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DOI: 10.5445/IR/1000078155
Veröffentlicht am 20.12.2017

Semilinear and quasilinear stochastic evolution equations in Banach spaces

Hornung, Luca

Abstract (englisch):
In this thesis, we investigate the Cauchy problem for the quasilinear stochastic evolution equation
\begin{equation*}
\begin{cases}
u(t)=[{-}A(u(t))u(t)+f(t)]\operatorname{dt}+B(u(t))dW(t),\quad t\in [0,T],\\
u(0)=u_0
\end{cases}
\end{equation*}
in a Banach space $ X. $

In the first part of the thesis, we concentrate on the parabolic situation, i.e. we assume that $ {-}A(u(t)) $ is for every $ t $ a generator of an analytic semigroup and that $ A(u(t)) $ has a bounded $ H^{\infty} $-calculus. Under a local Lipschitz assumption on $ u\mapsto A(u) $ we prove existence and uniqueness of a local strong solution up to a maximal stopping time that can be characterised by a blow-up alternative. We apply our local well-posedness result to a second order parabolic partial differential equation on $ \mathbb{R} ^d $, to a generalised Navier-Stokes equation describing non-Newtonian fluids and to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions. In the last situation, we can even show that the solution exists globally.

In the second part of the thesis, we go to a special hyper ... mehr


Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Jahr 2017
Sprache Englisch
Identifikator URN: urn:nbn:de:swb:90-781556
KITopen ID: 1000078155
Verlag Karlsruhe
Umfang IV, 173 S.
Abschlussart Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 13.12.2017
Referent/Betreuer Prof. L. Weis
Schlagworte Stochastic evolution equations, quasilinear parabolic, maximal regularity, functional calculus, nonlinear Maxwell equation, non-Newtonian fluids
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