6,386 research outputs found
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations
We study the global regularity, for all time and all initial data in
, of a recently introduced decimated version of the incompressible 3D
Navier-Stokes (dNS) equations. The model is based on a projection of the
dynamical evolution of Navier-Stokes (NS) equations into the subspace where
helicity (the scalar product of velocity and vorticity) is sign-definite.
The presence of a second (beside energy) sign-definite inviscid conserved
quadratic quantity, which is equivalent to the Sobolev norm, allows
us to demonstrate global existence and uniqueness, of space-periodic solutions,
together with continuity with respect to the initial conditions, for this
decimated 3D model. This is achieved thanks to the establishment of two new
estimates, for this 3D model, which show that the and the time
average of the square of the norms of the velocity field remain
finite. Such two additional bounds are known, in the spirit of the work of H.
Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing
well-posedness for the 3D NS equations. Furthermore, they are directly linked
to the helicity evolution for the dNS model, and therefore with a clear
physical meaning and consequences
Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach
We study the persistence of the Gevrey class regularity of solutions to
nonlinear wave equations with real analytic nonlinearity. Specifically, it is
proven that the solution remains in a Gevrey class, with respect to some of its
spatial variables, during its whole life-span, provided the initial data is
from the same Gevrey class with respect to these spatial variables. In
addition, for the special Gevrey class of analytic functions, we find a lower
bound for the radius of the spatial analyticity of the solution that might
shrink either algebraically or exponentially, in time, depending on the
structure of the nonlinearity. The standard theory for the Gevrey class
regularity is employed; we also employ energy-like methods for a generalized
version of Gevrey classes based on the norm of Fourier transforms
(Wiener algebra). After careful comparisons, we observe an indication that the
approach provides a better lower bound for the radius of analyticity
of the solutions than the approach. We present our results in the case of
period boundary conditions, however, by employing exactly the same tools and
proofs one can obtain similar results for the nonlinear wave equations and the
nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain
domains and manifolds without physical boundaries, such as the whole space
, or on the sphere
Entire solutions of hydrodynamical equations with exponential dissipation
We consider a modification of the three-dimensional Navier--Stokes equations
and other hydrodynamical evolution equations with space-periodic initial
conditions in which the usual Laplacian of the dissipation operator is replaced
by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at
high wavenumbers . Using estimates in suitable classes of analytic
functions, we show that the solutions with initially finite energy become
immediately entire in the space variables and that the Fourier coefficients
decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any . The
same result holds for the one-dimensional Burgers equation with exponential
dissipation but can be improved: heuristic arguments and very precise
simulations, analyzed by the method of asymptotic extrapolation of van der
Hoeven, indicate that the leading-order asymptotics is precisely of the above
form with . The same behavior with a universal constant
is conjectured for the Navier--Stokes equations with exponential
dissipation in any space dimension. This universality prevents the strong
growth of intermittency in the far dissipation range which is obtained for
ordinary Navier--Stokes turbulence. Possible applications to improved spectral
simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres
Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
We consider the Cauchy problem for the incompressible Navier-Stokes equations
in for a one-parameter family of explicit scale-invariant
axi-symmetric initial data, which is smooth away from the origin and invariant
under the reflection with respect to the -plane. Working in the class of
axi-symmetric fields, we calculate numerically scale-invariant solutions of the
Cauchy problem in terms of their profile functions, which are smooth. The
solutions are necessarily unique for small data, but for large data we observe
a breaking of the reflection symmetry of the initial data through a
pitchfork-type bifurcation. By a variation of previous results by Jia &
\v{S}ver\'ak (2013) it is known rigorously that if the behavior seen here
numerically can be proved, optimal non-uniqueness examples for the Cauchy
problem can be established, and two different solutions can exists for the same
initial datum which is divergence-free, smooth away from the origin, compactly
supported, and locally -homogeneous near the origin. In particular,
assuming our (finite-dimensional) numerics represents faithfully the behavior
of the full (infinite-dimensional) system, the problem of uniqueness of the
Leray-Hopf solutions (with non-smooth initial data) has a negative answer and,
in addition, the perturbative arguments such those by Kato (1984) and Koch &
Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin,
Ladyzhenskaya and others, already give essentially optimal results. There are
no singularities involved in the numerics, as we work only with smooth profile
functions. It is conceivable that our calculations could be upgraded to a
computer-assisted proof, although this would involve a substantial amount of
additional work and calculations, including a much more detailed analysis of
the asymptotic expansions of the solutions at large distances.Comment: 31 pages, 19 figure
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