## Tuesday Seminar of Analysis

Seminar information archive ～05/18｜Next seminar｜Future seminars 05/19～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

### 2022/04/12

16:00-17:30 Online

Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)

https://forms.gle/QbQKex12dbQrt2Lw6

**Amru Hussein**(Technische Universität Kaiserslautern)Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)

[ Abstract ]

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962

[ Reference URL ]Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962

https://forms.gle/QbQKex12dbQrt2Lw6