# Generalized regularized long wave equation with white noise dispersion

Generalized regularized long wave equation with white noise dispersion In this article, we address the generalized BBM equation with white noise dispersion which reads \begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned} d u - d u x x + u x ∘ d W + u p u x d t = 0 , in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the $$L^\infty _x$$ L x ∞ norm of the solutions decays to zero at $$O(t^{-\frac{1}{6}})$$ O ( t - 1 6 ) as t approaches to $$+\infty$$ + ∞ , by assuming that $$p>8$$ p > 8 and that the initial data is small in $$L^1_x\cap H^4_x$$ L x 1 ∩ H x 4 . This decay rate matches the one for solutions of the linear equation with white noise dispersion. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Stochastical Partial Differential Equations Springer Journals

# Generalized regularized long wave equation with white noise dispersion

, Volume 5 (3) – Jan 9, 2017
24 pages

/lp/springer_journal/generalized-regularized-long-wave-equation-with-white-noise-dispersion-EpTg0g09OB
Publisher
Springer US
Subject
Mathematics; Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis
ISSN
2194-0401
eISSN
2194-041X
D.O.I.
10.1007/s40072-016-0089-7
Publisher site
See Article on Publisher Site

### Abstract

In this article, we address the generalized BBM equation with white noise dispersion which reads \begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned} d u - d u x x + u x ∘ d W + u p u x d t = 0 , in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the $$L^\infty _x$$ L x ∞ norm of the solutions decays to zero at $$O(t^{-\frac{1}{6}})$$ O ( t - 1 6 ) as t approaches to $$+\infty$$ + ∞ , by assuming that $$p>8$$ p > 8 and that the initial data is small in $$L^1_x\cap H^4_x$$ L x 1 ∩ H x 4 . This decay rate matches the one for solutions of the linear equation with white noise dispersion.

### Journal

Stochastical Partial Differential EquationsSpringer Journals

Published: Jan 9, 2017

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