Abstract

We consider the system governing the evolution of pressureless viscous gases in dimension two in the case where the initial density is just bounded and bounded away from zero. Assuming that the initial velocity is sufficiently small compared to the viscosity in the critical Lorentz space $L^{2,1}$ (a large subspace of the natural energy space $L^2$), we prove the global existence and uniqueness of a solution with Lipschitz flow. This improves our recent work (2021), which, in a different functional framework, established a global result under the assumption that the density variations are small.

The main difficulty to get a global result lies in the fact that the density is just transported by the flow, with no diffusion, and does not decay to the reference density for large time. Our approach consists in proving time weighted energy estimates for the velocity (in the spirit of the work by Hoff (1995) on the compressible Navier–Stokes equations), then in taking advantage of a “dynamic” interpolation argument so as to establish that the gradient of the velocity field belongs to $L^1({ℝ}_{+};L^{\infty })$. This latter property ensures the uniqueness of the solution, and the control of the lower and upper bounds of the density.

To the best of our knowledge, this is the first global existence and uniqueness result for the system of pressureless gases with large density variations. The strategy is valid indistinctly in ${ℝ}^2$ or in smooth bounded domains of ${ℝ}^2$ and might be extendable to other models of nonhomogeneous viscous flows.

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