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On the existence of forward self-similar solutions to the two-dimensional Navier-Stokes equations

Changfeng Gui, Hao Liu, Chunjing Xie

TL;DR

This work addresses the forward self-similar solutions of the two-dimensional Navier–Stokes equations with large, non-$L^2$ initial data that are divergence-free and $-1$ homogeneous. The authors introduce an energy-perturbed framework by decomposing the solution into a caloric part $e^{t riangle}u_0$ and a finite-energy remainder, then derive global $H^1$-type energy estimates for the remainder via a crucial cancellation identity and a carefully constructed modified linear profile. They further develop weighted energy estimates to obtain pointwise decay, upgrade regularity to $H^2$ (and higher) and deduce optimal decay in the presence of additional regularity of the initial data. The existence is established through a Leray–Schauder fixed point argument in a weighted function space, yielding a global self-similar solution with explicit decay rates, demonstrating a sharp contrast to the 3D case where similar initial data can be small or require different methods. The results contribute a robust framework for forward self-similar behavior in 2D flows with singular, scale-invariant initial data and provide precise decay profiles tied to the initial data regularity.

Abstract

We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity $u_0$ that is homogeneous of degree $-1$ and locally Hölder continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the 2D problem is critical in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable. Consequently, the classical local energy estimates are not available. To overcome this, we decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part. By exploiting a kind of inherent cancellation relation between the linear part and the perturbation part, we can control interaction terms and establish the $H^1$-estimates for the perturbation part. Further investigating the corresponding Leray system in weighted Sobolev space, we can derive an optimal pointwise estimate. This gives the faster decay of the perturbation part at infinity and enables us to construct global-in-time self-similar solutions.

On the existence of forward self-similar solutions to the two-dimensional Navier-Stokes equations

TL;DR

This work addresses the forward self-similar solutions of the two-dimensional Navier–Stokes equations with large, non- initial data that are divergence-free and homogeneous. The authors introduce an energy-perturbed framework by decomposing the solution into a caloric part and a finite-energy remainder, then derive global -type energy estimates for the remainder via a crucial cancellation identity and a carefully constructed modified linear profile. They further develop weighted energy estimates to obtain pointwise decay, upgrade regularity to (and higher) and deduce optimal decay in the presence of additional regularity of the initial data. The existence is established through a Leray–Schauder fixed point argument in a weighted function space, yielding a global self-similar solution with explicit decay rates, demonstrating a sharp contrast to the 3D case where similar initial data can be small or require different methods. The results contribute a robust framework for forward self-similar behavior in 2D flows with singular, scale-invariant initial data and provide precise decay profiles tied to the initial data regularity.

Abstract

We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree and locally Hölder continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where -homogeneous vector fields are locally square-integrable, the 2D problem is critical in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable. Consequently, the classical local energy estimates are not available. To overcome this, we decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part. By exploiting a kind of inherent cancellation relation between the linear part and the perturbation part, we can control interaction terms and establish the -estimates for the perturbation part. Further investigating the corresponding Leray system in weighted Sobolev space, we can derive an optimal pointwise estimate. This gives the faster decay of the perturbation part at infinity and enables us to construct global-in-time self-similar solutions.
Paper Structure (12 sections, 16 theorems, 259 equations)

This paper contains 12 sections, 16 theorems, 259 equations.

Table of Contents

  1. Introduction and Main Results
  2. The Leray equations and the Energy estimates
  3. Self-similar variables and the Leray equations
  4. Main difficulties, key ideas and the outline for the energy estimates
  5. Proof of the energy estimates
  6. Weighted Energy Estimates and Pointwise Decay
  7. Weighted $H^1$-Estimates
  8. Weighted $H^2$ Estimates for $v_{\textnormal{re}}$
  9. Weighted $H^2$-Estimates for $\nabla v_{\textnormal{re}}$
  10. Pointwise Decay estimates for the self-similar solution
  11. Proof of the main theorem
  12. Proof of Lemma \ref{['lem:lineardecay']}

Key Result

Theorem 1.1

Assume $u_0(x)\in C^{0,\beta}_{loc}$ for some $0 < \beta\leq 1$ is a divergence-free, self-similar initial data in $\mathbb{R}^2\setminus\{0\}$, and that $\int_{\partial S^1} u_0\cdot n d\sigma = 0$. Suppose that here $S^1$ is the unit circle on $\mathbb{R}^2$. Then there is a global smooth self-similar solution $u (x,t)\in C^{\infty}(\mathbb{R}^2\times (0,\infty))$ to the Cauchy problem eq:NS-eq

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2: Estimates for $v_1$
  • proof
  • Theorem 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:energyestkey']}
  • Proposition 3.1: Weighted $H^1$ Estimate
  • ...and 22 more