[Paper Review] Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

Zhengzheng Chen, Lin He+1 more|arXiv (Cornell University)|Oct 18, 2015

Navier-Stokes equation solutions24 references3 citations

TL;DR

This paper establishes the global existence and time-asymptotic behavior of smooth, non-vacuum solutions to the one-dimensional nonisothermal compressible Korteweg fluid system with large initial data, under general density- and temperature-dependent viscosity, capillarity, and heat-conductivity coefficients. Using the energy method, Kanel's technique, and maximum principle, it proves long-time existence and decay to equilibrium for initial data close to constant states, extending prior results limited to isothermal or small-data cases.

ABSTRACT

The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type has been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient $μ(ρ)=ρ^α$, the capillarity coefficient $κ(ρ)=ρ^β$, and the heat-conductivity coefficient $ ildeα(θ)=θ^λ$ for some parameters $α,β,λ\in \mathbb{R}$ is considered. Under some assumptions on $α,β$ and $λ$, we prove the global existence and time-asymptotic behavior of large solutions around constant states. The proofs are given by the elementary energy method combined with the technique developed by Y. Kanel' \cite{Y. Kanel} and the maximum principle.

Motivation & Objective

• To address the lack of global large solution results for nonisothermal compressible Korteweg fluids, which had been largely restricted to isothermal or small-data settings.
• To establish the existence of smooth, non-vacuum solutions for the Cauchy problem with large initial data in one dimension.
• To analyze the time-asymptotic behavior of solutions around constant equilibrium states under general physical coefficients.
• To extend the applicability of Korteweg-type models to nonisothermal, large-data regimes with realistic, nonlinear coefficient dependencies.

Proposed method

• Adopting the Navier-Stokes-Korteweg system with density-dependent viscosity $\mu(\rho) = \rho^\alpha$, capillarity $\kappa(\rho) = \rho^\beta$, and temperature-dependent heat conductivity $\tilde{\alpha}(\theta) = \theta^\lambda$.
• Applying the elementary energy method combined with Y. Kanel’s technique to control nonlinear terms and establish a priori estimates.
• Using the maximum principle to bound the temperature and avoid vacuum formation, ensuring non-vacuum solutions.
• Establishing a priori bounds on $L^2$ and $H^1$ norms of density, velocity, and temperature perturbations from equilibrium.
• Performing energy estimates in Sobolev spaces to control higher-order derivatives and ensure regularity.
• Employing a continuation argument to extend local solutions globally in time.

Experimental results

Research questions

• RQ1 Can global smooth, non-vacuum solutions exist for the nonisothermal compressible Korteweg system with large initial data?
• RQ2 How do general, nonlinear dependencies of viscosity, capillarity, and heat-conductivity coefficients affect the long-time behavior of solutions?
• RQ3 What conditions on the exponents $\alpha$, $\beta$, and $\lambda$ ensure global existence and decay to equilibrium?
• RQ4 Can the maximum principle and Kanel’s technique be adapted to control temperature and avoid vacuum formation in nonisothermal settings?
• RQ5 What is the time-asymptotic behavior of solutions when initial data are large but close to a constant state?

Key findings

• Global smooth solutions exist for the nonisothermal Korteweg system with large initial data in one dimension, provided the physical coefficients are $\mu(\rho) = \rho^\alpha$, $\kappa(\rho) = \rho^\beta$, and $\tilde{\alpha}(\theta) = \theta^\lambda$ under suitable assumptions on $\alpha$, $\beta$, and $\lambda$.
• The solutions remain non-vacuum for all time, with temperature uniformly bounded away from zero and infinity under smallness conditions on $\gamma-1$ and initial perturbations.
• The $L^2$ and $H^1$ norms of the perturbations from equilibrium are uniformly bounded in time, with decay estimates established via energy estimates.
• Higher-order derivatives (up to $H^3$) are controlled in $L^2$ and $L^2$-in-time norms, ensuring global regularity.
• The solution converges to the constant equilibrium state as $t \to \infty$, with decay rates implied by the energy decay structure.
• The analysis closes the priori assumptions through a smallness condition on $\gamma-1$ and initial data, enabling the use of the maximum principle to control temperature bounds.