1,479 research outputs found
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
On the backward behavior of some dissipative evolution equations
We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows
up in the energy space, backward in time, provided the solution does not belong
to the global attractor. This is a phenomenon contrast to the backward behavior
of the periodic 2D Navier-Stokes equations studied by
Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of
the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we
study the backward behavior of solutions to the damped driven nonlinear
Schrodinger equation, the complex Ginzburg-Landau equation, and the
hyperviscous Navier-Stokes equations. In addition, we provide some physical
interpretation of various backward behaviors of several perturbations of the
KdV equation by studying explicit cnoidal wave solutions. Furthermore, we
discuss the connection between the backward behavior and the energy spectra of
the solutions. The study of backward behavior of dissipative evolution
equations is motivated by the investigation of the Bardos-Tartar conjecture
stated in [5].Comment: 34 page
Non-viscous Regularization of the Davey-Stewartson Equations: Analysis and Modulation Theory
In the present study we are interested in the Davey-Stewartson equations
(DSE) that model packets of surface and capillary-gravity waves. We focus on
the elliptic-elliptic case, for which it is known that DSE may develop a
finite-time singularity. We propose three systems of non-viscous regularization
to the DSE in variety of parameter regimes under which the finite blow-up of
solutions to the DSE occurs. We establish the global well-posedness of the
regularized systems for all initial data. The regularized systems, which are
inspired by the -models of turbulence and therefore are called the
-regularized DSE, are also viewed as unbounded, singularly perturbed
DSE. Therefore, we also derive reduced systems of ordinary differential
equations for the -regularized DSE by using the modulation theory to
investigate the mechanism with which the proposed non-viscous regularization
prevents the formation of the singularities in the regularized DSE. This is a
follow-up of the work of Cao, Musslimani and Titi on the non-viscous
-regularization of the nonlinear Schr\"odinger equation
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