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Nonuniqueness of weak solutions for the transport equation at critical space regularity

Alexey Cheskidov, Xiaoyutao Luo

TL;DR

The paper proves that for the linear transport equation with incompressible velocity in dimensions d≥3, weak- solution uniqueness in the class $L^1_t L^p$ fails in the sharp DiPerna–Lions regime when $\frac{1}{p}+\frac{1}{q}>1$, by constructing nonunique solutions via a space-time intermittent convex integration scheme.Key methodology combines space-time intermittency with oscillating stationary Mikado building blocks, enabling weak cancellation of the defect $R$ in $L^1_{t,x}$ and allowing $u$ to gain Sobolev regularity in space at the cost of time regularity.The construction defines a sequence solving the continuity-defect equation, designs perturbations $(\theta,w)$ and a refined defect $R_1$ through space-time convex integration, and establishes precise perturbation and defect estimates that close the iteration.The result sharpens the understanding of DiPerna–Lions sharpness and demonstrates that temporal intermittency is essential to reach the full nonuniqueness regime in higher dimensions.

Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

Nonuniqueness of weak solutions for the transport equation at critical space regularity

TL;DR

The paper proves that for the linear transport equation with incompressible velocity in dimensions d≥3, weak- solution uniqueness in the class $L^1_t L^p$ fails in the sharp DiPerna–Lions regime when $\frac{1}{p}+\frac{1}{q}>1$, by constructing nonunique solutions via a space-time intermittent convex integration scheme.Key methodology combines space-time intermittency with oscillating stationary Mikado building blocks, enabling weak cancellation of the defect $R$ in $L^1_{t,x}$ and allowing $u$ to gain Sobolev regularity in space at the cost of time regularity.The construction defines a sequence solving the continuity-defect equation, designs perturbations $(\theta,w)$ and a refined defect $R_1$ through space-time convex integration, and establishes precise perturbation and defect estimates that close the iteration.The result sharpens the understanding of DiPerna–Lions sharpness and demonstrates that temporal intermittency is essential to reach the full nonuniqueness regime in higher dimensions.

Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions . For divergence-free vector fields , the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class when . For such vector fields, we show that in the regime , weak solutions are not unique in the class . One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

Paper Structure

This paper contains 40 sections, 29 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Continuity-defect equation and the convex integration scheme
  4. Space-time intermittency in the convex integration
  5. Temporal intermittency via oscillating stationary solutions
  6. Organization of the paper
  7. Preliminaries
  8. Notations
  9. Antidivergence operators $\mathcal{R}$ and $\mathcal{B}$ on $\mathbb{T}^d$
  10. Improved Hölder's inequality on $\mathbb{T}^d$
  11. Mean values and oscillations
  12. The main proposition and proof of Theorem \ref{['thm:main_thm']}
  13. Time-periodic continuity-defect equation
  14. Proof of Theorem \ref{['thm:main_thm']}
  15. Stationary Mikado density and Mikado fields
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $p, q \in [1,\infty]$ and let $u \in L^1(0,T; W^{1, q} (\mathbb{T}^d))$ be a divergence-free vector field. For any $\rho_0 \in L^p(\mathbb{T}^d)$, there exists a unique renormalized solution $\rho \in C( [0,T]; L^{p}(\mathbb{T}^d)$ to eq:the_equation. Moreover, if then this solution $\rho$ is unique among all weak solutions in class $L^\infty (0,T; L^p(\mathbb{T}^d)$.

Theorems & Definitions (58)

  • Theorem 1.1: DiPerna-Lions MR1022305
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['thm:main_thm_short']}
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 48 more