundary problem.

Proposed method

• A flattening coordinate transformation is applied to map the time-dependent free boundary domain to a fixed reference domain, simplifying the analysis of the moving boundary.
• The problem is reformulated in Eulerian coordinates on the fixed domain using harmonic extension of the free surface height function to define the transformation.
• A high-order energy functional is constructed to control the solution in Sobolev spaces, incorporating fluid velocity, magnetic field, pressure, and surface height.
• Nonlinear terms are estimated using Sobolev interpolation inequalities and smallness assumptions on initial data to close the energy estimates.
• Elliptic estimates and trace inequalities are applied to control boundary terms and ensure regularity of the solution in the bulk and on the boundaries.
• A differential inequality is derived for the energy functional, leading to decay estimates via Grönwall-type arguments under smallness conditions.

Experimental results

Research questions

• RQ1 What are the decay rates of solutions to the 3D viscous incompressible MHD system with a free surface under periodic boundary conditions?
• RQ2 How does the presence or absence of surface tension affect the long-time decay behavior of the MHD system?
• RQ3 Can global well-posedness be established for the MHD system with a free surface in a horizontally periodic domain?
• RQ4 What role does the harmonic extension of the free surface play in the analysis of the moving boundary problem?
• RQ5 How do nonlinear terms and boundary stresses influence the energy decay rate in the absence of surface tension?

Key findings

• Global well-posedness is established for the viscous incompressible MHD system with a free surface, both with and without surface tension, under small initial data in Sobolev spaces.
• When surface tension is present, the global solution decays to equilibrium at an exponential rate.
• In the absence of surface tension, the solution decays at an almost exponential rate, specifically like $ (1+t)^{-(4N-8)} $ for large $ t $, where $ N $ is the regularity index.
• The decay rate is quantified via a differential inequality involving the energy functional, leading to a polynomial decay estimate of order $ 4N-8 $ in the absence of surface tension.
• The energy estimates are closed under a smallness condition on the initial data, ensuring the validity of the decay rates.
• The analysis relies on a flattening transformation and careful control of nonlinear and boundary terms using Sobolev and elliptic estimates.