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Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain

Siran Li

Abstract

We establish the inviscid limit of the Yudovich solution to the heat conductive Boussinesq equation with initial velocity and temperature/buoyancy in $L^2$ and initial vorticity in $L^\infty$ on the two-dimensional periodic domain ${\bf T}^2$. Given any finite time $T>0$ and $p \in [1,\infty[$, we show that the solution to the diffusive Boussinesq equation converges in $L^\infty(0,T; W^{1,p}({\bf T}^2))$ to the solution to the Euler--Boussinesq equation as the viscosity tends to zero, provided that the initial vorticity, velocity, and temperature/buoyancy converge strongly in $L^2$. Our proof adapts and extends the arguments in [P. Constantin, T. D. Drivas, and T. M. Elgindi, Comm. Pure Appl. Math. 75 (2022), 60--82] to forcing terms in $L^1(0,T; L^\infty({\bf T}^2))$.

Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain

Abstract

We establish the inviscid limit of the Yudovich solution to the heat conductive Boussinesq equation with initial velocity and temperature/buoyancy in and initial vorticity in on the two-dimensional periodic domain . Given any finite time and , we show that the solution to the diffusive Boussinesq equation converges in to the solution to the Euler--Boussinesq equation as the viscosity tends to zero, provided that the initial vorticity, velocity, and temperature/buoyancy converge strongly in . Our proof adapts and extends the arguments in [P. Constantin, T. D. Drivas, and T. M. Elgindi, Comm. Pure Appl. Math. 75 (2022), 60--82] to forcing terms in .
Paper Structure (7 sections, 8 theorems, 79 equations)

This paper contains 7 sections, 8 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two key estimates
  4. Quantified loss of $L^p$-regularity for scalars transported by Yudovich velocity
  5. Propagation of smallness for $u^\nu-u$
  6. Conclusion
  7. Comments

Key Result

Theorem 1

Fix any $T>0$. Consider the Euler--Boussinesq Equation eq: bk0 in $[0,T]\times{\mathbf{T}^2}$ with initial data $(u_0,\theta_0) \in L^2 \times L^2$ and $\omega_0 \in L^\infty$. There exists a unique solution $(u,\theta)$ such that

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • ...and 3 more