Global existence of weak solutions for the Maxwell--Stefan system in the whole space
We prove the global-in-time existence of weak solutions to the isothermal Maxwell--Stefan system on the whole space $\mathbb R^3$. The main difficulty is that, unlike in bounded domains, the concentrations generally have infinite mass and the standard mixing entropy is not finite. We therefore work with the relative entropy with respect to a strictly positive constant equilibrium state. The proof proceeds by solving the problem on balls $B_R$ with no-flux boundary conditions, using the bounded-domain entropy theory, and deriving estimates independent of $R$. These estimates yield uniform control of the relative entropy, the gradients $\nabla\sqrt{c_i}$, and the fluxes $J_i$. Passing to the limit $R\to\infty$ is achieved by local compactness and a diagonal argument. The resulting weak solution satisfies the Maxwell--Stefan system in the sense of distributions and obeys a global relative entropy inequality.