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# Documents  35B45 | enregistrements trouvés : 4

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## Post-edited  A spectral inequality for the bi-Laplace operator Robbiano, Luc (Auteur de la Conférence) | CIRM (Editeur )

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.
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 ...

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## Post-edited  $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains Kozono, Hideo (Auteur de la Conférence) | CIRM (Editeur )

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.).
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...

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## Multi angle  The wave equation on a model convex domain revisited Planchon, Fabrice (Auteur de la Conférence) | CIRM (Editeur )

We detail how the new parametrix construction that was developped for the general case allows in turn for a simplified approach for the model case and helps in sharpening both positive and negative results for Strichartz estimates.

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## Multi angle  Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes Li, Jinkai (Auteur de la Conférence) | CIRM (Editeur )

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 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-S...

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