The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids
Matthias Hieber Miho Murata
Evolution Equations & Control Theory 2015, 4(1): 69-87 doi: 10.3934/eect.2015.4.69
Consider the system of equations describing the motion of a rigid body immersed in a viscous, compressible fluid within the barotropic regime. It is shown that this system admits a unique, local strong solution within the $L^p$-setting.
keywords: moving domain strong $L^p$-solution Fluid rigid body interaction Lagragian formulation. compressible fluid
Remarks on the $L^p$-approach to the Stokes equation on unbounded domains
Matthias Geissert Horst Heck Matthias Hieber Okihiro Sawada
Discrete & Continuous Dynamical Systems - S 2010, 3(2): 291-297 doi: 10.3934/dcdss.2010.3.291
Consider a domain $\Omega \subset \R^n$ with uniform $C^3$-boundary and assume that the Helmholtz projection $P$ exists on $L^p(\Omega)$ for some $ 1 < p < \infty$. Of concern are recent results on the Stokes operator in $L^p(\Omega)$ generating an analytic semigroup on $L^p(\Omega)$ and admitting maximal $L^p$-$L^q$-regularity.
keywords: Stokes equation unbounded domains noncompact boundary.
Well-posedness results for the Navier-Stokes equations in the rotational framework
Matthias Hieber Sylvie Monniaux
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5143-5151 doi: 10.3934/dcds.2013.33.5143
Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
keywords: mild solutions. Coriolis force Navier-Stokes equations Dirichlet boundary conditions Stokes-Coriolis semigroup
Remarks on the theory of Oldroyd-B fluids in exterior domains
Matthias Hieber
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1307-1313 doi: 10.3934/dcdss.2013.6.1307
Consider the set of equations describing Oldroyd-B fluids with finite Weissenberg numbers in exterior domains. In this note, we describe the main ideas of the proofs of two recent results on global existence for this set of equations on exterior domains subject to Dirichlet boundary conditions. The methods described here are quite different from the techniques used in the Lagrangian approach which is often used in the case of infinite Weissenberg numbers.
keywords: Oldroyd-B fluids exterior domains global solution.
Local and global existence results for the Navier-Stokes equations in the rotational framework
Daoyuan Fang Bin Han Matthias Hieber
Communications on Pure & Applied Analysis 2015, 14(2): 609-622 doi: 10.3934/cpaa.2015.14.609
Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
keywords: global solution Rotational flows Fourier-Besov space Chemin-Lerner space. Littlewood-Paley decomposition
Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity
Michele Campiti Giovanni P. Galdi Matthias Hieber
Communications on Pure & Applied Analysis 2014, 13(4): 1613-1627 doi: 10.3934/cpaa.2014.13.1613
It is proved the existence of a unique, global strong solution to the two-dimensional Navier-Stokes initial-value problem in exterior domains in the case where the velocity field tends, at large spatial distance, to a prescribed velocity field that is allowed to grow linearly.
keywords: exterior domains. Navier-Stokes with linearly growing data
$L^\infty$-estimates for parabolic systems with VMO-coefficients
Horst Heck Matthias Hieber Kyriakos Stavrakidis
Discrete & Continuous Dynamical Systems - S 2010, 3(2): 299-309 doi: 10.3934/dcdss.2010.3.299
In this paper we prove $L^\infty$-a priori estimates for parabolic evolution equations in non-divergence form on all of $\R^n$ for bounded coefficients having only vanishing mean oscillation, thus allowing in particular non continuous coefficients.
keywords: $L^\infty$. Parabolic systems VMO

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