## Section: New Results

### Functional analysis of PDE models in Fluid Mechanics

Participants : Bilal Al Taki, Boris Haspot.

#### New functional inequality and its application

In [22], we prove by simple arguments a new kind of Logarithmic Sobolev inequalities generalizing two known inequalities founded in some papers related to fluid dynamics models. As a by product, we show how our inequality can help in obtaining some important a priori estimates for the solution of the Navier-Stokes-Korteweg system.

#### Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density

We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum ${\rho}_{0}{u}_{0}$ belongs to the set of the finite measure $\mathcal{M}\left(\mathbf{R}\right)$ and when the initial density ${\rho}_{0}$ is in the set of bounded variation functions $BV\left(\mathbf{R}\right)$. In particular it allows to deal with initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong then the initial density which admits initially shocks is instantaneously regularized and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity $v=u+\frac{\mu \left(\rho \right)}{{\rho}^{2}}{\psi}_{x}\rho $ with $\mu \left(\rho \right)$ the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is too weak (typically ${\rho}_{0}{v}_{0}\in \mathcal{M}\left(\mathbf{R}\right)$) then we prove the existence of weak energy in finite time but the density remains a priori discontinuous on the time interval of existence.

#### Strong solution for Korteweg system

In this paper we investigate the question of the local existence of strong solution for the Korteweg system in critical spaces when $N\ge 1$ provided that the initial data are small. More precisely the initial momentum ${\rho}_{0}{u}_{0}$ belongs to ${\text{bmo}}_{T}^{-1}\left({\mathbf{R}}^{N}\right)$ for $T>0$ and the initial density ${\rho}_{0}$ is in ${L}^{\infty}\left({\mathbf{R}}^{N}\right)$ and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the incompressible Navier-Stokes equations to the case of the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any $\rho (t,\xb7)$ with $t>0$. We also prove the existence of global strong solution for small initial data $({\rho}_{0}-1,{\rho}_{0}{u}_{0})$ in the homogeneous Besov spaces $({\dot{B}}_{2,\infty}^{\mathbf{N}-1}({\mathbf{R}}^{\mathbf{N}}\cap {\dot{B}}_{2,\infty}^{\mathbf{N}}({\mathbf{R}}^{\mathbf{N}}\cap {L}^{\infty}\left({\mathbf{R}}^{\mathbf{N}}\right))\times {\left({\dot{B}}_{2,\infty}^{\mathbf{N}-1}\left({\mathbf{R}}^{\mathbf{N}}\right)\right)}^{\mathbf{N}}$. This result allows in particular to extend in dimension $N=2$ the notion of Oseen solutions defined for incompressible Navier-Stokes equations to the case of the Korteweg system when the vorticity of the momentum ${\rho}_{0}{u}_{0}$ is a Dirac mass $\alpha {\delta}_{0}$ with $\alpha $ sufficiently small. However unlike the Navier Stokes equations