arXiv subjects

Ricardo A. Pastrán

Publications and source records attributed to Ricardo A. Pastrán.

3 recordsLinked to original sources

arXiv2023

Spatial decay properties for a model in shear flows posed on the cylinder

We study spatial decay properties for solutions of the Pelinovski-Stepanyants equation posed on the cylinder. We establish the maximum polynomial decay admissible for solutions of such a model. It is verified that the equation on the cylinder propagates polynomial weights with different restrictions than the model set in $\mathbb{R}^2$. For example, a local well-posedness theory is deduced which contains the line solitary wave provided by solutions of the Benjamin-Ono equation extended to the cylinder. Key results for our analysis are obtained from a detailed study of the dispersive effects of the linear equation in weighted Sobolev spaces. The results in this paper appear to be one of the first studies of polynomial decay for nonlocal models in the cylinder.

math.AP↗

arXiv2019

Well-posedness for Fractional Growth-Dissipative Benjamin-Ono Equations

This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}u_{xx}-(D_x^α-D_x^β)u+uu_x=0$, $0< α< β$. When $1<β<2$, we prove GWP in $H^s(\mathbb{R})$, $s>-β/4$. For $β\geq 2$, we show GWP in $H^s(\mathbb{R})$, $s>\max\{3/2-β, \, -β/2\}$. We establish that our results are sharp in the sense that the flow map $u_0\mapsto u$ fails to be $C^2$ in $H^s(\mathbb{R})$, for $s<-β/2$, and it fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<\min\{3/2-β, \, -β/4\}$. When $0< β<1$, we show ill-posedness in $H^s(\mathbb{R})$, $s\in \mathbb{R}$. Finally, if $β>3/2$, we prove GWP in $H^s(\mathbb{T})$, $s>\max\{3/2-β, \, -β/2\}$, and we deduce lack of $C^2$ regularity in $H^s(\mathbb{T})$ when $s<-β/2$, in particular we get sharp results when $β\geq 3$.

math.AP↗

arXiv2016

On the Well-posedness for the Chen-Lee equation in periodic Sobolev spaces

We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation $u_t+uu_x+β\mathcal{H}u_{xx}+η(\mathcal{H}u_x - u_{xx})=0$, where $x\in \mathbb{T}$, $t> 0$, $η>0$ and $\mathcal{H}$ denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for any $s>-\frac{1}{2}$. We also prove some ill-posedness issues when $s<-1$.

math.AP↗