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. Stochastical Partial Differential Equations Springer Journals

Generalized regularized long wave equation with white noise dispersion

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Springer US
Copyright © 2017 by Springer Science+Business Media New York
Mathematics; Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis
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