534 research outputs found
Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion
The three--dimensional incompressible viscous Boussinesq equations, under the
assumption of hydrostatic balance, govern the large scale dynamics of
atmospheric and oceanic motion, and are commonly called the primitive
equations. To overcome the turbulence mixing a partial vertical diffusion is
usually added to the temperature advection (or density stratification)
equation. In this paper we prove the global regularity of strong solutions to
this model in a three-dimensional infinite horizontal channel, subject to
periodic boundary conditions in the horizontal directions, and with
no-penetration and stress-free boundary conditions on the solid, top and
bottom, boundaries. Specifically, we show that short time strong solutions to
the above problem exist globally in time, and that they depend continuously on
the initial data
Determining Projections and Functionals for Weak Solutions of the Navier-Stokes Equations
In this paper we prove that an operator which projects weak solutions of the
two- or three-dimensional Navier-Stokes equations onto a finite-dimensional
space is determining if it annihilates the difference of two "nearby" weak
solutions asymptotically, and if it satisfies a single appoximation inequality.
We then apply this result to show that the long-time behavior of weak solutions
to the Navier-Stokes equations, in both two- and three-dimensions, is
determined by the long-time behavior of a finite set of bounded linear
functionals. These functionals are constructed by local surface averages of
solutions over certain simplex volume elements, and are therefore well-defined
for weak solutions. Moreover, these functionals define a projection operator
which satisfies the necessary approximation inequality for our theory. We use
the general theory to establish lower bounds on the simplex diameters in both
two- and three-dimensions. Furthermore, in the three dimensional case we make a
connection between their diameters and the Kolmogoroff dissipation small scale
in turbulent flows.Comment: Version of frequently requested articl
Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations
In light of the question of finite-time blow-up vs. global well-posedness of
solutions to problems involving nonlinear partial differential equations, we
provide several cautionary examples which indicate that modifications to the
boundary conditions or to the nonlinearity of the equations can effect whether
the equations develop finite-time singularities. In particular, we aim to
underscore the idea that in analytical and computational investigations of the
blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary
conditions may need to be taken into greater account. We also examine a
perturbation of the nonlinearity by dropping the advection term in the
evolution of the derivative of the solutions to the viscous Burgers equation,
which leads to the development of singularities not present in the original
equation, and indicates that there is a regularizing mechanism in part of the
nonlinearity. This simple analytical example corroborates recent computational
observations in the singularity formation of fluid equations
Large dispersion, averaging and attractors: three 1D paradigms
The effect of rapid oscillations, related to large dispersion terms, on the
dynamics of dissipative evolution equations is studied for the model examples
of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three
different scenarios of this effect are demonstrated. According to the first
scenario, the dissipation mechanism is not affected and the diameter of the
global attractor remains uniformly bounded with respect to the very large
dispersion coefficient. However, the limit equation, as the dispersion
parameter tends to infinity, becomes a gradient system. Therefore, adding the
large dispersion term actually suppresses the non-trivial dynamics. According
to the second scenario, neither the dissipation mechanism, nor the dynamics are
essentially affected by the large dispersion and the limit dynamics remains
complicated (chaotic). Finally, it is demonstrated in the third scenario that
the dissipation mechanism is completely destroyed by the large dispersion, and
that the diameter of the global attractor grows together with the growth of the
dispersion parameter
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
Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
The goal of this note is to show that, also in a bounded domain , with , any weak solution,
, of the Euler equations of ideal incompressible fluid in
, with the
impermeability boundary condition: on
, is of constant energy on the interval
provided the velocity field , with $\alpha>\frac13\,.
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