It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u$, there exist $h\in X^{r}_{har}\left ( \Omega \right )$, $w\in H^{1,r}\left ( \Omega \right )^{3}$ with div $w= 0$ and $p\in H^{1,r}\left ( \Omega \right )$ such that $u$ is uniquely decomposed as $u= h$ + rot $w$ + $\bigtriangledown p$.
On the other hand, if for the given $L^{r}$-vector field $u$ we choose its harmonic part $h$ from $V^{r}_{har}\left ( \Omega \right )$, then we have a similar decomposition to above, while the unique expression of $u$ holds only for $1< r< 3$. Furthermore, the choice of $p$ in $H^{1,r}\left ( \Omega \right )$ is determined in accordance with the threshold $r= 3/2$.
Our result is based on the joint work with Matthias Hieber, Anton Seyferd (TU Darmstadt), Senjo Shimizu (Kyoto Univ.) and Taku Yanagisawa (Nara Women Univ.).[-]

In this talk, I will present the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model, which has been used by Klein–Majda in [1] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [2] and Grabowski–Smolarkiewicz [3], contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. The global well-posedness was proved by combining the technique used in Hittmeir–Klein–Li–Titi [4], where global well-posedness was established for the pure moisture system for given velocity, and the ideas of Cao–Titi [5], who succeeded in proving the global solvability of the primitive equations without coupling to the moisture.[-]

In many physical fluid systems, the constituent particles present a parity-breaking intrinsic angular momentum: this is the case, for instance, of quantum fluids and super-fluids, polyatomic gases, chiral active matter and vortex dynamics. In such situations, only the skew-symmetric component of the total viscous stress tensor, often dubbed odd viscosity, is non-zero, implying that the viscosity becomes non-dissipative.
At the level of the mathematical model, the odd viscosity term is responsible for a loss of regularity, as it involves higher order space derivatives of the velocity field and, in the case of non-homogeneous fluids, of the density.
In this talk we consider the dynamics of non-homogeneous incompressible fluids having odd viscosity and we set up a well-posedness theory in Sobolev spaces for the related system of equations. The proof is based on the introduction of a set of suitable 'good unknowns' for the system, which allow to put in evidence an underlying hyperbolic structure and to circumvent, in this way, the loss of derivatives created by the odd viscosity term.
The talk is based on a joint work with Rafael Granero-Belinchón (Universidad de Cantabria) and Stefano Scrobogna (Università degli Studi di Trieste).[-]

In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to obtain a Carleman estimate up to the boundary.[-]