[Paper Review] Well-posedness of regular solutions for 3-D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity

Qin Duan, Zhouping Xin+1 more|arXiv (Cornell University)|Jul 13, 2023

Navier-Stokes equation solutionsMathematics3 citations

TL;DR

This paper establishes the local-in-time well-posedness of regular solutions for the 3D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity, where viscosity and thermal conductivity depend on temperature as a power law. By reformulating the system in terms of density, velocity, and entropy (ρ, u, S), and introducing a singular-degenerate weighted energy framework, the authors prove that the entropy maintains uniformly high-order regularity (S − S̄ ∈ L⁶ ∩ Ḣ³) near vacuum, overcoming the challenges posed by singular time evolution and spatial dissipation in the presence of vacuum.

ABSTRACT

For the degenerate viscous and heat conductive compressible fluids, the momentum equations and the energy equation are degenerate both in the time evolution and spatial dissipation when vacuum appears, and then the physical entropy S behaves singularly, which make it challenging to study the corresponding well-posedness of regular solutions with high order regularities of S near the vacuum. In this paper, for the physically important case that the coefficients of viscosities and heat conductivity depend on the absolute temperature θin a power law of Chapman-Enskog, we identify a class of initial data admitting a local-in-time regular solution with far field vacuum to the Cauchy problem of the 3-D full CNS, and such a solution possesses the uniformly high order regularities for S near the vacuum. The key idea here is to study the vacuum problem in terms of the mass density ρ, velocity u and S instead of (ρ, u,θ), which makes it possible to compare the orders of the degeneracy of the time evolution and the spatial dissipations near the vacuum in terms of the powers of ρ. However, for heat conductive fluids, both a degenerate spatial dissipation and a source term related to riangle ρ^{γ-1}, will appear in the time evolution equation for S, which makes it formidable to study the propagation of regularities of S. Fortunately, based on some elaborate analysis of the intrinsic degenerate-singular structures of the 3-D full CNS, we can choose proper weights to control the behaviors of (ρ, u,S) by introducing an enlarged reformulated system, which includes a singular parabolic system for u, and one degenerate-singular parabolic equation for S. Then one can carry out a series of weighted energy estimates carefully designed for this reformulated system, which provides an effective propagation mechanism for S's high order regularities near the vacuum.

Motivation & Objective

• To establish the local-in-time well-posedness of regular solutions for the 3D full compressible Navier-Stokes equations with degenerate viscosities and heat conductivity.
• To address the challenge of maintaining high-order regularity of the entropy S near vacuum, where both time evolution and spatial dissipation degenerate.
• To overcome the singular behavior of entropy in the presence of vacuum by reformulating the system in terms of (ρ, u, S) instead of (ρ, u, θ).
• To develop a new weighted energy estimate framework that controls the degenerate-singular structure of the entropy equation.
• To prove that the entropy deviation S − S̄ retains uniformly high-order regularity (L⁶ ∩ Ḣ³) throughout the solution's lifespan.

Proposed method

• Reformulate the 3D full compressible Navier-Stokes equations in terms of mass density ρ, velocity u, and specific entropy S, avoiding explicit dependence on temperature θ.
• Introduce an enlarged system that includes a singular parabolic equation for u and a degenerate-singular parabolic equation for S, capturing the intrinsic degeneracy near vacuum.
• Design a novel class of weighted energy estimates tailored to the degenerate-singular structure of the system, using weights derived from ρ and its powers.
• Apply carefully constructed energy estimates to control the high-order regularity of S, particularly near vacuum, by balancing time evolution and spatial dissipation terms.
• Use the Aubin-Lions compactness lemma and Gagliardo-Nirenberg inequalities to establish convergence and regularity in the approximation process.
• Employ a vanishing artificial dissipation method to pass to the limit and obtain a solution for the original degenerate system with far-field vacuum.

Experimental results

Research questions

• RQ1 Can regular solutions to the 3D full compressible Navier-Stokes equations be well-posed when viscosities and heat conductivity degenerate with temperature and vacuum is present?
• RQ2 How can the entropy S maintain high-order regularity (e.g., in Ḣ³) when its time evolution and spatial dissipation become singular near vacuum?
• RQ3 What reformulation of the system enables effective control of the degenerate-singular behavior of the entropy equation?
• RQ4 Can a weighted energy estimate framework be constructed to handle the competing degeneracies in time evolution and spatial dissipation of S?
• RQ5 What is the role of the (ρ, u, S) formulation in preserving regularity and enabling the proof of well-posedness in the presence of vacuum?

Key findings

• The paper establishes the local-in-time existence of a regular solution (ρ, u, S) to the 3D full compressible Navier-Stokes equations with far-field vacuum.
• The solution satisfies uniformly high-order regularity for the entropy deviation: S − S̄ ∈ L⁶(R³) ∩ Ḣ³(R³) for some constant S̄, throughout its lifespan.
• The authors identify a class of initial data (ρ₀ > 0, u₀, S₀) with finite total mass and energy that admit such a solution.
• The total mass, momentum, and energy are conserved over time, as shown via integration of the equations and use of regularity assumptions.
• The solution preserves the far-field behavior (ρ, u, S) → (0, 0, S̄) as |x| → ∞ for all t ≥ 0.
• The proof relies on a novel weighted energy estimate strategy applied to a reformulated system in (ρ, u, S), which isolates and controls the degenerate-singular structure of the entropy equation.