[Paper Review] Energy equality in compressible fluids with physical boundaries

Robin Ming Chen, Zhilei Liang+2 more|arXiv (Cornell University)|Aug 18, 2018

Navier-Stokes equation solutions26 references3 citations

TL;DR

This paper establishes an $L^p$-$L^q$ regularity condition on the velocity field that guarantees energy equality for weak solutions of the 3D compressible Navier–Stokes equations in a bounded domain with no-slip boundary conditions. Using a global mollification combined with boundary cut-off techniques, the authors prove energy conservation under the assumptions that density is bounded and $\sqrt{\rho} \in L^\infty_t H^1_x$, extending energy conservation results to compressible flows with physical boundaries.

ABSTRACT

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrtρ \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.

Motivation & Objective

• To establish sufficient conditions for energy equality in weak solutions of the 3D compressible Navier–Stokes equations with physical boundaries.
• To address the lack of energy equality in weak solutions, which is a critical issue in turbulent flow modeling and the Onsager conjecture framework.
• To extend existing energy conservation results to compressible flows with solid walls, where boundary effects complicate the analysis.
• To provide a regularity condition on velocity that ensures the absence of anomalous energy dissipation in viscous flows.
• To develop a rigorous analytical framework combining global mollification and boundary-localized cut-off to handle singularities near the boundary.

Proposed method

• Construct a global mollification of the velocity field combined with a boundary-localized cut-off function to isolate and control boundary effects.
• Apply a double limit procedure: first mollify the velocity and density fields, then take the boundary cut-off parameter to zero.
• Use the renormalized formulation of the continuity equation and energy inequality to control the regularized energy flux.
• Employ Hardy-type inequalities and $L^p$-$L^q$ estimates to control the convergence of nonlinear terms involving velocity and pressure.
• Decompose the energy flux into bulk and boundary-adjacent regions, proving that boundary contributions vanish in the limit via decay of cut-off gradients.
• Leverage the assumption $\sqrt{\rho} \in L^\infty_t H^1_x$ and bounded density to control pressure and its mollified versions in $L^2$-based estimates.

Experimental results

Research questions

• RQ1 Under what regularity conditions on the velocity field does energy equality hold for weak solutions of the 3D compressible Navier–Stokes equations with physical boundaries?
• RQ2 Can the energy equality be preserved in the presence of solid walls and vacuum regions, given the challenges posed by boundary layers and low regularity?
• RQ3 How can the nonlinearity and boundary effects in the momentum equation be controlled to ensure convergence of the resolved energy flux?
• RQ4 What is the minimal regularity requirement on velocity (in $L^p_t L^q_x$) to guarantee energy conservation in bounded domains?
• RQ5 Can a mollification-cut-off strategy effectively handle the singular behavior near the boundary while preserving energy balance?

Key findings

• Energy equality holds for weak solutions of the 3D compressible Navier–Stokes equations in a bounded domain if the velocity satisfies $u \in L^p_t L^q_x$ with $\frac{1}{q} + \frac{1}{p} \leq \frac{1}{2}$ and $q \geq 4$, under the assumptions $\rho \in L^\infty_t L^\infty_x$ and $\sqrt{\rho} \in L^\infty_t H^1_x$.
• The boundary contribution to the energy flux vanishes in the limit due to the decay of the gradient of the cut-off function and the Hardy inequality, ensuring no anomalous dissipation at the wall.
• The pressure term in the energy flux converges to zero in the double limit $\varepsilon \to 0$, $\delta \to 0$, due to uniform $L^2$ bounds on $P$ and the localization of mollified gradients near the boundary.
• The nonlinear convective term $\text{div}(u \otimes u)$ is controlled via $L^4$ and $L^2$ estimates on $u$ and $\nabla u$, with convergence of the mollified flux to zero under the stated $L^p$-$L^q$ condition.
• The method successfully handles the interaction between velocity regularity, density bounds, and boundary effects, generalizing previous results to the compressible, bounded case.
• The proof establishes that the energy dissipation rate vanishes in the limit, confirming the absence of interior or boundary-induced anomalous dissipation under the given regularity framework.