Raphaël Danchin

We are concerned with the barotropic compressible Navier–Stokes equations on the real line. Our primary goal is to establish the global well-posedness in a critical regularity framework in the case where the initial data are small perturbations of a stable constant state. Surprisingly, even though the result in the multidimensional case is by now classical, the one-dimensional case has not been elucidated yet as far as we know. This is due to the fact that in the critical framework, the regularity of the velocity is so negative that some nonlinear terms are out of control. Here, we overcome the difficulty by considering the equations in the mass Lagrangian coordinates system. Granted with a global well-posedness statement, we then establish optimal time decay estimates and investigate the high viscosity limit, pointing out the convergence of the specific volume to the solution of some ordinary differential equation, after time and space rescaling.

We study the large‐time behavior of finite‐energy weak solutions for the Vlasov–Navier–Stokes equations in a two‐dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable‐density case. In both cases, large‐time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han‐Kwan et al. [16] dedicated to the homogeneous, three‐dimensional case, where an additional smallness condition on the velocity was required. In the nonhomogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.

We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L 2 , we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a "dynamic interpolation" argument. We show that the constructed solutions have a uniformly C 1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain, or the torus.

We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L^2 , we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a 'dynamic interpolation' argument. We show that the constructed solutions have a uniformly C^1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data, and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold true in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain or the torus.

We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L^2 , we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a 'dynamic interpolation' argument. We show that the constructed solutions have a uniformly C^1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data, and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold true in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain or the torus.