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Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations

Šárka Nečasová, Tong Tang, Emil Wiedemann, Lu Zhu

TL;DR

The paper extends Onsager-type energy-conservation results to the inviscid Primitive Equations in a bounded cylinder with slip boundaries by (i) proving local energy conservation under precise regularity in an anisotropic Hölder framework, (ii) deriving a global energy conservation via a smooth boundary-approximation and a boundary-term decay condition, and (iii) establishing a conditional vanishing-viscosity result showing absence of anomalous dissipation when uniform hypotheses hold in the viscous regime. The methods hinge on pressure regularity, vertical integration, mollification, and anisotropic commutator estimates, adapted to the geometric challenges of corners and the vertical velocity's limited information. The results provide a rigorous criterion for energy conservation in bounded geophysical flows and a pathway to control dissipation in vanishing viscosity limits, with implications for understanding boundary-layer effects in the PE system. Overall, the work delivers a first Onsager-type energy conservation theory for the Primitive Equations with physical boundaries and outlines conditions ensuring stable inviscid limits in bounded domains.

Abstract

In this paper, we consider the problem of energy conservation for weak solutions of the inviscid Primitive Equations (PE) in a bounded domain. Based on the work [Bardos et al., Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 2019, 291-310], we prove the energy conservation for PE with boundary condition under suitable Onsager-type assumptions. But due to the special structure of PE system and its domain, some new challenging difficulties arise: the lack of information about the vertical velocity, and existing corner points in the domain. We introduce some new ideas to overcome the above obstacles. As a byproduct, we give a sufficient condition for absence of anomalous energy dissipation in the vanishing viscosity limit.

Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations

TL;DR

The paper extends Onsager-type energy-conservation results to the inviscid Primitive Equations in a bounded cylinder with slip boundaries by (i) proving local energy conservation under precise regularity in an anisotropic Hölder framework, (ii) deriving a global energy conservation via a smooth boundary-approximation and a boundary-term decay condition, and (iii) establishing a conditional vanishing-viscosity result showing absence of anomalous dissipation when uniform hypotheses hold in the viscous regime. The methods hinge on pressure regularity, vertical integration, mollification, and anisotropic commutator estimates, adapted to the geometric challenges of corners and the vertical velocity's limited information. The results provide a rigorous criterion for energy conservation in bounded geophysical flows and a pathway to control dissipation in vanishing viscosity limits, with implications for understanding boundary-layer effects in the PE system. Overall, the work delivers a first Onsager-type energy conservation theory for the Primitive Equations with physical boundaries and outlines conditions ensuring stable inviscid limits in bounded domains.

Abstract

In this paper, we consider the problem of energy conservation for weak solutions of the inviscid Primitive Equations (PE) in a bounded domain. Based on the work [Bardos et al., Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 2019, 291-310], we prove the energy conservation for PE with boundary condition under suitable Onsager-type assumptions. But due to the special structure of PE system and its domain, some new challenging difficulties arise: the lack of information about the vertical velocity, and existing corner points in the domain. We introduce some new ideas to overcome the above obstacles. As a byproduct, we give a sufficient condition for absence of anomalous energy dissipation in the vanishing viscosity limit.

Paper Structure

This paper contains 8 sections, 5 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of weak solutions for PE
  4. Notation
  5. Main results
  6. Proof of Theorem \ref{['t2.1']}
  7. Proof of Theorem \ref{['t2.2']}
  8. Vanishing Viscosity Limit

Key Result

Proposition 2.1

[b2] For a scalar or tensor-valued function $h\in L^p(Q_1)$$(1\leq p\leq\infty)$ and for any $f\in C^\infty$ satisfying $f(0)=0$ and $|f(\xi)|\leq C(1+|\xi|^p)$, write $\overline{f}:=I_{2,\eta}f$. Then, if $\Psi\in\mathcal{D}(\mathbb{R}^m)$ with support in $Q_3$, Furthermore, for any multi-order derivative $D^\alpha$, Finally, for every $\sigma>0$ small enough such that one has

Figures (2)

  • Figure 1: relation of $\alpha$ and $\beta$
  • Figure 2: regularize the cylinder

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Theorem 5.1
  • ...and 1 more