[Paper Review] Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces

Tobias Barker|arXiv (Cornell University)|Mar 20, 2017

Navier-Stokes equation solutions26 references3 citations

TL;DR

This paper establishes the existence of global solutions to the 3D Navier-Stokes equations with initial data in the critical Besov space $\dot{B}^{-1/4}_{4,\infty}$, proving weak* stability under convergence of initial data. It extends the notion of global $L_3$ solutions to a broader critical space, introducing a novel decomposition of homogeneous Besov spaces to overcome limitations in interpolation theory, and proves local existence for initial data in $\cdot{B}^{-1+3/p}_{p,\infty}$ for $p>4$, with applications to regularity criteria.

ABSTRACT

In 2016, Seregin and uSverák, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$ solutions'. A key feature of global $L_3$ solutions is continuity with respect to weak convergence of a sequence of solenoidal $L_3$ initial data. The first aim of this paper is to show that a similar notion of ' global $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ solutions' exists for solenoidal initial data in the wider critical space $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ and satisfies certain continuity properties with respect to weak* convergence of a sequence of solenoidal $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ initial data. This is the widest such critical space if one requires the solution to the Navier-Stokes equations minus the caloric extension of the initial data to be in the global energy class. For the case of initial values in the wider class of $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$ initial data ($p>4)$, we prove that for any $0

Motivation & Objective

• To extend the notion of global $L_3$ solutions to a wider critical space, specifically $\cdot{B}^{-1/4}_{4,\infty}$, preserving weak* stability under initial data convergence.
• To establish local existence for initial data in $\cdot{B}^{-1+3/p}_{p,\infty}$ with $p>4$, a broader class than $L_3$.
• To develop a new decomposition technique for homogeneous Besov spaces $\cdot{B}^{-1+3/p}_{p,\infty}$ that does not follow from standard real interpolation theory.
• To derive a new regularity criterion for 3D weak Leray-Hopf solutions based on the norm $\cdot{B}^{-1+3/p}_{p,\infty}$ of the velocity field.
• To provide a framework for analyzing potential blow-up scenarios via rescaling and ancient solution construction, leveraging weak* stability.

Proposed method

• Introduces the concept of 'global $\cdot{B}^{-1/4}_{4,\infty}$ solutions' as a generalization of global $L_3$ solutions, ensuring continuity under weak* convergence of initial data.
• Employs a dyadic decomposition of the initial data $u_0 = \bar{u}_0^N + \tilde{u}_0^N$ into a low-frequency part in $\dot{B}^{s_{p_2}+\delta_2}_{p_2,p_2}$ and a high-frequency part in $L_2$, with controlled norms.
• Applies Proposition 6.1 to construct a mild solution $W$ for the low-frequency part $\bar{u}_0^N$ in $\dot{B}^{s_{p_2}+\delta_2}_{p_2,p_2}$, satisfying a time-weighted $L_{p_2}$ estimate.
• Uses Proposition 6.2 to construct a solution $v = W + u$ for the full initial data, where $u$ solves a perturbed Navier-Stokes system with initial data $\tilde{u}_0^N$ in $L_2$, satisfying an energy inequality.
• Establishes weak* stability by showing that if $u_0^{(n)} \rightharpoonup^* u_0$ in $\dot{B}^{-1/4}_{4,\infty}$, then the corresponding solutions $u^{(n)}$ converge up to a subsequence in the sense of distributions.
• Relies on a novel decomposition of homogeneous Besov spaces $\dot{B}^{-1+3/p}_{p,\infty}$ that is not derivable from classical real interpolation, enabling the extension to wider critical spaces.

Experimental results

Research questions

• RQ1 Can the notion of global $L_3$ solutions with weak* stability be extended to the critical Besov space $\dot{B}^{-1/4}_{4,\infty}$?
• RQ2 Does there exist a global solution to the Navier-Stokes system for initial data in $\dot{B}^{-1+3/p}_{p,\infty}$ with $p>4$, and does it satisfy weak* stability?
• RQ3 Can a new regularity criterion for 3D weak Leray-Hopf solutions be derived based on the $\dot{B}^{-1+3/p}_{p,\infty}$ norm of the velocity field?
• RQ4 Is there a decomposition of $\dot{B}^{-1+3/p}_{p,\infty}$ that avoids standard real interpolation and enables solution construction in broader critical spaces?
• RQ5 How can weak* stability be leveraged to analyze potential blow-up scenarios via rescaling and ancient solution theory?

Key findings

• The paper proves the existence of global solutions in $\dot{B}^{-1/4}_{4,\infty}$ for solenoidal initial data, with continuity under weak* convergence of initial data.
• For any $0<T<\infty$, a solution exists on $\mathbb{R}^3 \times ]0,T[$ for initial data in $\dot{B}^{-1+3/p}_{p,\infty}$ with $p>4$, via a decomposition into $\dot{B}^{s_{p_2}+\delta_2}_{p_2,p_2}$ and $L_2$ components.
• A novel decomposition of homogeneous Besov spaces $\dot{B}^{-1+3/p}_{p,\infty}$ is constructed that does not follow from real interpolation, enabling the extension to wider critical spaces.
• The solution $v = W + u$ satisfies a time-weighted $L_{p_2}$ estimate: $\sup_{0<t<T} t^{\frac{s_{p_2}}{2} - \frac{\delta_2}{2}} \|w(\cdot,t)\|_{L_{p_2}} \leq 2M^{(0)}$, with $M^{(0)} = \sup_{0<t<T} t^{\frac{s_{p_2}}{2} - \frac{\delta_2}{2}} \|S(t)u_0\|_{L_{p_2}}$.
• The solution $u$ satisfies the energy inequality: $\|u(\cdot,t)\|_{L_2}^2 + 2\int_0^t \|\nabla u(\cdot,t')\|_{L_2}^2 dt' \leq 2\int_0^t \int (W \otimes u) : \nabla u \, dx dt' + \|V_0\|_{L_2}^2$, ensuring stability.
• The local energy inequality holds for $v$ and $q$, confirming the solution is a weak Leray-Hopf solution in the sense of distributions.