[Paper Review] Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $ L^2_{ loc}(\Bbb R^{3})$

Dongho Chae, Joerg Wolf|arXiv (Cornell University)|Oct 5, 2016

Navier-Stokes equation solutions9 references3 citations

TL;DR

This paper establishes the existence of forward discretely self-similar (DSS) solutions to the 3D Navier-Stokes equations for initial data in $L^2_{ ext{loc}}(R^3)$, a broader class than previously considered. Using a novel local Leray solution framework with projected pressure, the authors prove global existence via a weak compactness argument and self-similarity constraints, extending prior results beyond $L^3_ ext{weak}$ or Morrey spaces.

ABSTRACT

We prove the existence of a forward discretely self-similar solutions to the Navier-Stokes equations in $ \Bbb R^{3} imes (0,+\infty)$ for a discretely self-similar initial velocity belonging to $ L^2_{ loc}(\Bbb R^{3})$.

Motivation & Objective

• To extend the existence theory of self-similar solutions to the 3D Navier-Stokes equations beyond $L^3_ ext{weak}$ and Morrey spaces.
• To establish global existence of forward discretely self-similar (DSS) solutions for initial data in $L^2_{ ext{loc}}(R^3)$, which includes non-decaying and slowly decaying profiles.
• To introduce a new notion of local Leray solution satisfying a local energy inequality with projected pressure to handle the lack of global energy balance.
• To overcome limitations of prior methods that fail in $L^2_{ ext{loc}}$ due to insufficient decay or spatial integrability.
• To prove the uniqueness and weak continuity of the solution in time in $L^2_{ ext{loc}}(R^3)$, ensuring the solution is well-defined for all $t \geq 0$.

Proposed method

• Introduce a new class of local Leray solutions satisfying a local energy inequality with a projected pressure term $E^*_G$, derived from the steady Stokes system.
• Use a weak compactness argument in $L^{2}{ ext{loc}}$ to extract a limit solution from a sequence of approximate solutions with DSS symmetry.
• Apply a transformation formula under scaling $x \mapsto \lambda^k x$, $t \mapsto \lambda^{2k}t$ to preserve DSS structure and prove time regularity of the solution set.
• Establish weak continuity in time, $u \in C_w([0,\infty); L^2_{ ext{loc}}(R^3))$, via sequential convergence and harmonic function uniqueness in weak limits.
• Leverage lower semicontinuity of the $L^2$-norm and energy estimates to control the behavior of the solution at all times.
• Use the projected pressure operator $E^*_G$ to control pressure terms in the local energy inequality, ensuring consistency with the solenoidal structure.

Experimental results

Research questions

• RQ1 Can discretely self-similar solutions to the 3D Navier-Stokes equations be constructed for initial data in $L^2_{ ext{loc}}(R^3)$, a space not requiring decay at infinity?
• RQ2 Does the absence of global energy balance in $L^2_{ ext{loc}}$ initial data preclude the existence of DSS solutions, or can a modified energy inequality ensure existence?
• RQ3 Can a new local Leray solution framework with projected pressure handle the lack of integrability and maintain self-similarity under scaling?
• RQ4 Is the solution unique and weakly continuous in time when the initial data is discretely self-similar and in $L^2_{ ext{loc}}$?
• RQ5 Can the method used for $L^3_ ext{weak}$ initial data be generalized to the larger space $L^2_{ ext{loc}}$ without requiring stronger decay or integrability?

Key findings

• The paper proves the existence of a global forward discretely self-similar solution to the 3D Navier-Stokes equations for any initial velocity $u_0 \in L^2_{ ext{loc}}(R^3)$ with $\nabla \cdot u_0 = 0$.
• The solution satisfies a local energy inequality with projected pressure and belongs to $C_w([0,\infty); L^2_{ ext{loc}}(R^3))$, ensuring weak continuity in time.
• The solution set $M(u)$ of times where the solution is weakly continuous is shown to be all of $[0,\infty)$, confirming global regularity in the weak topology.
• The method overcomes the failure of prior approaches in $L^2_{ ext{loc}}$ by introducing a projected pressure framework and using weak compactness in $L^2_{ ext{loc}}$.
• The solution satisfies the self-similarity relation $u(x,t) = \lambda^k u(\lambda^k x, \lambda^{2k}t)$ for all $\lambda > 1$ and almost every $(x,t) \in \bR^3 \times (0,\infty)$, confirming DSS structure.
• The uniqueness of the weak limit in $L^2_{ ext{loc}}$ is established via harmonic function arguments and energy estimates, ensuring consistency across time sequences.