[Paper Review] Existence of solutions for compressible fluid models of Korteweg type

Boris Haspot|ArXiv.org|Mar 13, 2008

Navier-Stokes equation solutions4 citations

TL;DR

This paper establishes the existence and uniqueness of solutions for a non-isothermal compressible fluid model of Korteweg type, incorporating capillarity and temperature-dependent coefficients. Using critical scaling spaces and advanced function spaces (Besov-type), it proves global existence for small data near equilibrium and local existence for general data, with uniqueness and regularity control via energy estimates and paradifferential calculus in anisotropic settings.

ABSTRACT

This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general data, existence and uniqueness is stated on a short time interval. In the general case with physical coefficients depending on density and on temperature, additional regularity is required to control the temperature in $L^{\infty}$ norm. We prove global existence of solution close to a stable equilibrium and local in time existence of solution with more general data. Uniqueness is also obtained.

Motivation & Objective

• To analyze the initial boundary value problem for a general non-isothermal capillary fluid model derived from Dunn and Serrin's framework.
• To address the existence and uniqueness of solutions when physical coefficients depend only on density versus when they depend on both density and temperature.
• To establish global existence for small initial data near a stable equilibrium and local existence for general data, with appropriate regularity control.
• To ensure $ L^∞ $-norm control of temperature in the general case through additional regularity assumptions.

Proposed method

• Formulation of the Korteweg model using extended thermodynamics with free energy including a gradient term $ \frac{1}{2}\kappa|\nabla\rho|^2 $.
• Derivation of the full system: mass, momentum, and energy conservation with Korteweg tensor, viscous stress, and interstitial work.
• Employment of critical scaling spaces and anisotropic Besov-type function spaces $ \widetilde{L}^\rho_T(B^{s}_{p}) $ to handle low regularity and scaling invariance.
• Application of paradifferential calculus and commutator estimates to control nonlinear terms in the equations.
• Use of energy estimates and fixed-point arguments in function spaces to prove existence and uniqueness.
• Introduction of a modified energy functional and $ L^\infty $-control of temperature via additional regularity assumptions in the general case.

Experimental results

Research questions

• RQ1 Under what conditions does the initial boundary value problem for a non-isothermal Korteweg-type fluid model admit global solutions?
• RQ2 How does the dependence of physical coefficients (viscosity, capillarity, thermal conductivity) on both density and temperature affect the existence and regularity of solutions?
• RQ3 Can global existence be established for small initial data near a stable equilibrium in critical regularity spaces?
• RQ4 What additional regularity is required to control the temperature in $ L^\infty $-norm in the general case with temperature-dependent coefficients?
• RQ5 How do the nonlinearities in the momentum and energy equations affect the well-posedness of the system in anisotropic function spaces?

Key findings

• Global existence and uniqueness of solutions are established in critical scaling spaces for data close to a stable equilibrium when coefficients depend only on density.
• Local existence and uniqueness are proven for general initial data in the same case, under the same functional framework.
• In the general case with temperature-dependent coefficients, global existence is shown near equilibrium, requiring additional regularity to control the temperature in $ L^\infty $-norm.
• Local existence with general data is obtained in the general case, again with uniqueness and $ L^\infty $-control of temperature ensured by higher regularity.
• The proof relies on sharp estimates in anisotropic Besov spaces and paradifferential calculus, with bounds depending on the $ \widetilde{L}^\rho_T(B^{s}_{p}) $-norms of the velocity and density gradients.
• The framework allows for a unified treatment of compressible fluids with capillarity and non-isothermal effects, extending previous results to more general physical coefficients.