Abstract. Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole n -dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space ofḂ

We investigate the incompressible Navier-Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2, 3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. Let us emphasize that all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n-dimension if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper. MSC: 35Q30, 76D05

We consider the Cucker-Smale flocking model with a singular communication weight ψ(s) = s −α with α > 0. We provide a critical value of the exponent α in the communication weight leading to global regularity of solutions or finite-time collision between *

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier‐Stokes equations supplemented with H1 initial velocity and only bounded nonnegative density. In contrast to all the previous works on those topics, we do not require regularity or a positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions. Those solutions are global in the two‐dimensional case for general data, and in the three‐dimensional case if the velocity satisfies a suitable scaling‐invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in his 1996 book Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models, concerning the evolution of a drop of incompressible viscous fluid in the vacuum. © 2018 Wiley Periodicals, Inc.

Abstract. We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the NavierStokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory. -IntroductionThe purpose of this paper is to analyze the model of flow of compressible viscous fluid with variable entropy. Such flow can be described by the compressible Navier-Stokes equations coupled with an additional equation describing the evolution of the entropy. In case when the conductivity is neglected, the changes of the entropy are solely due to the transport and the whole system can be written as:(1.1a) 1c) where the unknowns are the density ̺ : (0, T ) × Ω → R + ∪ {0}, the entropy s : (0, T ) × Ω → R + and the velocity of fluid u : (0, T ) × Ω → R 3 , and where Ω is a three dimensional domain with a smooth boundary ∂Ω. The momentum, the continuity and the entropy equations are additionally coupled by the form of the pressure p, we assume thatwhere T (·) is a given smooth and strictly monotone function from R + to R + , in particular T (s) > 0 for s > 0. We assume that the fluid is Newtonian and that the viscous part of the stress tensor is of the following formwith D(u) = 1 2 (∇u + ∇u T ). Viscosity coefficients µ and η are assumed to be constant, hence we can write div S(∇u) = µ∆u + (µ + λ)∇ div u with λ = η − 2 3 µ. To keep the ellipticity of the Lamé operator we require that µ > 0, 3λ + 2µ > 0.(1.3)