[Paper Review] On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large $L_3$-initial data

G. Serëgin, Vladimír Šverák|arXiv (Cornell University)|Jan 12, 2016

Navier-Stokes equation solutions15 references3 citations

TL;DR

This paper establishes the existence of global weak solutions to the 3D Navier-Stokes equations for initial data in $L^3(\mathbb{R}^3)$, a critical space where classical methods fail. By decomposing the solution as $v = v^1 + v^2$, where $v^1$ solves the linear heat equation and $v^2$ satisfies a nonlinear correction equation, the authors prove existence via a fixed-point argument in $L^5$ and show that the solution satisfies the energy inequality, thus qualifying as a weak $L^3$-solution.

ABSTRACT

The aim of the note is to discuss different definitions of solutions to the Cauchy problem for the Navier-Stokes equations with the initial data belonging to the Lebesgue space $L_3(\mathbb R^3)$

Motivation & Objective

• To address the open problem of constructing global weak solutions for the 3D Navier-Stokes equations when initial data lies in the critical space $L^3(\mathbb{R}^3)$, which is outside the scope of classical $L^2$-based energy methods.
• To extend the theory of weak solutions beyond $L^2$ initial data by introducing a novel decomposition method that separates the linear and nonlinear contributions.
• To establish the existence of mild solutions in $L^5(Q_T)$ for small time intervals, leveraging smallness of the linear part and contraction mapping in a critical function space.
• To prove that the constructed mild solution satisfies the local energy inequality and thus qualifies as a weak $L^3$-solution globally in time.

Proposed method

• Decompose the solution as $v = v^1 + v^2$, where $v^1$ is the solution to the linear heat equation with initial data $v_0$, and $v^2$ is a correction term solving a nonlinear perturbation equation.
• Use the heat kernel $\Gamma$ and a derived kernel $K$ to express the mild solution via integral formulation involving the nonlinear term $v \otimes v$.
• Apply a fixed-point argument in the space $L^5(Q_T) \cap L^{\infty}(0,T; L^3(\mathbb{R}^3))$ to construct a mild solution for small time $T$, relying on smallness of the linear part $v^1$.
• Establish convergence of the iterative scheme $v^{(k+1)} = \mathcal{G}(v^{(k)} \otimes v^{(k)})$ in $L^5(Q_T)$ and $C([0,T]; L^3(\mathbb{R}^3))$ via contraction estimates.
• Prove that the limit solution satisfies the local energy inequality by estimating all terms involving $w = v - v^1$, including pressure and gradient terms, using cut-off functions and $L^p$-bounds.
• Use mollification of $v_0$ to control the smallness of $\kappa(T)$, ensuring the contraction condition $\kappa(T) \leq 1/(16c)$ holds for small $T$.

Experimental results

Research questions

• RQ1 Can global weak solutions to the 3D Navier-Stokes equations be constructed for initial data in $L^3(\mathbb{R}^3)$, a critical space not covered by classical $L^2$-based energy methods?
• RQ2 Does the decomposition $v = v^1 + v^2$, where $v^1$ solves the linear heat equation and $v^2$ corrects the nonlinearity, yield a viable method for constructing solutions in critical spaces?
• RQ3 Can a mild solution in $L^5(Q_T) \cap C([0,T]; L^3(\mathbb{R}^3))$ be constructed via fixed-point iteration, and does it satisfy the energy inequality?
• RQ4 Is the constructed mild solution a weak $L^3$-solution globally in time, and can uniqueness be established in this class?

Key findings

• A global weak solution exists for any divergence-free initial data $v_0 \in L^3(\mathbb{R}^3)$, extending the classical theory beyond $L^2$.
• The solution is constructed as $v = v^1 + v^2$, where $v^1$ is the linear solution and $v^2$ is a correction in the energy class on every bounded time interval.
• A mild solution exists locally in time in $L^5(Q_T) \cap C([0,T]; L^3(\mathbb{R}^3))$ via a contraction mapping argument, provided $\kappa(T) \leq 1/(16c)$.
• The constructed mild solution satisfies the local energy inequality, confirming it as a weak $L^3$-solution in $Q_T$.
• The solution is unique in the class of mild solutions in $L^5(Q_T) \cap C([0,T]; L^3(\mathbb{R}^3))$, and any weak $L^3$-solution coincides with the mild solution locally in time.
• The method is robust and extendable to unbounded domains with boundaries, unlike more complex approaches such as those in Lemarie-Rieusset (2016).