305 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

### 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

### A Blow-Up Criterion for the 3D Euler Equations Via the Euler-Voigt Inviscid Regularization

We propose a new blow-up criterion for the 3D Euler equations of
incompressible fluid flows, based on the 3D Euler-Voigt inviscid
regularization. This criterion is similar in character to a criterion proposed
in a previous work by the authors, but it is stronger, and better adapted for
computational tests. The 3D Euler-Voigt equations enjoy global well-posedness,
and moreover are more tractable to simulate than the 3D Euler equations. A
major advantage of these new criteria is that one only needs to simulate the 3D
Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for
the 3D Euler equations, computationally

### 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 $L^2$ theory for the Gevrey class
regularity is employed; we also employ energy-like methods for a generalized
version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms
(Wiener algebra). After careful comparisons, we observe an indication that the
$\ell^1$ approach provides a better lower bound for the radius of analyticity
of the solutions than the $L^2$ 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
$\mathbb{R}^n$, or on the sphere $\mathbb{S}^{n-1}$

### 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 $\Omega
\subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution,
$(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in
$\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t$, with the
impermeability boundary condition: $u\cdot \vec n =0$ on
$\partial\Omega\times(0,T)$, is of constant energy on the interval $(0,T)$
provided the velocity field $u \in L^3((0,T);
C^{0,\alpha}(\overline{\Omega}))$, with $\alpha>\frac13\,.

### 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
$H^{1/2}$, 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 $L^2-$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 $H^{1/2}-$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 $H^{1/2}$ and the time
average of the square of the $H^{3/2}$ 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

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