[Paper Review] Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations

Dongho Chae|ArXiv.org|Nov 28, 2008

Navier-Stokes equation solutions2 references3 citations

TL;DR

This paper establishes weighted Liouville-type theorems for the Navier-Stokes and Euler equations on $\mathbb{R}^{N}$, $N\geq 2$, and the MHD equations on $\mathbb{R}^{N}$, $N\geq 3$, proving that weak solutions vanish identically if the velocity and pressure satisfy integrability conditions with specific weight functions and non-negativity of weighted pressure. The results generalize prior work by incorporating general weights and extending to the Euler and MHD systems.

ABSTRACT

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb R^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on $\Bbb R^N imes (0, T)$. Similar results hold for the MHD Equations on $\Bbb R^N$, $N\geq3$.

Motivation & Objective

• To extend Liouville-type results for the Navier-Stokes and Euler equations on $\ mathbb{R}^{N}$, $N\geq 2$, by incorporating general weight functions in the integrability conditions.
• To establish the first such theorems for the Euler equations and the MHD equations on $\ mathbb{R}^{N}$, $N\geq 3$, under weighted integrability and pressure sign constraints.
• To unify and generalize previous results by [1] using a broader class of weight functions and weaker assumptions on pressure and velocity integrability.
• To prove that weak solutions vanish almost everywhere when the weighted $L^1$ norms of $|v|^2 + |p|$ and the weighted pressure integral are non-negative and finite.

Proposed method

• The proof uses a weighted energy estimate with radial weight functions $w(|x|)$, $\frac{1}{|x|}\int_0^{|x|} w(s)ds$, and $\frac{1}{|x|^2}\int_0^{|x|}\int_0^r w(s)dsdr$ to control the solution behavior.
• A key step involves deriving a differential inequality from the weak formulation of the equations, using test functions adapted to the weight structure.
• The non-negativity of the weighted pressure integral $\int_{\mathbb{R}^N} p(x,t)\left[w(|x|)+\frac{N-1}{|x|}\int_0^{|x|} w(s)ds\right]dx \geq 0$ is used to prevent energy accumulation.
• The method relies on Calderón-Zygmund estimates to control the pressure in terms of velocity and magnetic field $L^q$ norms for $2<q<\frac{2N}{N-1}$.
• For the MHD case, the pressure is decomposed using Riesz transforms and harmonic functions, with decay assumptions ensuring the harmonic part vanishes.
• The analysis handles two cases: integrability of $\frac{|v|^2+|p|}{1+|x|}$ or decay of pressure at infinity combined with $L^q$ integrability of velocity and magnetic field.

Experimental results

Research questions

• RQ1 Under what weighted integrability conditions on $|v|^2 + |p|$ and sign conditions on the weighted pressure does a weak solution to the Navier-Stokes equations vanish?
• RQ2 Can Liouville-type theorems be extended to the Euler equations, which lack viscosity, using weighted norms?
• RQ3 Can similar results be established for the MHD equations, which involve coupled velocity and magnetic fields?
• RQ4 How do general weight functions $w(r)$ affect the decay and integrability conditions needed to force trivial solutions?
• RQ5 What is the minimal integrability condition on $v$ and $p$ that, combined with a non-negative weighted pressure integral, implies $v=0$ a.e.?

Key findings

• If $\int_0^T \int_{\mathbb{R}^N} (|v|^2 + |p|) \left[w(|x|) + \frac{1}{|x|}\int_0^{|x|} w(s)ds + \frac{1}{|x|^2}\int_0^{|x|}\int_0^r w(s)dsdr\right] dxdt < \infty$ and $\int_{\mathbb{R}^N} p(x,t)\left[w(|x|)+\frac{N-1}{|x|}\int_0^{|x|} w(s)ds\right]dx \geq 0$, then $v=0$ a.e. on $\mathbb{R}^N \times (0,T)$.
• For the Euler equations ($\nu=0$), the same conclusion holds under the same weighted integrability and pressure sign conditions, extending results to the inviscid case.
• For the MHD equations with $\mu,\nu \geq 0$, if either $\int_0^T \int_{\mathbb{R}^N} \frac{|v|^2 + |b|^2 + |p|}{1+|x|}dxdt < \infty$ or pressure decays at infinity with $|v|+|b| \in L^2(0,T;L^q(\mathbb{R}^N))$ for $2<q<\frac{2N}{N-1}$, and the pressure sign condition holds, then $v=b=0$ a.e.
• The result holds even when the weight $w(r)$ is not integrable, provided $w(r) \leq \frac{C}{1+r}$ and $w$ is non-increasing, allowing for slower decay than $L^1$.
• The pressure term is controlled via Calderón-Zygmund estimates, showing $\|p\|_{L^{q/2}} \leq C_q(\|v\|_{L^q}^2 + \|b\|_{L^q}^2)$ for $q \in (2,\infty)$, which enables the use of $L^q$ integrability in the proof.
• The harmonic part of the pressure vanishes under the decay assumption $p(x,t) \to 0$ as $|x| \to \infty$, which is crucial for the second case in Theorem 1.2 and Theorem 3.2.