Global existence of weak solutions for the Maxwell--Stefan system in the whole space

Technical Summary: Global Existence of Weak Solutions for the Maxwell–Stefan System in the Whole Space

Problem Statement
This paper addresses the mathematical analysis of the isothermal Maxwell–Stefan system, a strongly coupled cross-diffusion model describing multicomponent mass transport in fluid mixtures. The system is defined on the whole space $\mathbb{R}^3$ rather than a bounded domain. The governing equations consist of mass balance laws $\partial_t c_i + \text{div } J_i = 0$ coupled with a linear algebraic system relating species fluxes $J_i$ to chemical potential gradients $\nabla \mu_i$, subject to the constraints $\sum c_i = 1$ and $\sum J_i = 0$.

The primary mathematical challenge in the whole-space setting is that concentrations $c_i$ generally do not possess finite mass (i.e., $\int_{\mathbb{R}^3} c_i \, dx = \infty$) if they approach a strictly positive constant equilibrium state at spatial infinity. Consequently, the standard Boltzmann/mixing entropy functional, which serves as a Lyapunov function in bounded domains, becomes infinite and is inapplicable. Furthermore, the lack of compact embedding for $H^1(\mathbb{R}^3)$ into $L^2(\mathbb{R}^3)$ prevents the direct application of global compactness arguments used in bounded domain theories.

Methodology
The proof of global existence follows an approximation strategy based on the exhaustion of $\mathbb{R}^3$ by a sequence of balls $B_R$ with radius $R$, combined with a relative entropy framework.

1. Relative Entropy Formulation: Instead of the standard entropy, the author utilizes a relative entropy functional $H[c|\bar{c}]$ with respect to a fixed, strictly positive constant equilibrium state $\bar{c}$. The relative entropy density differs from the standard entropy only by an affine term in the concentrations. Crucially, this modification preserves the Maxwell–Stefan flux relations because the gradients of the chemical potentials remain unchanged ($\nabla \log(c_i/\bar{c}_i) = \nabla \log c_i$).
2. Approximation on Bounded Domains: The problem is first solved on balls $B_R$ with homogeneous no-flux boundary conditions ($J_i \cdot \nu = 0$). The existence of weak solutions on these bounded domains is established using the entropy method developed by Jüngel and Stelzer [8].
3. Uniform Estimates: A key step involves deriving estimates independent of the radius $R$.
   • The no-flux boundary condition ensures mass conservation on $B_R$, allowing the standard entropy inequality to be rewritten as a relative entropy inequality with respect to the constant state $\bar{c}$.
   • The entropy dissipation term controls the gradients of the square roots of the concentrations ($\nabla \sqrt{c_i}$) and the fluxes ($J_i$).
   • These estimates rely purely on entropic and algebraic properties, explicitly avoiding the use of Poincaré inequalities, which would fail as $R \to \infty$ due to the diverging Poincaré constant.
4. Limit Passage: To pass to the limit $R \to \infty$, the author employs local compactness arguments.
   • Extensions of the approximate solutions $c^R$ and $J^R$ are defined on the whole space (setting $c^R = \bar{c}$ and $J^R = 0$ outside $B_R$).
   • Uniform bounds in $L^\infty(0, T; L^2(\mathbb{R}^3))$ for $c^R - \bar{c}$ and in $L^2((0, T) \times \mathbb{R}^3)$ for $\nabla \sqrt{c^R}$ and $J^R$ allow for weak convergence.
   • Strong local convergence in $L^2_{loc}$ is obtained via the Aubin–Lions lemma on fixed subdomains $B_K$, utilizing a diagonal argument. This strong convergence is sufficient to pass to the limit in the nonlinear Maxwell–Stefan relations.
   • The global entropy inequality is recovered using Fatou's Lemma and the lower semicontinuity of the relative entropy and dissipation terms.

Key Contributions and Results
The paper establishes the following main theorem:

• Existence: Under the assumption that the initial data $c_0$ satisfies $0 \le c_{0,i} \le 1$, $\sum c_{0,i} = 1$, and has finite relative entropy $H[c_0|\bar{c}] < \infty$, there exists a global-in-time weak solution $(c, J)$ on $(0, T) \times \mathbb{R}^3$.
• Regularity: The solution satisfies $c_i - \bar{c}_i \in L^\infty(0, T; L^2(\mathbb{R}^3))$, $\nabla \sqrt{c_i} \in L^2((0, T) \times \mathbb{R}^3)$, and $J_i \in L^2((0, T) \times \mathbb{R}^3)$.
• Entropy Inequality: The solution obeys a global relative entropy inequality:
  $$H[c(t)|\bar{c}] + C \int_0^t \int_{\mathbb{R}^3} \sum_{i=1}^{n+1} |\nabla \sqrt{c_i}|^2 \, dx \, ds \le H[c_0|\bar{c}]$$
  for almost every $t > 0$, where $C$ depends only on the Maxwell–Stefan diffusivities.

Significance and Scope
The paper claims to provide the first proof of global existence of weak solutions for the isothermal Maxwell–Stefan system on the whole space $\mathbb{R}^3$. The significance lies in overcoming the dual difficulties of infinite mass and the lack of global compactness.

The author explicitly notes the limitations and scope of the result:

• The far-field condition is not imposed pointwise; rather, the finite relative entropy condition serves as the natural replacement for finite mass, ensuring the solution remains close to the equilibrium state in a relative-entropy sense.
• Unlike results in bounded domains, the proof does not rely on Poincaré-type inequalities and does not yield exponential convergence to equilibrium.
• Therefore, the result is strictly an existence theorem for global finite-entropy weak solutions, rather than a long-time relaxation or asymptotic stability result.