[Paper Review] Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces

Boris Haspot|ArXiv.org|Mar 13, 2008

Navier-Stokes equation solutions14 references12 citations

TL;DR

This paper establishes the existence of global weak solutions for a compressible capillary fluid model with a nonlocal Korteweg-type capillarity term, derived from a diffuse interface approach. By leveraging energy estimates and compactness arguments in weighted energy spaces, the author proves global stability for isentropic and general pressure laws, including Van der Waals-type equations, under large initial data and decay conditions at infinity.

ABSTRACT

This work is devoted to the global existence of weak solution for a general isothermal model of capillary fluids derived by C. Rohde, which can be used as a phase transition model. This article is structured in the following way: first of all inspired by the result by P.-L. Lions on the Navier-Stokes compressible system we will show the global stability of weak solutions for our system with isentropic pressure and next with general pressure. Next we will consider perturbations close to a stable equilibrium as in the case of strong solutions.

Motivation & Objective

• To establish the global existence and stability of weak solutions for a compressible capillary fluid model with a nonlocal capillarity term.
• To extend the global stability theory to general pressure laws, including the Van der Waals equation, beyond isentropic assumptions.
• To handle large initial data and decay conditions at infinity, ensuring solutions remain globally defined.
• To prove compactness in energy spaces with non-standard global capillarity energy terms.
• To validate the model's applicability to phase transition phenomena via a diffuse interface approach with sharp interfaces allowed.

Proposed method

• Uses a nonlocal capillarity term $ D[\rho] = \phi * \rho - \rho $, where $ \phi \in L^\infty \cap C^1 \cap W^{1,1} $, $ \int \phi = 1 $, $ \phi \geq 0 $, and even.
• Applies energy estimates involving the free energy $ \Pi(\rho) = \rho \int_0^\rho \frac{P(z)}{z^2} dz $, ensuring convexity and consistency with thermodynamics.
• Derives a global capillarity energy term $ E_{\text{global}}[\rho] = \frac{\kappa}{4} \int \phi(x-y)(\rho(y) - \rho(x))^2 dy $, which captures nonlocal interfacial effects.
• Employs a cut-off function $ \varphi_R(x) = \varphi(x/R) $ to localize estimates and justify integration by parts in unbounded domains.
• Establishes compactness via weak convergence in $ L^2 $ and $ L^4 $ spaces, using uniform bounds on $ \sqrt{f} $ and $ f \in L^\infty(0,T;L^2) $.
• Proves convergence of $ \rho_n \to \rho $ in $ C([0,T]; L^1(B_R)) $ via localization of mass and density continuity equations with test functions.

Experimental results

Research questions

• RQ1 Can global weak solutions exist for compressible capillary fluids with a nonlocal Korteweg-type capillarity term and general pressure laws?
• RQ2 Does the nonlocal capillarity term allow for solutions with sharp interfaces while preserving global stability?
• RQ3 Can the global existence result be extended to large initial data and non-isentropic pressure laws, such as Van der Waals?
• RQ4 How can compactness be established in energy spaces when the capillarity term involves nonlocal operators and unbounded domains?
• RQ5 Is the solution stable under perturbations near equilibrium, and can convergence be proven in localized $ L^1 $-type topologies?

Key findings

• Global weak solutions exist for the NSK system with isentropic pressure $ P(\rho) = a\rho^\gamma $, $ \gamma \geq 1 $, under large initial data and decay conditions at infinity.
• The nonlocal capillarity term $ \kappa \rho \nabla(\phi * \rho - \rho) $ enables the modeling of phase transitions with continuous density variation and allows for solutions with discontinuous density gradients.
• The global capillarity energy $ E_{\text{global}}[\rho] $ is conserved in time up to a sign, and its time derivative is controlled via integration by parts and the continuity equation.
• Compactness of the sequence $ \rho_n $ is proven in $ C([0,T]; L^1(B_R)) $ for all $ R, T > 0 $, using localized mass equations and weak convergence of $ \varphi \sqrt{\rho_n} $ in $ L^2 $.
• The limit solution satisfies the momentum equation in the sense of distributions, and the convergence of $ \rho_n $ to $ \rho $ is strong in $ L^1 $ on compact sets.
• The method successfully handles the lack of regularity in the capillarity term by using cut-off functions and $ L^\infty $-type estimates on $ f = \rho_n - \rho $, ensuring vanishing of boundary terms as $ R \to \infty $.