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Weak and parabolic solutions of advection-diffusion equations with rough velocity field

Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa

Abstract

We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = Δu$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

Weak and parabolic solutions of advection-diffusion equations with rough velocity field

Abstract

We study the Cauchy problem for the advection-diffusion equation associated with a merely integrable divergence-free vector field defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.
Paper Structure (8 sections, 7 theorems, 31 equations, 2 figures)

This paper contains 8 sections, 7 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Distributional and parabolic solutions
  3. Distributional solutions
  4. Parabolic solutions
  5. Existence
  6. Uniqueness of solutions in the parabolic class
  7. A regularity result
  8. Further remarks and open problems

Key Result

Proposition 2.2

Let $\bm b\in L^1([0,T]; L^p(\mathbb{T}^d))$ be a divergence-free vector field and $u_0\in L^q(\mathbb{T}^d)$ for $p,q$ such that $1/p +1/q \leq 1$. Then there exists a distributional solution $u\in L^{\infty}([0,T];L^q(\mathbb{T}^d))$ to eq:ad.

Figures (2)

  • Figure 1: Visual depiction of the existence results for distributional and parabolic solutions for vector fields $\bm b \in L^1_t L^p_x$ and initial datum $u_0 \in L^q$.
  • Figure 2: Visual depiction of the uniqueness and regularity results for distributional and parabolic solutions for fields $\bm b \in L^\alpha_t L^p_x$ and initial datum $u_0 \in L^q$. Distributional solutions $u\in L^{\infty}_tL^q_x$ are well defined in the black wedge. In the blue cube parabolic solutions are unique and in the red wedge every distributional solution is parabolic.

Theorems & Definitions (20)

  • Definition 2.1: Distributional solution
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6: Commutator estimates I
  • proof
  • Theorem 2.7: Uniqueness of parabolic solutions
  • ...and 10 more