[Paper Review] Liouville-type theorems for the planer stationary MHD equations with growth at infinity

Wendong Wang|arXiv (Cornell University)|Mar 14, 2019

Navier-Stokes equation solutions18 references4 citations

TL;DR

This paper establishes Liouville-type theorems for the 2D stationary incompressible MHD equations under growth conditions on velocity and smallness assumptions on the magnetic field. By introducing a decay condition on the magnetic field and using local energy estimates in annular domains, the authors prove that under these conditions, the velocity and pressure are constant and the magnetic field vanishes identically, extending classical Liouville results to the MHD setting with nontrivial growth behavior.

ABSTRACT

For the two dimensional stationary MHD equations, we proved that Liouville type theorems hold if the velocity is growing at infinity, where the magnetic field is assumed to be bounded under a smallness condition. The key point is to overcome the nonlinear terms, since no maximum principle holds for the MHD case with respect to the Navier-Stokes equations.

Motivation & Objective

• To extend Liouville-type theorems from the Navier-Stokes equations to the 2D stationary MHD equations under velocity growth conditions.
• To address the challenge of nonlinear coupling in MHD, where the maximum principle does not apply as in the Navier-Stokes case.
• To establish conditions under which solutions to the MHD system must be trivial (constant velocity, zero magnetic field) despite velocity growth at infinity.
• To generalize prior results on integrability and decay conditions to the MHD framework, particularly for $ \nabla u \in L^q(\mathbb{R}^2) $ with $ 1 < q < \infty $.

Proposed method

• Introduces a decay condition on the magnetic field: $ |b(x)| \leq c_0(1+|x|)^\beta $ with $ \beta < 0 $, to control nonlinear terms in energy estimates.
• Uses local energy estimates on annular domains to manage the growth of velocity and the non-vanishing nonlinearities from $ b \cdot \nabla w $ and $ H $.
• Applies weighted Sobolev and interpolation inequalities to control the $ L^p $ norms of vorticity and magnetic field strength.
• Employs a cutoff function $ \phi $ in the energy estimate framework to localize the analysis and derive decay estimates.
• Relies on the Gagliardo-Nirenberg inequality and Lemma 1.4 to derive pointwise decay of velocity from $ \nabla u \in L^{q_0}(\mathbb{R}^2) $.
• Imposes a smallness condition on $ \|b\|_{L^1} + \||h|^{1/3}\|_{L^1} \leq \varepsilon_0 $ to control nonlinear terms and ensure convergence to zero in the limit.

Experimental results

Research questions

• RQ1 Under what conditions does the 2D stationary MHD system admit only trivial solutions (constant velocity, zero magnetic field) when the velocity grows at infinity?
• RQ2 Can Liouville-type theorems for the Navier-Stokes equations be extended to the MHD case when the velocity grows sublinearly and the magnetic field is small?
• RQ3 How can the lack of a maximum principle in MHD be overcome when analyzing solutions with growing velocity and bounded magnetic field?
• RQ4 What smallness and decay conditions on the magnetic field are sufficient to ensure that the vorticity and magnetic field strength vanish at infinity?
• RQ5 Is the condition $ \nabla u \in L^q(\mathbb{R}^2) $ for $ 1 < q < \infty $ sharp for Liouville-type results in the MHD setting?

Key findings

• If the velocity satisfies $ |u(x)| \leq c_0(1+|x|)^\alpha $ with $ \alpha < \frac{1}{3} $ and the magnetic field satisfies $ |b(x)| \leq c_0(1+|x|)^\beta $ with $ \beta < -\alpha $, then $ u $, $ \pi $ are constant and $ b \equiv 0 $, provided $ \|b\|_{L^1} + \||h|^{1/3}\|_{L^1} \leq \varepsilon_0 $ for sufficiently small $ \varepsilon_0 $.
• The result generalizes the classical Liouville theorem for the 2D Navier-Stokes equations to the MHD case, showing that $ \nabla u \in L^q(\mathbb{R}^2) $ for $ 1 < q < \infty $ implies $ u $ and $ \pi $ are constant, which is sharp as the result fails for $ \nabla u \in L^\infty(\mathbb{R}^2) $.
• The smallness condition on $ \|b\|_{L^1} + \||h|^{1/3}\|_{L^1} $ is essential to control the nonlinear terms $ b \cdot \nabla w $ and $ H $, which otherwise prevent the derivation of decay in the vorticity and magnetic field strength.
• The proof establishes that $ \nabla(|w|^{q-1}) \equiv 0 $ and $ \nabla(|h|^{q-1}) \equiv 0 $ in the limit as $ R \to \infty $, implying $ w \equiv C $, $ h \equiv C $, and since $ C = 0 $ by Proposition 2.1, the solutions are trivial.
• For $ \nabla u \in L^{q_0}(\mathbb{R}^2) $ with $ 2 < q_0 < \infty $, the magnetic field and velocity gradients are shown to be in $ L^p(\mathbb{R}^2) $ for all $ p \geq q_0 $, under the same smallness condition.
• The result is sharp in the sense that counterexamples exist when $ \nabla u \in L^\infty(\mathbb{R}^2) $, such as the Couette flow, which violates the Liouville conclusion.