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Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations

Thomas Y. Hou, Peicong Song

Abstract

We study the forward self-similar solutions to the $2$D hypodissipative Navier-Stokes equation with fractional diffusion $(-Δ)^α$ for $\frac{1}{2}<α<1$. We first show that for arbitrarily large $(1-2α)$-homogeneous initial data which are locally Lipschitz, there exists at least one weak solution whose profile differs from the self-similar profile of the fractional heat equation by an element of $H^α(\mathbb{R}^2)$. Moreover, when $α\in(\frac{2}{3},1)$ we show that any such weak solution is actually smooth, hence a strong solution, and satisfies certain far field decay estimates.

Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations

Abstract

We study the forward self-similar solutions to the D hypodissipative Navier-Stokes equation with fractional diffusion for . We first show that for arbitrarily large -homogeneous initial data which are locally Lipschitz, there exists at least one weak solution whose profile differs from the self-similar profile of the fractional heat equation by an element of . Moreover, when we show that any such weak solution is actually smooth, hence a strong solution, and satisfies certain far field decay estimates.
Paper Structure (14 sections, 23 theorems, 370 equations)

This paper contains 14 sections, 23 theorems, 370 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Comparison with existing literature
  4. Forward self-similar solutions
  5. Connection with nonuniqueness
  6. Strategy of the proof
  7. Preliminaries
  8. Existence of solutions in $H^\alpha(\mathbb{R}^2)$
  9. Regularity upgrade and weighted energy estimates
  10. $H^{m+\alpha}$-estimates
  11. Weighted $H^\alpha$-estimate
  12. Weighted $H^{1+\alpha}$-estimate
  13. Decay estimates of $V$
  14. Appendix: Proofs of various estimates

Key Result

Theorem 1.1

Let $\alpha\in(\frac{1}{2},1)$ and $u_0(x)\in C^{0,1}_{loc}(\mathbb{R}^2\backslash\{0\})$ be a divergence free, $(1-2\alpha)$-homogeneous data. Then, there exists a self-similar solution $u$ to fractional NS with initial data $u_0$ and profile function $U(x) = u(x,1)$, such that $U-U_0$ belongs to $ When $u_0\in C^{1,\beta}_{loc}(\mathbb{R}^2\backslash\{0\})$ for some $\beta\in(0,1]$, (ii)(High r

Theorems & Definitions (45)

  • Theorem 1.1
  • Lemma 2.1: Estimates of $U_0$
  • proof
  • Lemma 2.2: Commutator estimates
  • proof
  • Lemma 2.3: Convergence of mollification
  • proof
  • Theorem 2.1: Hytonen2012SharpWeightedBound
  • Theorem 3.1
  • Proposition 3.1: Existence of solutions on $B_R$
  • ...and 35 more