Matthias Hieber's research while affiliated with Technische Universität Darmstadt and other places
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Publications (161)
This article investigates the primitive equations with nonlinear equations of state.
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L2\documentclass[12pt]{minimal} \usepackage{amsma...
In this paper we introduce and study the primitive equations with $\textit{non}$-isothermal turbulent pressure and transport noise. They are derived from the Navier-Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence...
Consider the set of equations modeling the motion of a rigid body enclosed in sea ice. Using Hibler's viscous-plastic model for describing sea ice, it is shown by a certain decoupling approach that this system admits a unique, local strong solution within the $\mathrm{L}^p$-setting.
This article establishes local strong well-posedness of a model coupling the primitive equations of the ocean and the atmosphere with a regularized version of Hibler's viscous-plastic sea ice model. It is assumed that the ocean force on the sea ice is proportional to the shear rate, that the velocity of the ocean and the ice coincide on their commo...
It is shown that the viscous-plastic Hibler sea ice model admits a unique, strong $T$-time periodic solution provided the given $T$-periodic forcing functions are small in suitable norms. This is in particular true for time periodic wind forces and time periodic ice growth rates.
This article develops for the first time a rigorous analysis of Hibler’s model of sea ice dynamics. Identifying Hibler’s ice stress as a quasilinear second-order operator and regarding Hibler’s model as a quasilinear evolution equation, it is shown that a regularized version of Hibler’s coupled sea ice model, i.e., the model coupling velocity, thic...
Consider the space of harmonic vector fields u in \(L^r(\Omega )\) for \(1<r<\infty \) for three dimensional exterior domains \(\Omega \) with smooth boundaries \(\partial \Omega \) subject to the boundary conditions \(u\cdot \nu =0\) or \(u\times \nu =0\), where \(\nu \) denotes the unit outward normal on \(\partial \Omega \). Denoting these space...
This article investigates the primitive equations with nonlinear equations of state. A global, strong well-posedness result for this set of equations is established for initial data lying in critical spaces provided that the density, depending on temperature, salinity and pressure, satisfies certain regularity assumptions. These assumptions are in...
In an exterior domain \(\Omega \subset \mathbb {R}^3\) having compact boundary \(\partial \Omega = \bigcup _{j=1}^L\Gamma _j\) with L disjoint smooth closed surfaces \(\Gamma _1, \ldots , \Gamma _L\), we consider the problem on the existence of weak solutions \(\varvec{v}\) of the stationary Navier–Stokes equations in \(\Omega \) satisfying \(\varv...
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal $L^2$-regularity, the theory of critical spaces f...
In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1<r<∞. In fact, given an Lr-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely asu=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is ch...
Consider the classical Keller–Segel system on a bounded convex domain Ω⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subset {\mathbb {R}}^3$$\end{docume...
This article develops for the first time a rigorous analysis of Hibler's model of sea ice dynamics. Identifying Hibler's ice stress as a quasilinear second order operator and regarding Hibler's model as a quasilinear evolution equation, it is shown that Hibler's coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of...
Considering the anisotropic Navier–Stokes equations as well as the primitive equations, it is shown that the horizontal velocity of the solution to the anisotropic Navier–Stokes equations in a cylindrical domain of height ɛ with initial data u 0 = ( v 0 , w 0 ) ∈ B q , p 2 − 2 / p , 1/ q + 1/ p ⩽ 1 if q ⩾ 2 and 4/3 q + 2/3 p ⩽ 1 if q ⩽ 2, converges...
This article concentrates on various operator semigroups arising in the study of viscous and incompressible flows. Of particular concern are the classical Stokes semigroup, the hydrostatic Stokes semigroup, the Oldroyd as well as the Ericksen–Leslie semigroup. Besides their intrinsic interest, the properties of these semigroups play an important ro...
In this article, an L_1 maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. Th...
This article studies the primitive equations for geophysical flows subject to stochastic wind driven boundary conditions modeled by a cylindrical Wiener process. A rigorous treatment of stochastic boundary conditions yields that these equations admit a unique global, strong, pathwise solution within the $L^q_t$-$L^p_x$-setting of critical spaces. C...
Let \(\Omega \) be a two-dimensional exterior domain with smooth boundary \(\partial \Omega \) and \(1< r < \infty \). Then \(L^r(\Omega )^2\) allows a Helmholtz–Weyl decomposition, i.e., for every \(\mathbf{u}\in L^r(\Omega )^2\) there exist \(\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )\), \(w \in {\dot{H}}^{1,r}(\Omega )\) and \(p \in {\do...
The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the theory of evolution equations. In particular, we discuss the hydrostatic Stokes operator, well-posedness results in critical spaces within the Lp(Lq)-sca...
In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal \({{L}}^p\)-regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo,
Aliev–Panfi...
This course of lectures discusses various aspects of viscous fluid flows ranging from boundary layers and fluid structure interaction problems over free boundary value problems and liquid crystal flow to the primitive equations of geophysical flows. We will be mainly interested in strong solutions to the underlying equations and choose as mathemati...
Consider the bidomain equations from electrophysiology with FitzHugh--Nagumo transport subject to current noise, i.e., subject to stochastic forcing modeled by a cylindrical Wiener process. It is shown that this set of equations admits a unique global, strong pathwise solution within the setting of critical spaces. The proof is based on combining m...
This article presents the maximal regularity approach to the primitive equations. It is proved that the $3D$ primitive equations on cylindrical domains admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space $B^{2/p}_{pq}$ for $p,q\in (1,\infty)$ with $1/p+1/q \leq 1$. This solution regularize instantane...
It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution. The proof given relies on the existence of strong T-periodic solutions for the linearized system, its characterization in terms of m...
In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the $L^r$-setting for $1<p<\infty$.
The bidomain operator \({{\mathbb {A}}}\), a nonlocal operator, is studied in a bounded domain \(\Omega \subset {{\mathbb {R}}}^d\) with boundary \(\partial \Omega \) of class \(C^{2-}\) within the \(L_q\)-setting for \(1<q<\infty \). Assuming a fairly general framework, it is shown that this operator is sectorial, invertible on functions with mean...
In this article, the non-isothermal compressible Ericksen–Leslie system for nematic liquid crystals subject to general Leslie stress is considered. It is shown that this system is locally well-posed within the \(L_q\)-setting and that for initial data close to equilibria points (which are identical with the ones for the incompressible situation), t...
Consider the space of harmonic vector fields h in \(L^r(\Omega )\) for \(1<r<\infty \) in the two-dimensional exterior domain \(\Omega \) with the smooth boundary \(\partial \Omega \) subject to the boundary conditions \(h\cdot \nu =0\) or \(h\wedge \nu =0\), where \(\nu \) denotes the unit outward normal to \(\partial \Omega \). Denoting these spa...
Consider the Cauchy problem for the incompressible Oldroyd-B model in R ³ . For the case a = 0, global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math. B 21 13146), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for a -= 0. In this article,...
In diesem Kapitel erweitern wir die Differentialrechnung von Funktionen einer Variablen auf solche mit mehreren Veränderlichen. Wiederum lassen wir uns beim Begriff der Differenzierbarkeit einer Funktion von der zentralen Idee der linearen Approximierbarkeit leiten. Im Vergleich zu unseren bisherigen Untersuchungen ist allerdings die Situation im F...
In diesem Kapitel beginnen wir mit der Untersuchung von Kurven in \(\mathbb{R}^{n}\) und lassen uns hierbei von einem Kurvenbegriff, der aus der Physik, genauer aus der Kinematik herrührt, leiten. Er beschreibt die Abstraktion der Bewegung eines Punktes im Raum, die durch die Angabe des Ortes \(\gamma(t)\) zum Zeitpunkt \(t\) gegeben ist. Dieser An...
Welche Motive veranlassen Mathematiker, Räume von unendlicher Dimension einzuführen, eine Folge reeller Zahlen als einen Punkt in einem Folgenraum und eine Funktion als einen Punkt in einem Funktionenraum anzusehen?
In diesem Kapitel untersuchen wir mehrere, eng miteinander zusammenhängende Themenkomplexe. Wir beginnen mit der zentralen Frage, wann eine stetig differenzierbare Funktion \(f\) eine ebensolche Umkehrfunktion besitzt. Der Satz über die Umkehrfunktion gibt darauf eine befriedigende Antwort: Eine stetig differenzierbare Funktion, deren Differential...
Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $\varepsilon $ with initial data $u_0=(v_0,w_0)\in B^{2-2/p}_{q,p}$, $1/q+1/p\le 1$ if $q\ge 2$ and $4/3q+2/3p\le 1$ if $q\le 2$, c...
Consider the 3D incompressible Boussinesq equations for rotating stably stratified fluids. It is shown that this set of equations possesses a unique time periodic or almost time periodic solutions for external forces satisfying these properties, which, however, do not necessarily need to be small. An explicit bound on the size of the external force...
The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energ...
This article discusses the Stokes equation in various classes of domains Ω C Rⁿ within the Lp-setting for 1 ≤ p ≤ ∞ from the point of view of evolution equations. Classical as well as modern approaches to well-posedness results for strong solutions to the Stokes equation, to the Helmholtz decomposition, to the Stokes semigroup, and to mixed maximal...
Consider the $3$-d primitive equations in a layer domain $\Omega=G \times (-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundary conditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form $a=a_1 + a_2$,...
Die auf Leibniz und Newton zurückgehende Differential- und Integralrechnung bildet den inhaltlichen Kern jeder Einführung in die Analysis. Wir beginnen mit dem Begriff der Differenzierbarkeit einer Funktion, welcher durch den Wunsch geleitet ist, das lokale Verhalten von Funktionen genauer zu beschreiben.
Zentrales Thema des vorliegenden Kapitels sind die reellen Zahlen. Diese bilden das Fundament, auf welchem wir die Analysis schrittweise aufbauen werden. Um diese Zahlen präzise und rigoros definieren zu können, beginnen wir mit einer Einführung in die Grundbausteine der mathematischen Sprache (Mengen, Aussagen, Abbildungen, logische Symbole).
Die Bestimmung von Flächen, Volumen und Kurvenlängen gehört zu den historisch gesehen ältesten Problemen der Mathematik, und viele dieser Fragestellungen sind aus heutiger Sicht klassische Themen der Integrationstheorie. Um den Flächeninhalt einer krummlinig begrenzten Figur zu bestimmen, wurde dieser „von innen“ und „von außen“ durch einfachere Ob...
Dieses Lehrbuch zeichnet sich durch einen klaren und modernen Aufbau aus und ist auf eine breit angelegte Grundausbildung ausgerichtet. Es ist der erste Band einer zweiteiligen Einführung in die Analysis, die Studierende der Mathematik und verwandter Studienrichtungen (etwa Physik, Informatik und Ingenieurwissenschaften) sowie deren Dozenten anspri...
Im Zentrum dieses Kapitels steht die Entwicklung und Diskussion des Grenzwertbegriffs. Dieser bildet das Fundament der Analysis und wird für alle unseren weiteren Untersuchungen, speziell für die Differential- und Integralrechnung, unentbehrlich sein.
Im Zentrum dieses Kapitels stehen stetige Funktionen und ihre Eigenschaften. Ausgehend von unseren Überlegungen über die Konvergenz von Folgen in Kapitel I, definieren wir die Stetigkeit einer Funktion zunächst über die Folgenstetigkeit und zeigen dann anschließend, dass diese äquivalent ist zur sogenannten $(\ve$-$\delta)$--Formulierung der Stetig...
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally...
It is shown that a large class of semilinear evolution equations on the whole line with periodic or almost periodic forces admit periodic or almost periodic mild solutions. The approach presented generalizes the method described in [28] to the case of the whole line and to forces which are almost periodic in the sense of H. Bohr. It relies on inter...
Consider the primitive equations on $\R^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1 \in BUC_\sigma(\R^2;L^1(z_0,z_1))$ and $a_2 \in L^\infty_\sigma(\R^2;L^1(z_0,z_1))$ and where $BUC_\sigma(L^1)$ and $L^\infty_\sigma(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded f...
Consider the bidomain equations subject to ionic transport described by the models of FitzHugh-Nagumo, Aliev-Panfilov, or Rogers-McCulloch. It is proved that this set of equations admits a unique, strong T-periodic solution provided it is innervated by T-periodic intra- and extracellular currents. The approach relies on a new periodic version of th...
Consider the two-phase free boundary problem subject to surface tension and
gravitational forces for a class of non-Newtonian fluids with stress tensors
$T_i$ of the form $T_i=-\pi I+\mu_i(|D(v)|^2)D(v)$ for $i=1,2$, respectively,
and where the viscosity functions $\mu_i$ satisfy $\mu_i(s)\in C^3([0,\infty))$
and $\mu_i(0)>0$ for $i=1,2$. It is sho...
It is shown that the hydrostatic Stokes operator on Lp/σ(Ω), where Ω ⊂ ℝ³ is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded H∞-calculus on Lp/σ(Ω) for p ∈ (1, ∞) of H∞-angle 0. In particular, maximal Lq − Lp-regularity estimates for the linearized primitive equations are obtained.
This article discusses the Stokes equation in various classes of domains \(\Omega \subset \mathbb{R}^{n}\) within the L
p
-setting for 1 ≤ p ≤ ∞ from the point of view of evolution equations. Classical as well as modern approaches to well-posedness results for strong solutions to the Stokes equation, to the Helmholtz decomposition, to the Stokes se...
The general Ericksen-Leslie model for the flow of nematic liquid crystals is reconsidered in the non-isothermal case aiming for thermodynamically consistent models. The non-isothermal simplified model is then investigated analytically. A fairly complete dynamic theory is developed by analyzing these systems as quasilinear parabolic evolution equati...
This survey article discusses various aspects of modeling and analysis of the Ericksen-Leslie equations describing nematic liquid crystal flow both in the isothermal and non-isothermal situation. Of special interest is the development of thermodynamically consistent Ericksen-Leslie models in the general situation based on the entropy principle. The...
This article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations. It yields, on the one hand, a unified approach to various classical problems in incompressible fluid flow and, on the other hand, gives new results for periodic solutions to the Navier–Stokes–Oseen flow, the Navier–Stokes...
Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as...
Let \(\Omega \subset \mathbb{R}^{n}\), n ≥ 3, be an exterior domain with smooth boundary. It is shown that the Stokes semigroup on \(L_{\sigma }^{\infty }(\Omega )\) is a bounded analytic semigroup on this space.
We show that the three-dimensional primitive equations admit a strong
time-periodic solution of period $T>0$, provided the forcing term $f\in
L^2(0,\mathcal T; L^2(\Omega))$ is a time-periodic function of the same period.
No restriction on the magnitude of $f$ is assumed. As a corollary, if, in
particular, $f$ is time-independent, the corresponding...
In this article, an $L^p$-approach to the primitive equations is developed.
In particular, it is shown that the three dimensional primitive equations admit
a unique, global strong solution for all initial data $a \in
[X_p,D(A_p)]_{1/p}$ provided $p \in [6/5,\infty)$. To this end, the hydrostatic
Stokes operator $A_p$ defined on $X_p$, the subspace...
The general Ericksen-Leslie system for the flow of nematic liquid crystals is
reconsidered in the non-isothermal case aiming for thermodynamically consistent
models. The non-isothermal model is then investigated analytically. A fairly
complete dynamic theory is developed by analyzing these systems as quasilinear
parabolic evolution equations in an...
Consider the Stokes semigrou T-infinity defined on L-sigma(infinity) (Omega) where Omega subset of R-n, n >= 3, denotes an exterior domain with smooth boundary. It is shown that T-infinity(z)u(0) for u(0) is an element of L-sigma(infinity) (Omega) and z is an element of Sigma(theta) with theta is an element of (0, pi/2) satisfies pointwise estimate...
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.
Consider the set of equations describing fluids of Oldroyd-B type on an exterior domain. It is shown that the solution of the linearized equation is governed by a bounded analytic semigroup on L-p (Omega) x W-1,W-p (Omega), 1 < p < infinity, which is strongly stable. Moreover, it is shown that the trivial solution of the full system is asymptotical...
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.
Consider the equations of Navier-Stokes in R3 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 {equation presented} (R3), where p ∈ [2,∞] and r∈ [1,∞)...
The Stokes equation on a domain $\Omega \subset R^n$ is well understood in
the $L^p$-setting for a large class of domains including bounded and exterior
domains with smooth boundaries provided $1<p<\infty$. The situation is very
different for the case $p=\infty$ since in this case the Helmholtz projection
does not act as a bounded operator anymore....
Consider the Navier-Stokes equations in the rotational framework either on R3 or on open sets R3 subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
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 th...
Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shearthinning or shear-thickening fluids of power-law type of exponent d ≥ 1. We develop a method to prove that this system admits a uniq...
Consider the (simplified) Leslie-Erickson model for the flow of nematic
liquid crystals in a bounded domain $\Omega \subset \mathbb{R}^n$ for n > 1$.
This article develops a complete dynamic theory for these equations, analyzing
the system as a quasilinear parabolic evolution equation in an
$L_p-L_q$-setting. First, the existence of a unique local...
We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements...
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 the initial data is small with respect to
the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(\R^3)$, where $p \in
(1,\infty]$ and $r \in [1,\infty]$.
In th...
Consider the set of equations describing Oldroyd-B fluids in an exterior
domain. It is shown that this set of equations admits a unique, global solution
in a certain function space provided the initial data, but not necessarily the
coupling constant, is small enough.
In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough.
Consider a domain Ω ⊂ ℝn with possibly non compact but uniform C3-boundary and assume that the Helmholtz projection P exists on Lp(Ω) for some 1 < p < ∞. It is shown that the Stokes operator in Lp(Ω) generates an analytic semigroup on admitting maximal Lq-Lp-regularity. Moreover, for there exists a unique local mild solution to the Navier–Stokes eq...
In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique...
In this chapter the emphasis of the discussion shifts from Laplace integrals \(\hat f \)(λ) and \( \mathop {dF}\limits^ \wedge \)
(λ) to the Laplace transform \( L:f \mapsto \hat f \) and to the Laplace-Stieltjes transform \( {L_S}:F \mapsto \mathop {dF}\limits^ \wedge \) . The Laplace transform is considered first as an operator acting on L
∞ (ℝ+,...
In this chapter we consider the Laplacian on spaces of continuous functions. If Ω ⊂ ℝn
is an open, bounded set with boundary ∂Ω which is Dirichlet regular, we will show that the Laplacian generates a holomorphic semigroup on the space $${C_0}(\Omega )\,: = \left\{ {u\, \in (\bar \Omega )} \right\}:\,u{|_{\partial \Omega }}\, = 0.$$ Furthermore, usi...
Frequently, convergence of a function
f : \mathbbR+ ® X f : \mathbb{R}_+ \rightarrow X
for
t ® ¥t \rightarrow \infty
implies convergence of an average of this function. Assertions of this type are called Abelian theorems. A theorem is called Tauberian if, conversely, convergence of the function is deduced from the convergence of an average.
The first three sections of this chapter are of a preliminary nature. There, we collect properties of the Bochner integral of functions of a real variable with values in a Banach space X. We then concentrate on the basic properties of the Laplace integral
$$\hat f(\lambda ): = \int_0^\infty {{e^{ - \lambda t}}} f(t)dt: = \mathop {\lim }\limits_{\ta...
In this chapter we consider differential operators with constant coefficients, and more generally pseudo-differential operators on Lp(ℝn). The realization Opp
(a) in Lp(ℝn) of such an operator is translation invariant. We assume that the “symbol” a satisfies certain smoothness and growth assumptions. In particular, when a is a polynomial, then Opp...
In this chapter we study systematically well-posedness of the Cauchy problem. Given a closed operator A on a Banach space
X we will see in Section 3.1 that the abstract Cauchy problem
{lcr u¢(t) = Au(t) (t ³ 0), u(0) = x, \left\{\begin{array}{lcr} u^\prime(t) = Au(t) \,\,(t\geq0), \\ u(0) = x, \end{array}\right.
is mildly well-posed (i.e., for ea...
In this chapter, we give various results concerning the long-time asymptotic behaviour of mild solutions of homogeneous and inhomogeneous Cauchy problems on \(\mathbb {R}_+\) (see Section 3.1 for the definitions and basic properties). For the most part, we shall assume that the homogeneous problem is well-posed, so that the operator A generates a C...
In this chapter we study the wave equation $${u_{tt}}\, = \,\Delta u$$ on an open subset Ω of ℝn
. We will use the theory of cosine functions and work on L
2 (Ω). We first consider the Laplacian with Dirichlet boundary conditions. This is a selfadjoint operator and well-posedness is a consequence of the spectral theorem. A further aim is to replace...
Consider the density-dependent incompressible Navier–Stokes equations in ℝ with linearly growing initial data at infinity. It is shown that under certain regularity and growth assumptions on the data, the above system admits a unique, local solution. Moreover, the solution can be extended for arbitrary T > 0, provided the data are small enough with...
Consider the initial value problem for the three-dimensional Navier-Stokes equations with rotation in the half-space R(+)(3) subject to Dirichlet boundary conditions as well as the Ekman spiral, which is a stationary solution to the above equations. It is proved that the Ekman spiral is nonlinearly stable with respect to L(2)-perturbations provided...
Consider a domain Ω⊂ℝ n with uniform C 3 -boundary and assume that the Helmholtz projection P exists on L p (Ω) for some 1<p<∞. Of concern are recent results on the Stokes operator in L p (Ω) generating an analytic semigroup on L p (Ω) and admitting maximal L p -L q -regularity.
In this paper we prove L ∞-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.
Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global
mild solution for arbitrary speed of rotation provided the initial data u
0 is small enough in the
H\frac12s(\mathbbR3){H^{\frac12}_{\sigma}(\mathbb{R}^3)} -norm.
Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal Lp – Lq-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an
R{\mathcal{R}}-sectorial operator in Lps(W)L^{p}_{\sigma}(\Omega), 1 < p < ¥1 < p < \infty, of R{\mathcal{R}}...
Consider the stochastic Navier-Stokes-Coriolis equations in an infinite layer subject to Dirichlet boundary conditions as well as the Ekman spiral which is a stationary solution to the de-terministic equations. It is proved that the stochastic Navier-Stokes-Coriolis equation admits a weak martingale solution. Moreover, as an stochastic analogue of...
We show that the realization Ap of the elliptic operator Au=div(Q∇u)+F⋅∇u+Vu in Lp(RN,RN), p∈[1,+∞[, generates a strongly continuous semigroup, and we determine its domain if 1<p<+∞. The diffusion coefficients Q=(qij) are uniformly elliptic and bounded together with their first-order derivatives, the drift coefficients F can grow as |x|log|x|, and...
. Moreover, it is proved that this strong solution coincides with the known mild solution in the very weak sense.
Citations
... Here u = u(x, t) is the velocity vector field, ρ = ρ(x, t) is the scalar temperature or density of fluid, and W t is a cylindrical Wiener process valued in some separable Hilbert space and the corresponding stochastic integral is in the Itô sense. The term (b · ∇u, b · ∇ρ)dW t represents the transport noise and (σ (1) (u, ρ), σ (2) (u, ρ))dW t is the multiplicative noise. The main result of the paper is the global wellposedness of the system (1.1) in L p (Ω; L p (T 3 )) with p > 5 and small initial data. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/394211766620170-1470998817998_Q64/Amru-Hussein.jpg)
... [9,27,29,[34][35][36]41]. Unlike the equations describing atmospheric or oceanic dynamics as e.g. the primitive equations, see [3], rigorous analysis of the sea ice equations started only very recently by the works of Brandt, Disser, Haller-Dintelmann, Hieber [2] and Liu, Thomas and Titi [30]. The underlying set of equations is a coupled degenerate quasilinear parabolic-hyperbolic system, whose analysis is delicate. ...
... We emphasize that the coupling of the primitive equations and the sea ice is delicate, as the primitive equations are a 3D model, while the sea ice equations are two-dimensional. The sea ice equations are considered on the whole plane, while the primitive equations are studied on layer domains of the shape the primitive equations with nonlinear equations of state in [7]. The situation of an infinite layer domain for the primitive equations was also considered by Giga, Gries, Hieber, Hussein and Kashiwabara [22], where the authors study the primitive equations in the scaling-invariant space L ∞ (L 1 ), and by Hussein, Saal and Sawada [29], where the primitive equations with linearly growing initial data in the horizontal direction are investigated. ...
... On the other hand, the structure of vector fields in exterior domains is more complicated than that in bounded domains. In our previous paper [8,Theorem 2.4], we proved that every b ∈ L r ( ) for 3/2 < r < 3 with div b = 0 in the sense of distributions in admits a unique representation b = h + rot w, (1.3) where h ∈ L r ( ) is a harmonic vector field and w ∈Ḣ 1,r ( )(Ḣ 1,r ( ) denotes homogeneous Sobolev space). In the present paper, we first show that although there are infinitely many solenoidal extensions b into of β j on j , j = 1, . . . ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/326397869477888-1454830724024_Q64/Hideo-Kozono.jpg)
... The rigorous mathematical justification of the small aspect ratio from the Navier-Stokes equtions to the primitive equations, i.e., hydrostatic approximation, was studied in [1], where the weak convergences were established. The strong convergences, which are global and uniform in time, and the convergence rate were established in [29] and [15] by different ways. ...
... [18,27] and also [17,23,26,29], where in particular for applications in fluid mechanics semigroup theory plays an important role, cf. [22]. For a similar approach where the stationary part is treated separately, and in particular applications to quasi-and semilinear problems see also the works of Kyed and co-authors, cf. ...
... Sometimes we will refer to the noise in (1.5) as the general noise. The noise of this type is consistent with the one typically used in the abstract theory of SPDEs by Da Prato and Zabczyk [7,8] when studying the long time behavior and invariant measures for stochastic PDEs, both in local [23][24][25] and nonlocal [11] settings. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/301528188506117-1448901330213_Q64/Oleksandr-Misiats.jpg)
... As we know, in a simply connected domain, a smooth irrotational vector field is a grad field. Although two-dimensional exterior domain is not simply connected, Kozono and his collaborators has established similar results in [19]. Lemma 3.4 (see [19]) The following equations ...
... This approach is based on the analysis of the hydrostatic Stokes operator and the corresponding hydrostatic Stokes semigroup. For a survey on results concerning the deterministic primitive equations using energy estimates, we refer to [38] and for a survey concerning the approach based on evolution equations to [29]. ...
... In relation to the existence of periodic solutions, there are some relevant recent papers. For example, in [32], M. Hieber et al. prove the periodic version of the Da Prato-Grisvard theorem as well as its extension to semi-linear evolution equations. That is to say, from this outcome they proved the existence of strong T -periodic solution of the bidomain model, that is, in the context of the corresponding abstract evolution problem. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/955892795179008-1604914010313_Q64/Patrick-Tolksdorf.jpg)
