Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L^{2}_{\rm loc} (\mathbb{R}^3)$

Dongho Chae; Joerg Wof

Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L^{2}_{\rm loc} (\mathbb{R}^3)$

Analysis of PDEs 2022-06-29 v2

Authors: Dongho Chae , Joerg Wof

Abstract

We consider the Navier-Stokes equations in $\mathbb{R}^3$ subject to the initial condition with initial velocity field in $L^{2}_{\rm loc} (\mathbb{R}^3)$ such that $\limsup_{R \to +\infty } R^{-1} \|u_{0} \|_{ L^{2}(B(R))} < +\infty$ . Our aim is to show the local existence of a weak solution, global existence of weak solution if $C=0$ and the partial regularity in the sense of Caffarelli-Kohn-Nirenberg.

Keywords

navier-stokes equations weak solutions local existence partial regularity initial data in l2 loc

Cite

@article{arxiv.2206.12115,
  title  = {Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L^{2}_{\rm loc} (\mathbb{R}^3)$},
  author = {Dongho Chae and Joerg Wof},
  journal= {arXiv preprint arXiv:2206.12115},
  year   = {2022}
}

Comments

Similar result is already obtained by Z. Bradshaw, I. Kukavica and T-P. Tsai (Indiana Univ. Math. J., vol. 7(2022), pp. 191-212

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