1,412 research outputs found

### 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

### 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

### 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

### 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

### A convergent family of approximate inertial manifolds

AbstractA new method of construction of approximate inertial manifolds (AIMs) is derived for a very general class of evolution partial differential equations. We construct a family (MN)NÏµN and show that when the spectral gap condition holds, it converges to an exact inertial manifold. When it does not hold, we prove that the attractorâ€”when it existsâ€”is contained in a thin neighborhood of the AIM MN and when N is large, the thinness decreases exponentially with respect to the dimension of MN

### Remarks on the Prandtl Equation for a Permeable Wall

The goal of this article is to study the boundary layer for a flow in a channel with permeable walls. Observing that the Prandtl equation can be solved almost exactly in this case, we are able to derive rigorously a number of results concerning the boundary layer and the convergence of the Navier-Stokes equations to the Euler equations. We indicate also how to derive higher order terms in the inner and outer expansions with respect to the kinematic viscosity v

### Slow manifolds and invariant sets of the primitive equations

The authors review, in a geophysical setting, several recent mathematical results on the forcedâ€“dissipative hydrostatic primitive equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence and regularity (smoothness) results and describing their implications for the long-time behavior of the solution. These results are used to show how the solution of the primitive equations in a periodic box comes close to geostrophic balance as t â†’ âˆž. Then a review follows of how geostrophic balance could be extended to higher orders in the Rossby number, and it is shown that the solution of the primitive equations also satisfies a higher-order balance up to an exponentially small error. Finally, the connection between balance dynamics in the primitive equations and its global attractor, which is the only known invariant set (for a sufficiently general forcing), is discussed

### Boundary Layers in Channel Flow with Injection and Suction

We present a rigorous result regarding the boundary layer associated with the incompressible Newtonian channel flow with injection and suction. Â© 2000 Elsevier Science Ltd. All rights reserved

### Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly non-characteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (non-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus non-characteristic. the form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e-Uz/Îµ where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and Îµ is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J. 35, No. 2, 209-230) where the convergence in L2 of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. in the two-dimensional case we are able to derive the physically relevant uniform in space (Lâˆž norm) estimates of the boundary layer. the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids. Â© 2002 Elsevier Science (USA)

### The Convergence of the Solutions of the Navier-Stokes Equations to that of the Euler Equations

In this article, we establish partial results concerning the convergence of the solutions of the Navier-Stokes equations to that of the Euler equations. Convergence is proved in space dimension two under a physically reasonable assumption, namely that the gradient of the pressure remains bounded at the boundary as the Reynolds number converges to infinity

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