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Discrete & Continuous Dynamical Systems - S

2010 , Volume 3 , Issue 2


A special issue
Dedicated to Professor Vsevolod Aleksevich Solonnikov on the Occasion of his 75th Birthday

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Preface
Šárka Nečasová, Reimund Rautmann and Werner Varnhorn
2010, 3(2): i-ii doi: 10.3934/dcdss.2010.3.2i +[Abstract](424) +[PDF](1949.6KB)
Abstract:
This Volume of Discrete and Continuous Dynamical Systems is dedicated to Professor Vsevolod Aleksevich Solonnikov on the occasion of his 75th birthday.
   He was born on June 8th, 1933 in Leningrad. Working at the Department of the Steklov Mathematical Institute in today St. Petersburg for more than 50 years he has published nearly 250 scientific papers on the theory of partial differential equations and function theory. Moreover, he was teaching for 20 years at the Chair of Mathematical Physics at the Faculty of Mathematics and Mechanics at Leningrad State University. He has many pupils. Under his supervision, two doctoral and eight candidate theses were defended. He regularly participates in international scientific conferences all over the world. He is speaking at least six European languages.

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Nonstationary Stokes system with variable viscosity in bounded and unbounded domains
Helmut Abels
2010, 3(2): 141-157 doi: 10.3934/dcdss.2010.3.141 +[Abstract](418) +[PDF](265.8KB)
Abstract:
We consider a generalization of the nonstationary Stokes system, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove unique solvability of the nonstationary system with optimal regularity in $L^q$-Sobolev spaces, in particular for an exterior force $f\in L^q(Q_T)$. Moreover, we characterize the domains of fractional powers of some associated Stokes operators $A_q$ and obtain a corresponding result for $f\in L^q(0,T;\mathcal{D}(A_q^\alpha))$. The result holds for a general class of domains including bounded domain, exterior domains, aperture domains, infinite cylinder and asymptotically flat layer with $W^{2-\frac1r}_r$-boundary for some $r>d$ with $r\geq \max(q,q')$.
On the very weak solution for the Oseen and Navier-Stokes equations
Chérif Amrouche and María Ángeles Rodríguez-Bellido
2010, 3(2): 159-183 doi: 10.3934/dcdss.2010.3.159 +[Abstract](400) +[PDF](333.9KB)
Abstract:
We study the existence of very weak solutions regularity for the Stokes, Oseen and Navier-Stokes system when non-smooth Dirichlet boundary data for the velocity are considered in domains of class $C^{1,1}$. In the Navier-Stokes case, the results will be valid for external forces non necessarily small. Regularity results for more regular data will be also discussed.
Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations
Claude Bardos and E. S. Titi
2010, 3(2): 185-197 doi: 10.3934/dcdss.2010.3.185 +[Abstract](360) +[PDF](213.6KB)
Abstract:
A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.
An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.
Luigi C. Berselli
2010, 3(2): 199-219 doi: 10.3934/dcdss.2010.3.199 +[Abstract](417) +[PDF](272.7KB)
Abstract:
We study with elementary tools the stationary 3D Navier-Stokes equations in a flat domain, equipped with Navier (slip without friction) boundary conditions. We prove existence and uniqueness of weak, strong, and very-weak solutions in appropriate Banach spaces and most of the result hold true without restrictions on the size of the data. Results are partially known, but our approach allows us to give rather elementary and self-contained proofs.
The Dirichlet to Neumann map - An application to the Stokes problem in half space
Ihsane Bikri, Ronald B. Guenther and Enrique A. Thomann
2010, 3(2): 221-230 doi: 10.3934/dcdss.2010.3.221 +[Abstract](366) +[PDF](154.4KB)
Abstract:
We illustrate the use of the Dirichlet to Neumann map for elliptic and parabolic problems in the context of the Stokes problems. An analogous representation to that obtained by Solonnikov in [5] for the case of a sphere is given for the half space problem. The validity of this representation is obtained establishing properties of the $\DtN$ map for the Laplace and Heat operators.
A challenging open problem: The inviscid limit under slip-type boundary conditions.
Hugo Beirão da Veiga
2010, 3(2): 231-236 doi: 10.3934/dcdss.2010.3.231 +[Abstract](402) +[PDF](123.7KB)
Abstract:
In these notes we present some results proved in the forthcoming paper [3]. We consider the $\,3-D$ evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, and study the problem of the strong convergence ($ k > 1+\frac{3}{p},$ see below) of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. This problem is still open, except in the case of flat boundaries. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds. The cause of this different behavior is quite subtle.
A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies
Paul Deuring, Stanislav Kračmar and Šárka Nečasová
2010, 3(2): 237-253 doi: 10.3934/dcdss.2010.3.237 +[Abstract](332) +[PDF](237.2KB)
Abstract:
A Green's formula is proved for solutions of a linearized system describing the stationary flow of a viscous incompressible fluid around a rigid body which is rotating and translating. The formula in question is based on the fundamental solution obtained by integrating the time variable in the fundamental solution of the corresponding evolutionary problem.
An existence result for non-Newtonian fluids in non-regular domains
Lars Diening and Michael Růžička
2010, 3(2): 255-268 doi: 10.3934/dcdss.2010.3.255 +[Abstract](424) +[PDF](238.7KB)
Abstract:
We show the existence of weak solutions to the steady system describing the motion of certain non-Newtonian fluids in non-regular domains. This generalizes previous results for Lipschitz continuous domains. In the proof we combine a localization of the Lipschitz truncation method with a domain decomposition theorem, which enables to extend results known for nice domains to John domains.
Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations
Andrei Fursikov
2010, 3(2): 269-289 doi: 10.3934/dcdss.2010.3.269 +[Abstract](398) +[PDF](322.9KB)
Abstract:
Local existence theorem of smooth solution $v(t,\cdot), t\in \R_+$ for 3D Navier-Stokes equations is proved, when initial data belongs to a certain unbounded ellipsoid of suitable function space. Unboundedness of stable invariant manifolds for 3D Navier-Stokes equations is proved as well.
Remarks on the $L^p$-approach to the Stokes equation on unbounded domains
Matthias Geissert, Horst Heck, Matthias Hieber and Okihiro Sawada
2010, 3(2): 291-297 doi: 10.3934/dcdss.2010.3.291 +[Abstract](358) +[PDF](166.5KB)
Abstract:
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.
$L^\infty$-estimates for parabolic systems with VMO-coefficients
Horst Heck, Matthias Hieber and Kyriakos Stavrakidis
2010, 3(2): 299-309 doi: 10.3934/dcdss.2010.3.299 +[Abstract](451) +[PDF](187.8KB)
Abstract:
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.
On the local strong solutions for the FENE dumbbell model
Ondřej Kreml and Milan Pokorný
2010, 3(2): 311-324 doi: 10.3934/dcdss.2010.3.311 +[Abstract](327) +[PDF](201.6KB)
Abstract:
We consider a model for the polymeric fluid which has recently been studied in [12]. We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regularity assumptions on the initial data than in the mentioned paper. The main difference in our approach is the use of the $L^p$ theory for the Stokes system.
The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions
Petr Kučera
2010, 3(2): 325-337 doi: 10.3934/dcdss.2010.3.325 +[Abstract](371) +[PDF](196.7KB)
Abstract:
In this paper we deal with the system of periodic Navier-Stokes equations with mixed boundary conditions. We define Banach spaces XP and YP , respectively, the space of "possible'' solutions of this problem and the space of its data. We define the operator NP : Xp $\to$ YP and formulate our problem in terms of operator equations. Let u $\in$ XP and gP u : XP $\to$ YP be the Frechet derivative of NP at u . Denote by MR the set of all functions u such that gPu is one-to-one and onto YP . We prove that MR is weakly dense and weakly open.
On a comprehensive class of linear material laws in classical mathematical physics
Rainer Picard
2010, 3(2): 339-349 doi: 10.3934/dcdss.2010.3.339 +[Abstract](308) +[PDF](153.3KB)
Abstract:
A class of evolutionary problems is considered, which covers a number of diverse initial boundary value problems of classical mathematical physics. The claim that this class is indeed to a large extent sufficiently general is exemplified by some specific models for visco-elastic solids.
An alpha model for compressible fluids
Paolo Secchi
2010, 3(2): 351-359 doi: 10.3934/dcdss.2010.3.351 +[Abstract](337) +[PDF](153.0KB)
Abstract:
We consider a barotropic compressible generalization of the Lagrangian averaged Euler-alpha models, obtained by D.D. Holm in [2]. The model extends to the compressible case the Euler-alpha closure equations for incompressible ideal fluids. The alpha model that we consider is a coupled parabolic-elliptic system; we show that it admits local strong solutions defined for small time.
On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3
Zdeněk Skalák
2010, 3(2): 361-370 doi: 10.3934/dcdss.2010.3.361 +[Abstract](416) +[PDF](166.0KB)
Abstract:
Let $A$ be the Stokes operator. We show as the main result of the paper that if $w$ is a global weak solution to the Navier-Stokes equations satisfying the strong energy inequality, $\beta \in [0,1/2]$ and $\alpha \in [\beta,\infty)$, then there exist $t_0 \ge 1$, $C_1>1$ and $\delta_1 \in (0,1)$ such that

$ ||A^\alpha w(t)|| \le C_1 ||A^\beta w(t+\delta)|| $

for every $t \ge t_0$ and every $\delta \in [0,\delta_1]$.

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