scholarly journals On space-time properties of solutions for nonlinear evolutionary equations with random initial data

2021 ◽  
Vol 3 (1) ◽  
pp. 11-20
Author(s):  
Kyrill I. Vaninsky
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Nonlinear Wave Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Random Initial Data ◽  
Evolutionary Equations ◽  
Hölder Exponents ◽  
Kdv Type Equations

We consider space-time properties of periodic solutions of nonlinear wave equations, nonlinear Schrödinger equations and KdV-type equations with initial data from the support of the Gibbs’ measure. For the wave and Schrödinger equations we establish the best Hölder exponents. We also discuss KdV-type equations which are more difficult due to a presence of the derivative in the nonlinearity.

scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

scholarly journals On nonlinear Schrödinger equations with random initial data

2022 ◽  
Vol 4 (4) ◽  
pp. 1-14
Author(s):  
Mitia Duerinckx ◽  
◽  
Keyword(s):  
Initial Data ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Well Posedness ◽  
Schrödinger Equations ◽  
Random Initial Data ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Spatially Homogeneous ◽  
The Continuum

<abstract><p>This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.</p></abstract>

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

scholarly journals A SYSTEM OF COUPLED SCHRÖDINGER EQUATIONS WITH TIME-OSCILLATING NONLINEARITY

2012 ◽  
Vol 23 (11) ◽  
pp. 1250119 ◽  
Author(s):  
X. CARVAJAL ◽  
P. GAMBOA ◽  
M. PANTHEE
Keyword(s):  
Nonlinear Optics ◽  
Initial Data ◽  
Periodic Function ◽  
Initial Value Problem ◽  
Coupled System ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Periodic Functions ◽  
Initial Value ◽  
Physical Problems

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear Schrödinger equations [Formula: see text] where θ1 and θ2 are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data φ, ψ ∈ H1(ℝn), as |ω| → ∞, the solution (uω, vω) of the above IVP converges to the solution (U, V) of the IVP associated to [Formula: see text] with the same initial data, where I(g) is the average of the periodic function g. Moreover, if the solution (U, V) is global and bounded, then we prove that the solution (uω, vω) is also global provided |ω| ≫ 1.

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