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Non-unique stationary solutions of even active scalar equations

Mimi Dai, Chao Wu

Abstract

We study a class of active scalar equations with even non-local operator in the drift term. Non-trivial stationary weak solutions in the space $C^{0-}$ are constructed using the iterative convex integration approach.

Non-unique stationary solutions of even active scalar equations

Abstract

We study a class of active scalar equations with even non-local operator in the drift term. Non-trivial stationary weak solutions in the space are constructed using the iterative convex integration approach.
Paper Structure (10 sections, 4 theorems, 99 equations)

This paper contains 10 sections, 4 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Relevant previous work
  4. Organization of the paper
  5. Construction for the unforced active scalar equation
  6. Main iteration
  7. Proof of Theorem \ref{['thm']}
  8. The forced stationary equation
  9. Sum-difference formulation
  10. Heuristics for the proof of Proposition \ref{['prop-2']}

Key Result

Theorem 1.2

Let $0<\gamma<2-\alpha$ and $\alpha<1$. There exists a non-trivial weak solution $\theta$ of (S-ase) with $\Lambda^{-1}\theta\in C^{\alpha}(\mathbb T^2)$.

Theorems & Definitions (5)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 3.1