arXiv subjects
Publications and source records attributed to Gustavo Ponce.
We study persistence properties of solutions of the Benjamin-Ono equation in weighted Sobolev spaces. Roughly, we show that for $\beta<7/2$, the solution $u(x,t)$ of the BO remains in the space $L^2(|x|^{2\beta} dx)$ if and only if its data $u(x,0)$ belongs to this space and it is regular enough, i.e. $u_0\in H^{\beta}(\mathbb R)$.
We consider solutions to the initial value problem associated to the intermediate long wave (ILW) equation. We establish persistence properties of the solution flow in weighted Sobolev spaces, and show that they are sharp. We also deal with the long time dynamics of large solutions to the ILW equation. Using virial techniques, we describe regions of space where the energy of the solution must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.
We study unique continuation properties of solutions to the b-family of equations. This includes the Camassa-Holm and the Degasperi-Procesi models. We prove that for both, the initial value problem and the periodic boundary value problem, the unique continuation results found in \cite{LiPo} are optimal. More precisely, the result established there for the constant $c_0=0$ fails for any constant $c_0\neq 0$.
This work is concerned with the Benjamin-Bona-Mahony equation. This model was deduced as an approximation to the Korteweg-de Vries equation in the description of unidirectional propagation of long waves. Our goal here is to study unique continuation and regularity properties on solutions to the associated initial value problem and initial periodic boundary value problems.
We consider the long time dynamics of large solutions to the Benjamin-Ono equation. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.
We consider unique continuation properties of solutions to a family of evolution equations. Our interest is mainly on nonlinear non-local models. This class contains the Benjamin-Ono, the Intermediate Long Wave, the Camassa-Holm, the dispersion generalized Benjamin-Ono and non-local Schrödinger equations as well as their generalizations. We shall review, discuss, expand, and comment on several results. In addition, we shall state some open questions concerning these results and their techniques.
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,\,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$ such that for some non-empty open set $Ω\subset \mathbb R^n\times[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) \in Ω$, then $u_1(x,t)=u_2(x,t)$ for any $(x,t)\in\mathbb R^n\times[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in \cite{GhSaUh}, and some extensions obtained here.
We show that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the Intermediate Long Wave equation converge to zero locally in an increasing-in-time region of space of order $\,t/\log(t)$. Also, for solutions with a mild $L^1$-norm growth in time is established that its limit infimum converge to zero, as time goes to infinity. This confirms the non existence of breathers and other solutions for the ILW model moving with a speed "slower" than a soliton. We also prove that in the far field linearly dominated region, the $L^2$ norm of the solution also converges to zero as time approaches infinity. In addition, we deduced several scenarios for which the initial value problem associated to the generalized Benjamin-Ono and the generalized Intermediate Long Wave equations cannot possess time periodic solutions (breathers). Finally, as it was previously demonstrated in solutions of the KdV and BO equations, we establish the following propagation of regularity result : if the datum $u_0\in H^{3/2+}(\mathbb R)\cap H^m((x_0,\infty))$, for some $\;x_0\in\mathbb R,\,m\in Z^+,\,m\geq 2$, then the corresponding solution $u(t,\cdot)$ of the Intermediate Long Wave equation belongs to $H^m(β,\infty)$, for any $t>0$ and $β\in\mathbb R$.
It is shown that if $\,u(x,t)\,$ is a solution of the initial value problem for the Camassa-Holm equation which vanishes in an open set $\,Ω\subset \mathbb R\times [0,T]$, then $\,u(x,t)=0,\,(x,t)\in\mathbb R\times [0,T]$. This result also applies to solutions of the initial periodic boundary value problems associated to the Camassa-Holm equation. The argument of proof can be placed in a general setting to extend the above results to a class of non-linear non-local 1-dimensional models which includes the Degasperis-Procesi equation.
We show that for any uniformly bounded in time $H^1\cap L^1$ solution of the dispersive generalized Benjamin-Ono equation, the limit infimum, as time $t$ goes to infinity, converges to zero locally in an increasing-in-time region of space of order $t/\log t$. This result is in accordance with the one established by Muñoz and Ponce \cite{MP1} for solutions of the Benjamin-Ono equation. Similar to solutions of the Benjamin-Ono equation, for a solution of the dispersive generalized Benjamin-Ono equation, with a mild $L^1$-norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending on the rate of growth of its $L^1$-norm. As a consequence, the existence of breathers or any other solution for the dispersive generalized Benjamin-Ono equation moving with a speed "slower" than a soliton is discarded. In our analysis the use of commutators expansions is essential.
We prove that if $u_1,\,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)\in\R \times [0,T]$ which agree in an open set $Ω\subset \R \times [0,T]$, then $u_1\equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
In this note we shall continue our study on the initial value problem associated for the generalized derivative Schrödinger (gDNLS) equation $$ \partial_tu=i\partial_x^2u + μ\,|u|^α\partial_x u, \hskip10pt x,t\in\mathbb{R}, \hskip5pt 0<α\le 1\;\; {\rm and}\;\; |μ|=1. $$ Inspiring by Cazenave-Naumkin's works we shall establish the local well-posedness for a class of data of arbitrary size in an appropriate weighted Sobolev space, thus removing the size restriction on the data required in our previous work. The main new tool in the proof is the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation with lower order variable coefficients established by Kenig-Ponce-Vega.
We prove that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^1\cap L^1$ solution to the Benjamin-Ono equation converge to zero locally in an increasing-in-time region of space of order $\,t/\log t$. Also for a solution with a mild $L^1$-norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending of the rate of growth of its $L^1$-norm. In particular, we discard the existence of breathers and other solutions for the BO model moving with a speed \lq\lq slower" than a soliton.
In this paper our first aim is to identify a large class of non-linear functions $\,f(\cdot)\,$ for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or "small" breathers solutions. Also we prove that all small, uniformly in time $L^1\cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $x\ll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.
We consider the IVP associated to the generalized KdV equation with low degree of non-linearity \begin{equation*} \partial_t u + \partial_x^3 u \pm |u|^α\partial_x u = 0,\; x,t \in \mathbb{R},\;α\in (0,1). \end{equation*} By using an argument similar to that introduced by Cazenave and Naumkin [2] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [3] in solutions of the $k$-generalized KdV equation.
In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger equations \begin{equation*} \p_tu=i\p_x^2u + μ\,|u|^{\a}\p_xu, \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |μ|=1, \end{equation*} and \begin{equation*} \p_tu=i\p_x^2u + μ\,\p_x\big(|u|^{\a}u\big), \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |μ|=1. \end{equation*} Following the argument introduced by Cazenave and Naumkin \cite{Cazenave} we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega in \cite{KPV1}.
We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions". In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in a paper of Himonas, Misiołek, Ponce, and Zhou extend to the class of solutions considered here.
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $\,l\in \mathbb R^+$.