scholarly journals Solving the 4NLS with white noise initial data

2020 ◽  
Vol 8 ◽  
Author(s):  
Tadahiro Oh ◽  
Nikolay Tzvetkov ◽  
Yuzhao Wang
Keyword(s):  
Invariant Measure ◽  
Initial Data ◽  
White Noise ◽  
Fourth Order ◽  
Random Initial Data ◽  
Fourier Restriction ◽  
Singular Component ◽  
Gauge Transform ◽  
Nonlinear Decomposition ◽  
Noise Measure

Abstract We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

scholarly journals Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Three Dimensional ◽  
Space Time ◽  
Symmetry Analysis ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

scholarly journals A Characterization of White Noise Test Functionals

1991 ◽  
Vol 121 ◽  
pp. 185-194 ◽  
Author(s):  
H.-H. Kuo ◽  
J. Potthoff ◽  
L. Streit
Keyword(s):  
White Noise ◽  
Noise Test ◽  
Noise Measure

In a recent paper [PS 89], two of the present authors have found a characterization of a certain space (ℐ)* of generalized functionals of white noise, i.e. generalized functionals on ℐ′(R) equipped with the σ-algebra ℬ generated by its cyclinder sets and with the white noise measure μ given by

scholarly journals SCALING LIMITS FOR TIME-FRACTIONAL DIFFUSION-WAVE SYSTEMS WITH RANDOM INITIAL DATA

2010 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
GI-REN LIU ◽  
NARN-RUEIH SHIEH
Keyword(s):  
Initial Data ◽  
Initial Conditions ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Equation System ◽  
Random Initial Data ◽  
Diffusion Type ◽  
Diffusion Wave ◽  
Decoupling Method ◽  
Time Space

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

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