# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

June 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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2010, 3(2): i-ii doi: 10.3934/dcdss.2010.3.2i +[Abstract](1538) +[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.

2010, 3(2): 141-157 doi: 10.3934/dcdss.2010.3.141 +[Abstract](1796) +[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 $r\geq \max(q,q')$.
2010, 3(2): 159-183 doi: 10.3934/dcdss.2010.3.159 +[Abstract](1805) +[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.
2010, 3(2): 185-197 doi: 10.3934/dcdss.2010.3.185 +[Abstract](1569) +[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.
2010, 3(2): 199-219 doi: 10.3934/dcdss.2010.3.199 +[Abstract](1742) +[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.
2010, 3(2): 221-230 doi: 10.3934/dcdss.2010.3.221 +[Abstract](1654) +[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.
2010, 3(2): 231-236 doi: 10.3934/dcdss.2010.3.231 +[Abstract](1709) +[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.
2010, 3(2): 237-253 doi: 10.3934/dcdss.2010.3.237 +[Abstract](1553) +[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.
2010, 3(2): 255-268 doi: 10.3934/dcdss.2010.3.255 +[Abstract](1824) +[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.
2010, 3(2): 269-289 doi: 10.3934/dcdss.2010.3.269 +[Abstract](1545) +[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.
2010, 3(2): 291-297 doi: 10.3934/dcdss.2010.3.291 +[Abstract](1461) +[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.
2010, 3(2): 299-309 doi: 10.3934/dcdss.2010.3.299 +[Abstract](1856) +[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.
2010, 3(2): 311-324 doi: 10.3934/dcdss.2010.3.311 +[Abstract](1376) +[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.
2010, 3(2): 325-337 doi: 10.3934/dcdss.2010.3.325 +[Abstract](1791) +[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.
2010, 3(2): 339-349 doi: 10.3934/dcdss.2010.3.339 +[Abstract](1352) +[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.
2010, 3(2): 351-359 doi: 10.3934/dcdss.2010.3.351 +[Abstract](1592) +[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.
2010, 3(2): 361-370 doi: 10.3934/dcdss.2010.3.361 +[Abstract](1839) +[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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