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Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

Changxing Miao, Zhiwen Zhao

TL;DR

The paper proves sharp nonuniqueness of weak solutions to the Navier–Stokes equations in $C_t([0,T];L^{q}(\\mathbb{T}^{3}))$ for some $2<q\ll3$ by employing intermittent convex integration with $L^{q}$-normalized jets and a Navier–Stokes–Reynolds framework. The authors construct an iterative scheme that starts from the trivial solution and produces a sequence $(u_m,\mathring{R}_m)$ with decreasing Reynolds stress while prescribing an energy profile $e(t)$, yielding nonunique weak solutions starting from zero initial data. They further establish that the constructed solutions belong to $C_tW^{\alpha,q}$ for small $\alpha>0$, leveraging fractional Gagliardo–Nirenberg inequalities. The framework emphasizes the threshold behavior of $L^{q}$-normalized building blocks as $q$ approaches the critical value $3$ and provides a robust stepping stone for extending intermittent convex integration to broader classes of flows and energies.

Abstract

In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{α,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<α\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$.

Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

TL;DR

The paper proves sharp nonuniqueness of weak solutions to the Navier–Stokes equations in for some by employing intermittent convex integration with -normalized jets and a Navier–Stokes–Reynolds framework. The authors construct an iterative scheme that starts from the trivial solution and produces a sequence with decreasing Reynolds stress while prescribing an energy profile , yielding nonunique weak solutions starting from zero initial data. They further establish that the constructed solutions belong to for small , leveraging fractional Gagliardo–Nirenberg inequalities. The framework emphasizes the threshold behavior of -normalized building blocks as approaches the critical value and provides a robust stepping stone for extending intermittent convex integration to broader classes of flows and energies.

Abstract

In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with -normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in for some uniform without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of in virtue of the fractional Gagliardo-Nirenberg inequalities with some . More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that -normalized building blocks carry the threshold effect of the exponent arbitrarily close to the critical value .
Paper Structure (9 sections, 12 theorems, 149 equations)

This paper contains 9 sections, 12 theorems, 149 equations.

Table of Contents

  1. Introduction and main results
  2. Main iterative proposition
  3. Implementation for intermittent convex integration scheme
  4. Mollification
  5. $L^{q}$-normalized intermittent jets
  6. Construction on the perturbation
  7. Estimates on the perturbation
  8. Estimates on the Reynolds stress
  9. The energy iteration

Key Result

Theorem 1.2

There exists some $2<q\ll3$ such that for any given nonnegative smooth function $e(t):[0,T]\rightarrow[0,\infty)$, there is a weak solution $u\in C_{t}([0,T];L^{q}(\mathbb{T}^{3}))$ of the Navier-Stokes equation satisfying that $\int_{\mathbb{T}^{3}}|u(x,t)|^{2}dx=e(t)$ for all $t\in[0,T].$

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['pro01']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem01']}
  • Lemma 3.2
  • ...and 17 more