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Uniform Boundedness of Homogeneous Incompressible Flows in $\mathbb{R}^3$

Ulisse Iotti

TL;DR

This work analyzes the global extendability of local, finite-energy solutions to the 3D incompressible Euler and Navier–Stokes equations in $\mathbb{R}^3$ for initial data $\mathbf{u}_0\in H^s(\mathbb{R}^3)$. By introducing a geometric two-component partition of the configuration space based on the orthogonality between $\mathbf{u}$ and $\nabla p$ and coupling it with a Lagrangian flow that preserves the partition, the authors reduce the problem to two decoupled subproblems on $\mathbb{R}^3\setminus\Omega_t$ and $\Omega_t$ where $\Omega_t=\{x: \bm{\omega}(x,t)=0\}$. They prove uniform-in-time bounds for $\|\mathbf{u}\|_\infty$ in the exterior region and apply a maximum principle in the interior region to obtain global-in-time regularity: $\|\mathbf{u}(\cdot,t)\|_\infty=\|\mathbf{u}_0\|_\infty$ for Euler (when $s>\frac{3}{2}+2$) and $\|\mathbf{u}(\cdot,t)\|_\infty\le \|\mathbf{u}(\cdot,\varepsilon)\|_\infty$ for NSE (when $s>\frac{3}{2}-1$). The approach hinges on the Lamb-vector representation, BKM-type criteria, and a Lagrangian perspective that reveals a pressure–energy structure conducive to uniform bounds and global extendability.

Abstract

This paper investigates the extendability of local solutions for incompressible 3D Navier-Stokes and 3D Euler problems, with initial data $\mathbf{u}_0$ in the Sobolev space $H^s (\mathbb{R}^3)$, where $s$ ensures the existence and uniqueness of classical solutions. A geometric decomposition of the configuration space, identified by the orthogonality between the solution $\mathbf{u}$ and the pressure forces $\nabla p$, splits the problem into two simpler subproblems, which enable the uniform boundedness of the solution in each component of the partition, thereby ensuring the extendability of the solution.

Uniform Boundedness of Homogeneous Incompressible Flows in $\mathbb{R}^3$

TL;DR

This work analyzes the global extendability of local, finite-energy solutions to the 3D incompressible Euler and Navier–Stokes equations in for initial data . By introducing a geometric two-component partition of the configuration space based on the orthogonality between and and coupling it with a Lagrangian flow that preserves the partition, the authors reduce the problem to two decoupled subproblems on and where . They prove uniform-in-time bounds for in the exterior region and apply a maximum principle in the interior region to obtain global-in-time regularity: for Euler (when ) and for NSE (when ). The approach hinges on the Lamb-vector representation, BKM-type criteria, and a Lagrangian perspective that reveals a pressure–energy structure conducive to uniform bounds and global extendability.

Abstract

This paper investigates the extendability of local solutions for incompressible 3D Navier-Stokes and 3D Euler problems, with initial data in the Sobolev space , where ensures the existence and uniqueness of classical solutions. A geometric decomposition of the configuration space, identified by the orthogonality between the solution and the pressure forces , splits the problem into two simpler subproblems, which enable the uniform boundedness of the solution in each component of the partition, thereby ensuring the extendability of the solution.

Paper Structure

This paper contains 13 sections, 13 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Ideas of the proof
  4. Preliminaries and known results
  5. Local existence and uniqueness theorems in the Eulerian approach
  6. Sufficient conditions for extendability
  7. Representation by the Lamb Vector
  8. Foundations of the Lagrangian approach for incompressible fluids
  9. Partition of $\mathbb{R}^3 \times [0, T^*)$ leading to problem simplification
  10. The $(\textbf{u}, \nabla p)$- orthogonality set
  11. The zero-vorticity set
  12. Uniform boundedness of solutions for the Euler problem
  13. Uniform boundedness of solutions for the Navier-Stokes problem

Key Result

Theorem 2.1

Let $\textbf{u}(x,t)$ be a solution to Euler with initial condition $\textbf{u}_0 \in H^s(\mathbb{R}^3)$ where $s > \scriptsize{\hbox{$3$} \!\mathord{\left/{\newline}\right.\nulldelimiterspace}\!\hbox{$2$}} + 2$. Then, Hence, the solution $\textbf{u}$ can be extended for all $t \geq 0$.

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 3.1: Theorem of local existence of classical solutions for incompressible Navier-Stokes problems in $H^s(\mathbb{R}^n)$ for $s > \scriptsize{\hbox{$n$} \!\mathord{\left/{\newline}\right.\nulldelimiterspace}\!\hbox{$2$}} - 1$
  • Theorem 3.2: Theorem of local existence of classical solutions for incompressible Euler problems in $H^s(\mathbb{R}^n)$ for $s > \scriptsize{\hbox{$n$} \!\mathord{\left/{\newline}\right.\nulldelimiterspace}\!\hbox{$2$}} + 1$
  • Theorem 3.3: $L^\infty$ gradient control and global existence
  • Theorem 3.4: BKM. $L^\infty$ vorticity control and global existence
  • Definition 1
  • Remark 4.1
  • Remark 4.2
  • ...and 7 more