# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January & February 2004 , Volume 10 , Issue 1&2

Partial Differential Equations and Applications

A special volume in honor of Mark Vishik's 80th birthday
Guest Editors: Vladimir Chepyzhov, Messoud Efendiev, Alain Miranville and Roger Temam

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2004, 10(1&2): i-vi doi: 10.3934/dcds.2004.10.1i +[Abstract](1414) +[PDF](86.9KB)
Abstract:
The name of Mark Vishik is one of the very first names that I heard when I started research in mathematics in 1964. One year before in 1963, Mark published an article that had a very deep influence on the theory of nonlinear partial differential equations all along the 1960s, although this paper is nearly forgotten by now, and probably very few know about it. Later on I will describe in detail this part of his career that I witnessed during the preparation of my thesis.

2004, 10(1&2): 1-19 doi: 10.3934/dcds.2004.10.1 +[Abstract](1352) +[PDF](220.7KB)
Abstract:
Oscillations in a nonlinear, strongly spatially inhomogenious media are described by scalar semilinear hyperbolic equations with coefficients strongly depending on spatial variables. The spatial patterns of solutions may be very complex, we describe them in terms of binary lattice functions and show that the patterns are preserved by the dynamics. We prove that even when solutions of the equations are not unique, the large-scale spatial patterns of solutions are preserved. We consider arbitrary large spatial domains and show that the number of distinct invariant domains in the function space depends exponentially on the volume of the domain.
2004, 10(1&2): 21-30 doi: 10.3934/dcds.2004.10.21 +[Abstract](1694) +[PDF](126.0KB)
Abstract:
After briefly recalling the pseudo-holomorphic approach in contact form geometry and after sketching the ways with which this approach defines invariants, we introduce another approach, of more technical type, which starts with a variational problem on Legendrian curves. We show how this approach leads also to the definition of a homology.
Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].
Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.
This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].
2004, 10(1&2): 31-52 doi: 10.3934/dcds.2004.10.31 +[Abstract](4658) +[PDF](246.2KB)
Abstract:
The existence of a global attractor in the natural energy space is proved for the semilinear wave equation $u_{t t}+\beta u_t -\Delta u + f(u)=0$ on a bounded domain $\Omega\subset\mathbf R^n$ with Dirichlet boundary conditions. The nonlinear term $f$ is supposed to satisfy an exponential growth condition for $n=2$, and for $n\geq 3$ the growth condition $|f(u)|\leq c_0(|u|^{\gamma}+1)$, where $1\leq\gamma\leq\frac{n}{n-2}$. No Lipschitz condition on $f$ is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser's property, which implies the connectedness of the attractor.
In the case $n\geq 3$ and $\gamma>\frac{n}{n-2}$ the existence of a global attractor is proved under the (unproved) assumption that every weak solution satisfies the energy equation.
2004, 10(1&2): 53-74 doi: 10.3934/dcds.2004.10.53 +[Abstract](1408) +[PDF](252.5KB)
Abstract:
The Navier-Stokes equation driven by heat conduction is studied. It is proven that if the driving force is small then the solutions of the Navier-Stokes equation are ultimately regular. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl numer close to one, we prove the ultimate existence and regularity of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal $\mathcal B$-attractor. Examples of simple $\mathcal B$-attractors from pattern formation are given and a method to study their instabilities proposed.
2004, 10(1&2): 75-88 doi: 10.3934/dcds.2004.10.75 +[Abstract](1519) +[PDF](172.6KB)
Abstract:
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2004, 10(1&2): 89-116 doi: 10.3934/dcds.2004.10.89 +[Abstract](1556) +[PDF](270.5KB)
Abstract:
The Bénard problem, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the temperature is considered. Non-homogeneous boundary conditions, external force and heat source in dual function spaces, and an arbitrary spatial domain (possibly nonsmooth and unbounded) as long as the Poincaré inequality holds on it (channel-like domain) are allowed. Moreover our approach, unlike in previous works, avoids the use of the maximum principle which would be problematic in this context. The mathematical formulation of the problem, the existence of global solution and the existence and finite dimensionality of the global attractor are proved.
2004, 10(1&2): 117-135 doi: 10.3934/dcds.2004.10.117 +[Abstract](1649) +[PDF](210.6KB)
Abstract:
A semigroup $S_t$ of continuous operators in a Hilbert space $H$ is considered. It is shown that the fractal dimension of a compact strictly invariant set $X$ ($X\subset H, S_tX=X$) admits the same estimate as the Hausdorff dimension, namely, both are bounded from above by the Lyapunov dimension calculated in terms of the global Lyapunov exponents. Applications of the results so obtained to the two-dimensional Navier-Stokes equations are given.
2004, 10(1&2): 137-163 doi: 10.3934/dcds.2004.10.137 +[Abstract](1525) +[PDF](386.5KB)
Abstract:
We consider a two--dimensional coupled transmission problem with the conservation laws for compressible viscous flows, where in a subdomain $\Omega_1$ of the flow--field domain $\Omega$ the coefficients modelling the viscosity and heat conductivity are set equal to a small parameter $\varepsilon>0$. The viscous/viscous coupled problem, say $P_\varepsilon$, is equipped with specific boundary conditions and natural transmission conditions at the artificial interface $\Gamma$ separating $\Omega_1$ and $\Omega \setminus \Omega_1$. Here we choose $\Gamma$ to be a line segment. The solution of $P_\varepsilon$ can be viewed as a candidate for the approximation of the solution of the real physical problem for which the dissipative terms are strongly dominated by the convective part in $\Omega_1$. With respect to the norm of uniform convergence, $P_\varepsilon$ is in general a singular perturbation problem. Following the Vishik--Ljusternik method, we investigate here the boundary layer phenomenon at $\Gamma$. We represent the solution of $P_\varepsilon$ as an asymptotic expansion of order zero, including a boundary layer correction. We can show that the first term of the regular series satisfies a reduced problem, say $P_0$, which includes the inviscid/viscous conservation laws, the same initial conditions as $P_\varepsilon$, specific inviscid/viscous boundary conditions, and transmission conditions expressing the continuity of the normal flux at $\Gamma$. A detailed analysis of the problem for the vector--valued boundary layer correction indicates whether additional local continuity conditions at $\Gamma$ are necessary for $P_0$, defining herewith the reduced coupled problem completely. In addition, the solution of $P_0$ (which can be computed numerically) plus the boundary layer correction at $\Gamma$ (if any) provides an approximation of the solution of $P_\varepsilon$ and, hence, of the physical solution as well. In our asymptotic analysis we mainly use formal arguments, but we are able to develop a rigorous analysis for the separate problem defining the correctors. Numerical results are in agreement with our asymptotic analysis.
2004, 10(1&2): 165-176 doi: 10.3934/dcds.2004.10.165 +[Abstract](1381) +[PDF](151.3KB)
Abstract:
We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that relates the total vorticity to the gradient of the back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows). We obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number. Consequently, the change in vertical separation between fluid masses carried by the flow vanishes linearly with the maximal local Rossby number.
2004, 10(1&2): 177-192 doi: 10.3934/dcds.2004.10.177 +[Abstract](1078) +[PDF](183.3KB)
Abstract:
We consider a semilinear differential stochastic equation in a Hilbert space $H$ with a dissipative and Lipschitz nonlinearity. We study the corresponding transition semigroup in a space $L^2(H,\nu)$ where $\nu$ is an invariant measure.
2004, 10(1&2): 193-200 doi: 10.3934/dcds.2004.10.193 +[Abstract](1615) +[PDF](148.9KB)
Abstract:
We study the positivity, for large time, of the solutions to the heat equation $\mathcal Q_a(f,u^0)$:

$\mathcal Q_a(f,u^0)\qquad$ $\partial_tu-\Delta u=au+f(t,x),$ in $Q=]0,\infty [ \times \Omega,$

$u(t,x)=0\qquad$ $(t,x)\in ]0,\infty [ \times \partial \Omega,$

$u(0,x)=u^0(x), \qquad x\in \Omega,$

where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ and $a\in\mathbb R$. We obtain some sufficient conditions for having a finite time $t_p>0$ (depending on $a$ and on the data $u^0$ and $f$ which are not necessarily of the same sign) such that $u(t,x)>0 \forall t>t_p, a. e. x\in\Omega$.

2004, 10(1&2): 201-210 doi: 10.3934/dcds.2004.10.201 +[Abstract](1193) +[PDF](148.5KB)
Abstract:
In this article, we study the generalization of the the decomposition $W_p^m(G)=\mathcal O_p^m(G)\oplus\partial W_{p,0}^{m+1}(G), p>1,m=0,\pm 1,\cdots$ to the case of several complex variables. More precisely, we consider the Lebesgue space $L_2(G)$ and prove that the above decomposition is closely related to the solvability of a complex Neumann problem whose solvability is equivalent to the complex version of Poincaré's inequality.
2004, 10(1&2): 211-238 doi: 10.3934/dcds.2004.10.211 +[Abstract](1620) +[PDF](288.3KB)
Abstract:
Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
2004, 10(1&2): 239-252 doi: 10.3934/dcds.2004.10.239 +[Abstract](1625) +[PDF](169.5KB)
Abstract:
We prove convergence towards an equilibrium state for any globally defined solution for a conserved phase field model. A generalized version of the Lojasiewicz-Simon theorem is used.
2004, 10(1&2): 253-268 doi: 10.3934/dcds.2004.10.253 +[Abstract](1459) +[PDF](192.4KB)
Abstract:
Part of the Kolmogorov-Kraichnan-Batchelor theory of turbulence concerns the average enstrophy flux across wave numbers. To support that theory, rigorous relations involving both the net and one-way flux are established using ensemble averages in [9]. In this note we show that some of these relations hold recurrently, and provide explicit estimates for the time intervals of recurrence which are independent of the solution.
2004, 10(1&2): 269-288 doi: 10.3934/dcds.2004.10.269 +[Abstract](1783) +[PDF](220.5KB)
Abstract:
We discuss some historical background concerning a modified version of the Navier-Stokes equations for the motion of an incompressible fluid. The classical (Newtonian) linear relation between the Cauchy stress tensor and the rate of strain tensor yields the Navier-Stokes equations. Certain nonlinear relations are also consistent with basic physical principals and result in equations with "stronger" dissipation. We describe a class of models that has its genesis in Kolmogorov's similarity hypothesis for 3-dimensional isotropic turbulence and was formulated by Smagorinsky in the meteorological context of rapidly rotating fluids and more generally by Ladyzhenskaya. These models also describe the motion of fluids with shear dependent viscosities and have received considerable attention. We present a dyadic model for such modified Navier-Stokes equations. This model is an example of a hierarchical shell model. Following the treatment of a (non-physically motivated) linear hyper-dissipative model given by Katz-Pavlović, we prove for the dyadic model a bound for the Hausdorff dimension of the singular set at the first time of blow up. The result interpolates between the results of solvability for sufficiently strong dissipation of Ladyzhenskaya, (later strengthened by Nečas et al) and the bound for the dimension of the singular set for the Navier-Stokes equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications of this dyadic model for the modified Navier-Stokes equation themselves.
2004, 10(1&2): 289-314 doi: 10.3934/dcds.2004.10.289 +[Abstract](1827) +[PDF](264.8KB)
Abstract:
We study the problem of stabilization a solution to 3D Navier-Stokes system given in a bounded domain $\Omega$. This stabilization is carried out with help of feedback control defined on a part $\Gamma$ of boundary $\partial \Omega$. We assume that $\Gamma$ is closed 2D manifold without boundary. Here we continuer investigation begun in [6], [7] where stabilization problem for parabolic equation and for 2D Navier-Stokes system was studied.
2004, 10(1&2): 315-336 doi: 10.3934/dcds.2004.10.315 +[Abstract](1624) +[PDF](254.1KB)
Abstract:
We prove a priori estimates in $L^2(0,T;W^{1,2}(\Omega)) \cap L^\infty(Q)$, existence and uniqueness of solutions to Cauchy--Dirichlet problems for parabolic equations

$\frac {\partial \sigma(u)}{\partial t} - \sum_{i=1}^n \frac {\partial}{\partial x_i}${$\rho(u) b_i (t,x,\frac{\partial u}{\partial x})$} $+ a (t,x,u,\frac{\partial u}{\partial x}) = 0,$

$(t,x) \in Q = (0,T) \times \Omega$, where $\rho(u) = \frac{d }{du}\sigma(u)$. We consider solutions $u$ such that $\rho^{\frac{1}{2}}(u) | \frac{\partial u}{\partial x} | \in L^2 (0,T;L^2 (\Omega ) ), \frac {\partial }{\partial t}\sigma(u) \in L^2 ( 0,T;[ W^{1,2} ( \Omega ) ]^\star ).$
Our nonstandard assumption is that log$\rho (u)$ is concave. Such assumption is natural in view of drift diffusion processes for example in semiconductors and binary alloys, where $u$ has to be interpreted as chemical potential and $\sigma$ is a distribution function like $\sigma=e^u$ or $\sigma=\frac {1}{1+e^{-u}}$.

2004, 10(1&2): 337-348 doi: 10.3934/dcds.2004.10.337 +[Abstract](1408) +[PDF](162.9KB)
Abstract:
We consider a Timoshenko model of a viscoelastic beam fixed at the endpoints and subject to nonlinear external forces. The model consists of two coupled second order linear integrodifferential hyperbolic equations that govern the evolution of the lateral displacement $u$ and the total rotation angle $\phi$. We prove that these equations generate a dissipative dynamical system, whose trajectories are eventually confined in a uniform absorbing set, the dissipativity being due to the memory mechanism solely. This fact allows us to state the existence of a uniform compact attractor.
2004, 10(1&2): 349-366 doi: 10.3934/dcds.2004.10.349 +[Abstract](1685) +[PDF](202.0KB)
Abstract:
Recently fourth order equations of the form $u_t = -\nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed for noise reduction and simplification of two dimensional images. The operator $\mathcal G$ is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator $J_\sigma$ is a standard mollifier. Using ODE methods on Sobolev spaces, we prove existence and uniqueness of solutions of this problem for $H^1$ initial data.
2004, 10(1&2): 367-385 doi: 10.3934/dcds.2004.10.367 +[Abstract](1709) +[PDF](244.6KB)
Abstract:
In this paper we will give a proof of Kolmogorov's theorem on the conservation of invariant tori. This proof is close to the one given by Bennettin, Galgani, Giorgilli and Strelcyn in [2]; we follow the outline of their proof, but carry out the steps somewhat differently in several places. In particular, the use of balls rather than polydiscs simplifies several arguments and improves the estimates.
2004, 10(1&2): 387-396 doi: 10.3934/dcds.2004.10.387 +[Abstract](1639) +[PDF](154.1KB)
Abstract:
We establish soliton-like asymptotics for finite energy solutions to classical particle coupled to a scalar wave field. Any solution that goes to infinity as $t\to\infty$ converges to a sum of traveling wave and of outgoing free wave. The convergence holds in global energy norm. The proof uses a non-autonomous integral inequality method.
2004, 10(1&2): 397-422 doi: 10.3934/dcds.2004.10.397 +[Abstract](1501) +[PDF](305.6KB)
Abstract:
In this note we discuss a slight generalization of the following result by Alt and Caffarelli: if the logarithm of the Poisson kernel of a Reifenberg flat chord arc domain is Hölder continuous, then the domain can be locally represented as the area above the graph of a function whose gradient is Hölder continuous. In this note we show that if the Poisson kernel of an unbounded Reifenberg flat chord arc domain is 1 a.e. on the boundary then the domain is (modulo rotation and translation) the upper half plane. This result plays a key role in the study of regularity of the free boundary below the continuous threshold.
2004, 10(1&2): 423-433 doi: 10.3934/dcds.2004.10.423 +[Abstract](1781) +[PDF](162.7KB)
Abstract:
A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets of which converge upper semi continuously to their continuous time counterparts as the maximum time step decreases to zero. The proof involves a Lyapunov function characterizing the uniform pullback attractor of the original system.
2004, 10(1&2): 435-458 doi: 10.3934/dcds.2004.10.435 +[Abstract](2043) +[PDF](247.0KB)
Abstract:
In this paper we investigate a limiting system that arises from the study of steady-states of the Lotka-Volterra competition model with cross-diffusion. The main purpose here is to understand all possible solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done without the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.
2004, 10(1&2): 459-472 doi: 10.3934/dcds.2004.10.459 +[Abstract](1215) +[PDF](194.0KB)
Abstract:
The main objective of this article and the previous articles [2, 3, 7] is to provide a rigorous characterization of the boundary layer separation of 2-D incompressible viscous fluids. First we establish a simple equation linking the separation location and time with the Reynolds number, the external forcing the boundary curvature, and the initial velocity field. Second, we show that external forcing with reverse orientation to the initial velocity field leads to structural bifurcation at a degenerate singular point with integer index of the velocity field at the critical bifurcation time. Necessary and sufficient kinematic conditions are given to identify the case for boundary layer separation.
2004, 10(1&2): 473-496 doi: 10.3934/dcds.2004.10.473 +[Abstract](1646) +[PDF](252.4KB)
Abstract:
An extension to the nonautonomous case of the energy equation method for proving the existence of attractors for noncompact systems is presented. A suitable generalization of the asymptotic compactness property to the nonautonomous case, termed uniform asymptotic compactness, is given, and conditions on the energy equation associated with an abstract class of equations that assure the uniform asymptotic compactness are obtained. This general formulation is then applied to a nonautonomous Navier-Stokes system on an infinite channel past an obstacle, with time-dependent forcing and boundary conditions, and to a nonautonomous, weakly damped, forced Korteweg-de Vries equation on the real line.
2004, 10(1&2): 497-505 doi: 10.3934/dcds.2004.10.497 +[Abstract](1098) +[PDF](142.5KB)
Abstract:
It is shown that some well known functional equations in $\mathbb R^n, n\geq 2$, turn out to be overdetermined. This means that their solutions are uniquely defined if the corresponding relations are fulfilled not in the whole spaces $\mathbb R^n, n\geq 2$, but only at the points of some smooth submanifolds in $\mathbb R^n$.
2004, 10(1&2): 507-515 doi: 10.3934/dcds.2004.10.507 +[Abstract](1494) +[PDF](130.9KB)
Abstract:
This paper provides a new variational characterization of a spatially heteroclinic solution for a family of semilinear elliptic PDE's.
2004, 10(1&2): 517-542 doi: 10.3934/dcds.2004.10.517 +[Abstract](1400) +[PDF](292.8KB)
Abstract:
We present a number of results concerning large-time qualitative behavior of solutions for high-order hyperbolic equations and first-order hyperbolic systems. We discuss the properties of exponential stability and exponential dichotomy, construction of stable, unstable, and center manifolds, Grobman--Hartman type theorems on linearization of the phase portrait, and existence and uniqueness of time-bounded and almost periodic (AP) solutions.
2004, 10(1&2): 543-556 doi: 10.3934/dcds.2004.10.543 +[Abstract](2010) +[PDF](176.6KB)
Abstract:
We study regularity of general and axisymmetric weak solutions of the 3D MHD equations with dissipation and resistance. A general weak solution is shown to be smooth if it satisfies a Serrin condition. The regularity of axisymmetric weak solutions is analyzed through the MHD equations in cylindrical coordinates, whose concrete form is derived here using Gibbs' notion of dyadic product. We establish that it is sufficient to impose conditions on certain components (in cylindrical coordinates) of an axisymmetric weak solution in order for the solution to be regular.
2004, 10(1&2): 557-580 doi: 10.3934/dcds.2004.10.557 +[Abstract](1819) +[PDF](277.5KB)
Abstract:
We consider a nonlinear fourth order parabolic equation with a nonlocal term which describes the time evolution of a flame front. After having established the existence of a global attractor for a corresponding boundary value problem, we prove the existence of inertial sets.

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