Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation
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2026
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| author | Radu, Alexandru F. |
| author_facet | Radu, Alexandru F. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_02354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation Radu, Alexandru F. Analysis of PDEs 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. |
| title | Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.02354 |