CIRM - Videos & books Library - $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains

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Auteurs : Kozono, Hideo (Auteur de la Conférence)
CIRM (Editeur )

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Harmonic vector fields Betti number Scalar potential and vector potential A priori estimate Uniqueness

Résumé : 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.).

Keywords : Helmholtz-Weyl decomposition; harmonic vector field; Betti number

Codes MSC :
35A25 - Other special methods
35B45 - A priori estimates
35J25 - Boundary value problems for second-order elliptic equations
35Q30 - Stokes and Navier-Stokes equations
58A10 - Differential forms

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https://videos.cirm-math.fr/2019-10-28_Kozono.mp4

Informations sur la Rencontre

Nom de la rencontre : Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraites
Organisateurs de la rencontre : Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen
Dates : 28/10/2019 - 01/11/2019
Année de la rencontre : 2019
URL Congrès : https://conferences.cirm-math.fr/2071.html

Données de citation

DOI : 10.24350/CIRM.V.19575403
Citer cette vidéo: Kozono, Hideo (2019). $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19575403
URI : http://dx.doi.org/10.24350/CIRM.V.19575403

Bibliographie

KOZONO, Hideo et YANAGISAWA, Taku. $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains. Indiana Univ. Math. J, 2009, vol. 58, no 4, p. 1853-1920. - https://doi.org/10.1512/iumj.2009.58.3605

HIEBER, Matthias, KOZONO, Hideo, SEYFERT, Anton, et al. A Characterization of Harmonic $L^r$-Vector Fields in Two-Dimensional Exterior Domains. The Journal of Geometric Analysis, 2019, p. 1-18. - https://doi.org/10.1007/s12220-019-00216-0

HIEBER, Matthias, KOZONO, Hideo, SEYFERT, Anton, et al. $L^r$-Helmholtz-Weyl decomposition in three dimensional exterior domains, submitted. -

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