arXiv subjects

Oscar G. Riaño

Publications and source records attributed to Oscar G. Riaño.

3 recordsLinked to original sources

arXiv2019

On a higher dimensional version of the Benjamin--Ono equation

We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1Δu+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish sharp space--time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in $L^2$-Sobolev spaces $H^{s}(\mathbb{R}^d)$, with $s>5/3$ if $d=2$ and $s>d/2+1/2$ if $d\ge 3$. We also provide ill-posedness results.

math.AP↗

arXiv2019

The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

We study the initial value problem associated to a higher dimensional version of the Benjamin-Ono equation. Our purpose is to establish local well-posedness results in weighted Sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow. In consequence, optimal decay rate for this model is determined. A key ingredient is the deduction of a new commutator estimate involving Riesz transforms.

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↗