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Discrete & Continuous Dynamical Systems - A

2004 , Volume 11 , Issue 1

A special issue on Qualitative Properties of Some Evolution Equations
Guest Editors: D. Bresch, T. Colin, M. Ghil, S. Wang

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Qualitative properties of some evolution equations
D. Bresch , T. Colin , M. Ghil and  Shouhong Wang
2004, 11(1): i-ii doi: 10.3934/dcds.2004.11.1i +[Abstract](25) +[PDF](39.4KB)
Abstract:
This special issue consists of research and expository papers that deal with the qualitative description of solutions of time-dependent partial differential equations. These evolution equations arise from problems in plasma physics, water waves, shock waves, optics, geophysical flows, and nematic crystal polymers.

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From Bloch model to the rate equations
B. Bidégaray-Fesquet , F. Castella and  Pierre Degond
2004, 11(1): 1-26 doi: 10.3934/dcds.2004.11.1 +[Abstract](32) +[PDF](276.5KB)
Abstract:
We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is asymptotically solution to a linear Boltzmann equation, in which transition rates are appropriate time averages of the potential. This study provides a mathematical justification of the approximation of Bloch equations by rate equations, as described in e.g. [Lou91]. The techniques used stem from manipulations on the density matrix and the averaging theory for ordinary differential equations. Diophantine estimates play a key role in the analysis.
Singularity formation in the generalized Benjamin-Ono equation
Jerry Bona and  H. Kalisch
2004, 11(1): 27-45 doi: 10.3934/dcds.2004.11.27 +[Abstract](34) +[PDF](420.8KB)
Abstract:
A Fourier-collocation scheme is used to approximate solutions to the generalized Benjamin-Ono equation $u_t + u^pu_x - H u_{x x} = 0$. The numerical simulation suggests that the equation features smooth solutions that become unbounded in finite time.
Rotating fluids in a cylinder
D. Bresch , B. Desjardins and  D. Gérard-Varet
2004, 11(1): 47-82 doi: 10.3934/dcds.2004.11.47 +[Abstract](35) +[PDF](345.9KB)
Abstract:
We study various singularly perturbed models related to rotating flows in a cylinder. At first we consider the three dimensional incompressible Navier--Stokes equations with turbulent viscosity, in the low Rossby limit. We prove a strong convergence result for ill prepared data, under a geometrical assumption on the cylinder section and a genericity condition on the singular operator.
In a second section, we discuss the compressible Navier--Stokes equations with anisotropic viscosity tensor in the combined low Mach and low Rossby number limit. In the case of well prepared initial data, we prove that global weak solutions with Dirichlet boundary conditions converge to the solution of a two--dimensional quasi-geostrophic model taking into account the compressibility. In the case of ill prepared data, we only show that we can hope a strong convergence result under the same kind of assumptions as in the incompressible case.
Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems
T. Colin and  D. Lannes
2004, 11(1): 83-100 doi: 10.3934/dcds.2004.11.83 +[Abstract](100) +[PDF](268.8KB)
Abstract:
We prove that the Davey-Stewartson approximation (which degenerates into a cubic Schrödinger equation in $1D$) furnishes a good approximation for the exact solution of a wide class of quadratic hyperbolic systems. This approximation remains valid for large times of logarithmic order. We also consider the general case where the polarized component of the mean field needs not to be well-prepared. This is possible by adding to the Davey-Stewarston approximation a long-wave correction, which consists of a wave freely propagated by the long-wave operator associated to the original system.
Remarks on a Smoluchowski equation
Peter Constantin , Ioannis Kevrekidis and  E. S. Titi
2004, 11(1): 101-112 doi: 10.3934/dcds.2004.11.101 +[Abstract](24) +[PDF](206.9KB)
Abstract:
We study the long time dynamics of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. We prove uniform bounds for the long time average of gradients of the distribution function in terms of the nondimensional parameter characterizing the intensity of the potential. In the two dimensional case we obtain lower and upper bounds for the number of steady states. We prove that the system is dissipative and that the potential serves as unique determining mode of the system.
Global in time weak solutions for compressible barotropic self-gravitating fluids
Bernard Ducomet , Eduard Feireisl , Hana Petzeltová and  Ivan Straškraba
2004, 11(1): 113-130 doi: 10.3934/dcds.2004.11.113 +[Abstract](92) +[PDF](238.4KB)
Abstract:
The existence of global in time weak solutions to the Navier-Stokes-Poisson system of barotropic compressible flow is proved. The system takes into account the effect of self-gravitation. Moreover, the case of a non-monotone pressure important in certain applications in astrophysics and the theory of nuclear fluids is included.
Boundary layer and long time stability for multi-D viscous shocks
O. Guès , G. Métivier , M. Williams and  K. Zumbrun
2004, 11(1): 131-160 doi: 10.3934/dcds.2004.11.131 +[Abstract](30) +[PDF](316.0KB)
Abstract:
This is an expository paper whose goal is to provide a detailed survey without the full technicalities of the methods used recently in [GMWZ1, GMWZ2] to prove the existence of curved multi-D viscous shocks, to rigorously justify the small viscosity limit, and to prove long time stability of multidimensional planar viscous shocks.
Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach
Chiu-Ya Lan and  Chi-Kun Lin
2004, 11(1): 161-188 doi: 10.3934/dcds.2004.11.161 +[Abstract](32) +[PDF](264.2KB)
Abstract:
In this article we apply the renormalization group method to study the potential flows of a compressible viscous fluid at small Reynolds number. The derived renormalization equation of order one is a system of reaction convection diffusion equations. The global existence and uniqueness of the weak solutions satisfying the energy inequality are proved following the methodology of Leray. The comparison between the exact solution and its approximation is also discussed.
Asymptotic structure for solutions of the Navier--Stokes equations
Tian Ma and  Shouhong Wang
2004, 11(1): 189-204 doi: 10.3934/dcds.2004.11.189 +[Abstract](33) +[PDF](245.1KB)
Abstract:
We study in this article the large time asymptotic structural stability and structural evolution in the physical space for the solutions of the 2-D Navier-Stokes equations with the periodic boundary conditions. Both the Hamiltonian and block structural stabilities and structural evolutions are considered, and connections to the Lyapunov stability are also given.
Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems
G. Métivier and  K. Zumbrun
2004, 11(1): 205-220 doi: 10.3934/dcds.2004.11.205 +[Abstract](36) +[PDF](241.4KB)
Abstract:
In this paper we prove the continuity of stable subspaces associated to parabolic-hyperbolic boundary value problems, for limiting values of parameters. The analysis is based on the construction performed in [MZ] of Kreiss' type symmetrizers.
Long-time averaging for some conservative PDEs having quadratic nonlinearities
Steve Schochet
2004, 11(1): 221-233 doi: 10.3934/dcds.2004.11.221 +[Abstract](28) +[PDF](218.6KB)
Abstract:
The leading-order asymptotics through times of the order of the inverse square of the amplitude are determined for solutions to certain conservative PDEs having quadratic nonlinearities. This time scale is much longer than in standard averaging results.

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