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A unified theory of existence of suitable weak solutions to the 3D incompressible Navier-Stokes equations for non-decaying initial data

A. Balakrishna, I. Kukavica, W. S. Ożański

TL;DR

This work develops a unified framework for the existence of suitable weak solutions to the 3D incompressible Navier–Stokes equations with non-decaying initial data by introducing a Morrey-type space $M^{p,q}_{\mathscr{C}}$ built from ball covers $\mathscr{C}$. It proves local-in-time existence of local energy solutions under two growth regimes for the covering family, employing a novel non-divergence-free construction in the harder regime and a reduction to Leray theory in the easier regime, yielding solutions that are suitable in the Caffarelli–Kohn–Nirenberg sense. A key contribution is a robust a priori bound in the $M$-norm and a pressure decomposition that controls nonlocal interactions, complemented by a regularization lemma and a carefully designed iterative scheme. The results extend the class of admissible initial data and provide a flexible approach to non-decaying flows in 3D NSE, with implications for partial regularity via the CK/Nirenberg framework.

Abstract

We consider any cover $\mathscr{C}$ of $\mathbb{R}^3$ by balls of radius bigger or equal $1$ satisfying two conditions: (i) any ball intersects at most $σ>0$ other balls, and (ii) intersecting balls have comparable sizes. We consider a natural Morrey-type space such that the $L^2_{\mathrm{uloc}}$ setting of Lemarié-Rieusset (Recent Developments in the Navier-Stokes Problem, 2002) and the dyadic-type space considered by Bradshaw and Kukavica (J. Math. Fluid Mech., 22(1), 2020) are particular cases. We provide a priori estimates and prove local existence of weak solutions in two cases; first, when there exists $ε>0$ such that $|B|^{1/3} \lesssim |x_B|^{1-ε}$ for all $B\in \mathscr{C}$, where $x_B$ denotes the center of~$B$, or when $|B|^{1/3} \gtrsim 1+ |x_B|$ for all $B\in\mathscr{C}$. In particular, we introduce a new non-divergence-free approach to the construction of weak solutions, which simplifies the existence proof in the $L^2_{\mathrm{uloc}}$ setting. In addition, for the dyadic setting, we do not require vanishing at the spatial infinity. The constructed solutions are suitable in the sense of Caffarelli, Kohn, and Nirenberg, thus allowing an application of the partial regularity theory.

A unified theory of existence of suitable weak solutions to the 3D incompressible Navier-Stokes equations for non-decaying initial data

TL;DR

This work develops a unified framework for the existence of suitable weak solutions to the 3D incompressible Navier–Stokes equations with non-decaying initial data by introducing a Morrey-type space built from ball covers . It proves local-in-time existence of local energy solutions under two growth regimes for the covering family, employing a novel non-divergence-free construction in the harder regime and a reduction to Leray theory in the easier regime, yielding solutions that are suitable in the Caffarelli–Kohn–Nirenberg sense. A key contribution is a robust a priori bound in the -norm and a pressure decomposition that controls nonlocal interactions, complemented by a regularization lemma and a carefully designed iterative scheme. The results extend the class of admissible initial data and provide a flexible approach to non-decaying flows in 3D NSE, with implications for partial regularity via the CK/Nirenberg framework.

Abstract

We consider any cover of by balls of radius bigger or equal satisfying two conditions: (i) any ball intersects at most other balls, and (ii) intersecting balls have comparable sizes. We consider a natural Morrey-type space such that the setting of Lemarié-Rieusset (Recent Developments in the Navier-Stokes Problem, 2002) and the dyadic-type space considered by Bradshaw and Kukavica (J. Math. Fluid Mech., 22(1), 2020) are particular cases. We provide a priori estimates and prove local existence of weak solutions in two cases; first, when there exists such that for all , where denotes the center of~, or when for all . In particular, we introduce a new non-divergence-free approach to the construction of weak solutions, which simplifies the existence proof in the setting. In addition, for the dyadic setting, we do not require vanishing at the spatial infinity. The constructed solutions are suitable in the sense of Caffarelli, Kohn, and Nirenberg, thus allowing an application of the partial regularity theory.
Paper Structure (15 sections, 14 theorems, 166 equations, 3 figures)

This paper contains 15 sections, 14 theorems, 166 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Examples of configurations covered by Definition \ref{['Cr']}
  3. Sketch of the proof of Theorem \ref{['thm_main']}
  4. Notation and preliminaries
  5. Properties of $\mathscr{C}$
  6. A priori estimates
  7. The pressure estimate
  8. Construction of solutions
  9. A linear system with divergence
  10. Decay estimates on $w= \mathrm{div} \,\widetilde{u}$
  11. The pressure estimate
  12. Estimate in $M$
  13. A regularization lemma
  14. The conclusion of the proof of Theorem \ref{['thm_main']} in the case \ref{['ass_extra']}
  15. Proof of Theorem \ref{['thm_main']} in the case \ref{['ass2']}

Key Result

Theorem 1.4

Let $u_0\in M$ be divergence-free, and suppose that $(u,p)$ is a local energy solution with initial data $u_0$, where for a sufficiently large constant $C=C(\sigma,\eta)\geq 1$. Suppose that $\alpha_t$ and $\beta_t$ are continuous on $[0,T]$. Then,

Figures (3)

  • Figure 2: A sketch of the proof of \ref{['pita1']}.
  • Figure 3: A sketch of the construction of the set $\mathscr{D} = \cup_{n\geq 1} A_n$. Note that $\mathscr{D}$ is a family of cubes for a given $B\in \sc$.
  • Figure 4: A sketch of the collection $\mathcal{E}_{B'}$. Note that $\mathcal{E}_{B'}$ is a family of cubes for a given pair of $B,B'\in \sc$, such that $B'\not \subset B^{(3)}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: weighted space $M^{p,q}$
  • Theorem 1.4: A priori estimate in $M$
  • Theorem 1.5: Local existence in $M$
  • Remark 1.6
  • Definition 2.1
  • Lemma 3.1: Separation of layers
  • proof : Proof of Lemma \ref{['lem_sepa']}
  • Corollary 3.2
  • ...and 23 more