Navier-Stokes equations on the flat cylinder with vorticity production on the boundary

We study the two-dimensional Navier-Stokes system on a flat cylinder with the usual Dirichlet boundary conditions for the velocity field u. We formulate the problem as an infinite system of ODE's for the natural Fourier components of the vorticity, and the boundary conditions are taken into account by adding a vorticity production at the boundary. We prove equivalence to the original Navier-Stokes system and show that the decay of the Fourier modes is exponential for any positive time in the periodic direction, but it is only power-like in the other direction.Comment: 25 page

On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations

In our previous paper [12] (Rev. Math. Phys. 16, 383-420 (2004)), a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work the abstract framework of \cite{uno} is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper this extended framework is applied to theincompressible Navier-Stokes equations, on a torus T^d of any dimension. In this way a number of results are obtained in the setting of the Sobolev spaces H^n(T^d), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e., giving the values of all the necessary constants; this makes a difference with most of the previous literature). Nextly, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their H^n distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).Comment: LaTeX, 84 pages. The final version published in Reviews in Mathematical Physic

Vanishing viscosity limit of navier-stokes equations in gevrey class

In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$

Synchronization of extended systems from internal coherence

A condition for the synchronizability of a pair of PDE systems, coupled through a finite set of variables, is commonly the existence of internal synchronization or internal coherence in each system separately. The condition was previously illustrated in a forced-dissipative system, and is here extended to Hamiltonian systems, using an example from particle physics. Full synchronization is precluded by Liouville's theorem. A form of synchronization weaker than "measure synchronization" is manifest as the positional coincidence of coherent oscillations ("breathers" or "oscillons") in a pair of coupled scalar field models in an expanding universe with a nonlinear potential, and does not occur with a variant of the model that does not exhibit oscillons.Comment: version accepted for publication in PRE (paragraph beginning at the bottom of pg. 5 has been rewritten to suggest unifying principle for synchronizability, applying to both forced-dissipative and Hamiltonian systems; other minor changes

Breakdown of Conformal Invariance at Strongly Random Critical Points

We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system which has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the RG flow.Comment: 9 page

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.Comment: 11 pages, 11 figure

Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain

In this paper we deal with the local null controllability of the N-dimensional Navier-Stokes system with internal controls having one vanishing component. The novelty of this work is that no condition is imposed on the control domain

On a non-isothermal model for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the {\it absolute temperature} \teta, the {\it velocity field} \ub, and the {\it director field} \bd, representing preferred orientation of molecules in a neighborhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier-Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field \bd, where the transport (viscosity) coefficients vary with temperature. The dynamics of \bd is described by means of a parabolic equation of Ginzburg-Landau type, with a suitable penalization term to relax the constraint |\bd | = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field \bd. The proposed model is shown compatible with \emph{First and Second laws} of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data

Near-linear dynamics in KdV with periodic boundary conditions

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction