[Paper Review] Weak solution for compressible fluid models of Korteweg type

Boris Haspot|ArXiv.org|Mar 13, 2008

Navier-Stokes equation solutions8 references3 citations

TL;DR

This paper establishes the existence of global weak solutions for a compressible isothermal Korteweg fluid model in one and two spatial dimensions, avoiding the need to assume boundedness of $1/\rho$ (i.e., no vacuum control). In dimension $N=1$, it proves finite-time strong convergence in the energy space using a novel gain of derivative via fractional Laplacian estimates and compactness arguments, while in $N=2$, it requires $1/\rho \in L^\infty$ for global existence.

ABSTRACT

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) \cite{3DS}, which can be used as a phase transition model. We distinguish two cases when the dimension N=2 and N=1, in the first case we need that $\frac{1}ρ\in L^{\infty}$, when N=1 we get a weak solution in finite time in the energy space.

Motivation & Objective

• To establish global existence of weak solutions for a general isothermal Korteweg fluid model with capillarity.
• To remove the restrictive assumption $1/\rho \in L^\infty$ (i.e., vacuum control) in the existence theory, which was previously required in prior works.
• To prove finite-time strong convergence of approximate solutions in the energy space for the 1D case using refined regularity estimates.
• To develop a new approach to handle the nonlinear capillarity term $\kappa\rho\nabla\Delta\rho$ via Orlicz space and fractional Sobolev embeddings.

Proposed method

• Use of approximate solutions $(\rho_n, u_n)$ satisfying the Korteweg system (1.1) in weak form.
• Employment of energy estimates in Orlicz spaces to control $\Pi(\rho)$ and $\nabla\rho$ terms.
• Application of fractional Laplacian $\Lambda^s$ with $0 < s \leq 1$ to gain regularity on $\rho^2$, enabling control of $\partial_x\rho \otimes \partial_x\rho$.
• Use of Sobolev embedding and composition theorems to show $\rho^2 \in L^\infty(H^1)$ and $\Lambda^s\rho^2 \in L^\infty(L^\infty)$ for $s < 1/2$.
• Application of Aubin-Lions lemma and Dini's theorem to prove uniform convergence of $\partial_x\rho_n$ in $L^2$ and control of vacuum via cutoff functions.
• Use of compactness arguments on $\phi \partial_x\rho_n$ for $\phi \in C_0^\infty$ to pass to the limit in the nonlinear terms.

Experimental results

Research questions

• RQ1 Can global weak solutions exist for the Korteweg system without assuming $1/\rho \in L^\infty$?
• RQ2 What regularity gains can be achieved for $\rho$ via fractional Laplacian estimates in the 1D case?
• RQ3 How can compactness be established for $\partial_x\rho_n$ in the presence of vacuum?
• RQ4 Can the capillarity term $\kappa\rho\nabla\Delta\rho$ be handled in the distributional limit without $L^\infty$ control on $\rho$?
• RQ5 What conditions on initial data ensure strong convergence of approximate solutions in the energy space for $N=1$?

Key findings

• In dimension $N=1$, the paper proves existence of a weak solution on a finite time interval $(0,T)$ without requiring $1/\rho \in L^\infty$, using a gain of derivative via $\Lambda^s$ estimates.
• For $N=1$, the sequence $\partial_x\rho_n$ converges strongly in $L^2(\mathbb{R}\times\mathbb{R})$ to $\partial_x\rho$ under the energy space initial data assumption.
• The paper shows that $\rho \in L^\infty(H^1)$ and $\Lambda^s\rho^2 \in L^\infty(L^\infty)$ for $0 < s < 1/2$, enabling control of the nonlinear term $\int |\partial_x\rho|^2 |\Lambda^s\rho^2|$.
• For small initial data, the paper proves global existence in $\mathbb{R}\times\mathbb{R}$ with strong convergence of $\partial_x\rho_n$ to $\partial_x\rho$ in $L^2(\mathbb{R}\times\mathbb{R})$.
• The vacuum is handled via cutoff functions and Dini's theorem, showing $\|\partial_x\rho_n\|_{L^2}$ is uniformly bounded away from infinity.
• In $N=2$, global existence is established under the additional assumption $1/\rho \in L^\infty$, which is not required in the 1D case.