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Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation

Alexandru F. Radu

Abstract

We study uniqueness of mild solutions to the two--dimensional incompressible Navier-Stokes equations on the torus in borderline spatial classes. While Lorentz-space methods yield uniqueness in $C([0,T);L^{2,1}(\mathbb{T}^2))$ via real interpolation and weak $L^2$ control, extending such arguments to larger Lorentz spaces $L^{2,q}$, $1<q<2$, encounters endpoint obstructions. In this paper we prove that uniqueness in $C([0,T);L^{2,q}(\mathbb{T}^2))$ holds provided one assumes a short-time $L^\infty$ smoothing property at every restart time, namely \[ \lim_{δ\downarrow 0}\sup_{t\in(T_0,T_0+δ]}\sqrt{t-T_0}\,\|v(t)\|_{L^\infty(\mathbb{T}^2)}=0, \quad \text{for all } T_0\in[0,T). \] The proof combines the restart mild formulation, the $L^1$ bound for the periodic Oseen kernel of $e^{tΔ}\mathbb{P}\nabla\cdot$, and an explicit Beta-function computation yielding a strict $L^2$ contraction on short intervals. The smoothing assumption is natural in Kato and Koch-Tataru type critical well-posedness frameworks and clarifies how parabolic regularization can replace Lorentz endpoint structure in uniqueness arguments.

Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation

Abstract

We study uniqueness of mild solutions to the two--dimensional incompressible Navier-Stokes equations on the torus in borderline spatial classes. While Lorentz-space methods yield uniqueness in via real interpolation and weak control, extending such arguments to larger Lorentz spaces , , encounters endpoint obstructions. In this paper we prove that uniqueness in holds provided one assumes a short-time smoothing property at every restart time, namely \[ \lim_{δ\downarrow 0}\sup_{t\in(T_0,T_0+δ]}\sqrt{t-T_0}\,\|v(t)\|_{L^\infty(\mathbb{T}^2)}=0, \quad \text{for all } T_0\in[0,T). \] The proof combines the restart mild formulation, the bound for the periodic Oseen kernel of , and an explicit Beta-function computation yielding a strict contraction on short intervals. The smoothing assumption is natural in Kato and Koch-Tataru type critical well-posedness frameworks and clarifies how parabolic regularization can replace Lorentz endpoint structure in uniqueness arguments.
Paper Structure (13 sections, 6 theorems, 66 equations)

This paper contains 13 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basic functional-analytic setup
  4. Periodic mild formulation and time restart
  5. Kernel bounds for $e^{t\Delta}\mathbb P\nabla\cdot$ on $\mathbb{T}^2$
  6. Lorentz spaces and the embedding $L^{2,q}\hookrightarrow L^2$
  7. A basic $L^2$ product inequality used in the main proof
  8. Main Result
  9. Applications: consequences of the main result.
  10. Uniqueness among Koch--Tataru/Kato-type solutions
  11. Short-time stability in $L^2$
  12. Conflict of Interest Statement
  13. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

Key Result

Lemma 2.1

Let $K_{\mathrm{per}}(t,\cdot)$ be the periodic kernel for the operator $e^{t\Delta}\mathbb P\nabla\cdot$ on $\mathbb{T}^2$ as in eq:kernel-form. Then there exists a constant $C>0$ such that for all $t>0$, and

Theorems & Definitions (15)

  • Lemma 2.1: Periodic kernel bounds
  • proof : Proof sketch
  • Lemma 2.2: Embedding $L^{2,q}\hookrightarrow L^2$ for $q\le 2$
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Definition 4.1: A restart-smoothing path class
  • Corollary 4.2: Uniqueness in $C_tL_x^{2,q}$ within $\mathcal{X}_0$
  • ...and 5 more