## Section: New Results

### Functional analysis of PDE models in Fluid Mechanics

#### On the rigid-lid approximation of shallow water Bingham model

Member: J. Sainte-Marie

*Coll.: Bilal Al Taki, Khawla Msheik*

The paper [17] discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham’s constitutive law, we follow a similar approach to that introduced in [G. Duvaut and J.-L. Lions, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

#### Global bmo-1(${\mathbb{R}}^{N}$) radially symmetric solution for compressible Navier-Stokes equations with initial density in ${\mathbb{L}}^{\infty}\left({\mathbb{R}}^{N}\right)$

Member: B. Haspot

In [24], we investigate the question of the existence of global weak solutionfor the compressible Navier Stokes equations provided that the initial momentum belongs to bmo-1(${\mathbb{R}}^{N}$) with $N=2,3$ and is radially symmetric. We prove then a equivalent of the so-called Koch-Tataru theorem for the compressible Navier-Stokes equations. In addition we assume that the initial density is only bounded in ${\mathbb{L}}^{\infty}\left({\mathbb{R}}^{N}\right)$, it allows us in particular to consider initial density admitting shocks. 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 regularizinginasmuch as the density becomes Lipschitz. To finish we prove the global existence of strong solution for large initial data pro-vided that the initial data are radially symmetric and sufficiently regular in dimension $N=2,3$ for $\gamma -$law pressure.

#### New effective pressure and existence of global strong solution for compressible Navier-Stokes equations with general viscosity coefficient in one dimension

Member: Boris Haspot

*Coll.: Cosmin Burtea*

In this paper we prove the existence of global strong solution for the Navier-Stokesequations with general degenerate viscosity coefficients. The cornerstone of the proofis the introduction of a new effective pressure which allows to obtain an Oleinik-typeestimate for the so called effective velocity. In our proof we make use of additionalregularizing effects on the velocity which requires to extend the technics developedby Hoff for the constant viscosity case.