Search

It is shown that the uniform radius of spatial analyticity σ (t) of solutions at time t to the Ostrovsky equation with positive dispersion cannot decay faster than | t | − 4 3 as | t | → ∞ given initial data that is analytic with fixed radius σ 0 . The main ingredients in the proof are almost conservation law for the solution to the Ostrovsky equation in space of analytic functions and space-time dyadic bilinear estimates associated with the Ostrovsky equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the convergence problem of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and the stochastic continuity of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with random data. On the one hand, we establish the pointwise convergence related to the free reduced Ostrovsky equation in Fourier–Lebesgue spaces Ĥ 1 p , p 2 (R) (4 ≤ p < ∞) with rough data. In particular, we show that s ≥ 1 p is the necessary condition for the maximal function estimate in Ĥ s , p 2 (R) , which means that s = 1 p is optimal for rough data. On the other hand, we present the stochastic continuity of free reduced Ostrovsky equation at t = 0 in Fourier–Lebesgue spaces L ̂ r (R) (2 ≤ r < ∞) with random data. [ABSTRACT FROM AUTHOR]

In this paper we investigate the pointwise convergence problem for the Korteweg–de Vries–Benjamin–Ono equation u t + γ H (∂ x 2 u) − β ∂ x 3 u = 0 , (x , t) ∈ R × R , u (x , 0) = f (x) ∈ H s (R) , where γ β > 0. We prove that the solution u (x , t) = U t f (x) converges pointwisely to the initial data f (x) for a.e. x ∈ R when f ∈ H s (R) with s ≥ 1 4 , and that the Hausdorff dimension of the divergence set of points of the solution is α U (s) = 1 − 2 s when 1 4 ≤ s ≤ 1 2 . We also obtain the stochastic continuity for the initial data with much less regularity, i.e. for a large class of the initial data in L 2 (R) , via the randomization technique. [ABSTRACT FROM AUTHOR]