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Xavier Carvajal

Publications and source records attributed to Xavier Carvajal.

At least 19 recordsLinked to original sources
arXiv2026

Local well-posedness for a system of modified KdV equations in modulation spaces

In this work, we consider the initial value problem (IVP) for a system of modified Korteweg-de Vries (mKdV) equations \begin{equation} \begin{cases} \partial_t v + \partial_x^3 v+ \partial_x (v w^2) = 0, \hspace{0.98 cm} v(x,0)=ψ(x),\\ \partial_t w + α\partial_x^3 w+\partial_x (v^2 w) = 0,\hspace{0.5 cm} w(x,0)=ϕ(x). \end{cases} \end{equation} The main interest is in addressing the well-posedness issues of the IVP when the initial data are considered in the modulation space $M_s^{2,p}(\mathbb{R})$, $p\geq 2$. In the case when $0<α\ne 1$, we derive new trilinear estimates in these spaces and prove that the IVP is locally well-posed for data in $M_s^{2,p}(\mathbb{R})$ whenever $s> \frac14-\frac{1}{p}$ and $p\geq 2$. In deriving the trilinear estimate, the fact that the Fourier supports of the solution components $v$ and $w$ lie on distinct cubic curves, namely $τ= ξ^3$ and $τ= αξ^3$, introduces additional difficulties in handling the resonant case. This makes the analysis substantially different from what one encounters in the single-equation setting. To overcome the difficulties arising in the resonant case, it was necessary to impose the more restrictive condition $s> \frac14-\frac{1}{p}$ on the trilinear estimate, rather than the natural threshold $s> \frac14-\frac{3}{2p}$ , which would otherwise yield sharp local well-posedness for $s>-\frac12$ when $p=2$.

math.AP↗
arXiv2024

Well-posedness for a higher order water wave model on modulation spaces

Considered in this work is the initial value problem (IVP) associated to a higher order water wave model \begin{equation*} \begin{cases} η_t+η_x-γ_1 η_{xxt}+γ_2η_{xxx}+δ_1 η_{xxxxt}+δ_2η_{xxxxx}+\frac{3}{2}ηη_x+γ(η^2)_{xxx}-\frac{7}{48}(η_x^2)_x-\frac{1}{8}(η^3)_x=0,\\ η(x,0) = η_0(x). \end{cases} \end{equation*} The main interest is in addressing the well-posedness issues of the IVP when the given initial data are considered in the modulation space $M_s^{2,p}(\mathbb{R})$ or the $L^p$-based Sobolev spaces $H^{s,p}(\mathbb{R})$, $1\leq p<\infty$. We derive some multilinear estimates in these spaces and prove that the above IVP is locally well-posed for data in $M_s^{2,p}(\mathbb{R})$ whenever $s>1$ and $p\geq 1$, and in $H^{s,p}(\mathbb{R})$ whenever $p\in [1,\infty)$ and $s\geq \max\left\{ \frac1{p}+\frac12, 1 \right\}$. We also use a combination of high-low frequency technique and an {\em a priori estimate}, and prove that the local solution with data in the modulation spaces $M_s^{2,p}(\mathbb{R})$ can be extended globally to the time interval $[0, T]$ for any given $T\gg1$ if $1\leq \frac32-\frac1p <s<2$ or if $(s,p)\in [2, \infty]\times [2, \infty]$.

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arXiv2023

On propagation of regularities and evolution of radius of analyticity in the solution of the fifth order KdV-BBM model

We consider the initial value problem (IVP) associated to a fifth order KdV-BBM type model that describes the propagation of unidirectional water waves. We prove that the regularity in the initial data propagates in the solution, in other words no singularities can appear or disappear in the solution to this model. We also prove the local well-posedness of the IVP in the space of the analytic functions, the so called Gevrey class. Furthermore, we discuss the evolution of radius of analyticity in such class by providing explicit formulas for upper and lower bounds.

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arXiv2023

Sharp global well-posedness for the cubic nonlinear Schrödinger equation with third order dispersion

We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion \begin{equation*} \partial_{t}u+iα\partial^{2}_{x}u- \partial^{3}_{x}u+iβ|u|^{2}u = 0, \quad x,t \in \mathbb{R}, \end{equation*} for given data in the Sobolev space $H^s(\mathbb{R})$. This IVP is known to be locally well-posed for given data with Sobolev regularity $s>-\frac14$ and globally well-posed for $s\geq 0$ [3]. For given data in $H^s(\mathbb{R})$, $0>s> -\frac14$ no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in [3].

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arXiv2020

Sharp well-posedness for a coupled system of mKdV type equations

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $$ \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=ϕ(x), $$ $$ \partial_tw + α\partial_x^3w + \partial_x(v^2w) =0,\ \ w(x,0)=ψ(x),$$ and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\textrm{I}\!\textrm{R})\times H^{k}(\textrm{I}\!\textrm{R})$, $s,k> -\frac12$ and $|s-k|\leq 1/2$, for $α\neq 0,1$. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $C^3$ at the origin, when $s<-1/2$ or $k<-1/2$ or $|s-k|>2$; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1/2$ (b) $k-2s>1$ or $s<-1/2$; (c) $s=k=-1/2 $; (III) the local well-posedness result is sharp in a sense that we can not reduce the proof of the trilinear estimates, proving some related bilinear estimates (as in Tao [19]).

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arXiv2020

Sharp ill-posedness and well-posedness results for dissipative KdV equations on the real line

This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right. \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.

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arXiv2019

Sharp well-posedness for a coupled system of mKdV type equations

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,&v(x,0)=ϕ(x),\\ \partial_tw + α\partial_x^3w + \partial_x(v^2w) =0,& w(x,0)=ψ(x), \end{cases} \end{equation*} and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$, $s> -\frac12$, for $0<α<1$. Our result covers the whole scaling sub-critical range of Sobolev regularity contrary to the case $α=1$, where the local well-posedness holds only for $s\geq \frac14$. We also prove that the local well-posedness result is sharp in two different ways, viz., for $s<-\frac12$ the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow-map that takes initial data to the solution fails to be $C^3$ at the origin. These results hold for $α>1$ as well.

math.AP↗
arXiv2019

On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain

In this work we are interested in the well-posedness issues for the initial value problem associated with a higher order water wave model posed on a pe\-rio\-dic domain $\mathbb{T}$. We derive some multilinear estimates and use them in the contraction mapping argument to prove local well-posedness for initial data in the periodic Sobolev space $H^s(\mathbb{T})$, $s\geq 1$. With some restriction on the parameters appeared in the model, we use the conserved quantity to obtain global well-posedness for given data with Sobolev regularity $s\geq 2$. Also, we use splitting argument to improve the global well-posedness result in $H^s(\mathbb{T})$ for $1\leq s< 2$. Well-posedness result obtained in this work is sharp in the sense that the flow-map that takes initial data to the solution cannot to be continuous for given data in $H^s(\mathbb{T})$, $s< 1$. Finally, we prove a norm-inflation result by showing that the solution corresponding to a smooth initial data may have arbitrarily large $H^s(\mathbb{T})$ norm, with $s<1$, for arbitrarily short time.

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arXiv2019

On sharp global well-posedness and Ill-posedness for a fifth-order KdV-BBM type equation

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp.

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arXiv2015

On the well-posedness of higher order viscous Burgers' equations

We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well-posedness for data with Sobolev regularity below $L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation $p$, and nonlinearity $k+1$, satisfy a relation $p=2k+1$.

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arXiv2013

Sharp local well-posedness of KdV type equations with dissipative perturbations

In this work, we study the initial value problems associated to some linear perturbations of KdV equations. Our focus is in the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.

math.AP↗
arXiv2013

Operators That Attain their Minima

In this paper we study the theory of operators on complex Hilbert spaces, which attain their minimum in the unit sphere. We prove some important results concerning the characterization of the N*, and also AN* operators, see respectively Definition 1.1 and Definition 1.3. The injective property plays an important role in these operators, and shall be established by these classes.

math.FA↗
arXiv2013

Persistence property in weighted Sobolev spaces for nonlinear dispersive equations

We generalize the Abstract Interpolation Lemma proved by the authors in [2]. Using this extension, we show in a more general context, the persistence property for the generalized Korteweg-de Vries equation, see (1.2), in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear dispersive models, for instance the multidimensional nonlinear Schrodinger equation.

math.AP↗
arXiv2013

Well-posedness for a Family of Perturbations of the KDV Equation in Periodic Sobolev Spaces of Negative Order

We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+ηLu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $η>0$, $(Lu)^{\wedge}(k)=-Φ(k)\hat{u}(k)$, $k\in \mathbb{Z}$ and $Φ\in \mathbb{R}$ is bounded above. Particular cases of this problem are the Korteweg-de Vries-Burgers equation for $Φ(k)=-k^2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $Φ(k)=k^2-k^4$, and the Ostrovsky-Stepanyams-Tsimring equation for $Φ(k)=|k|-|k|^3$.

math.AP↗
arXiv2010

Operators that achieve the norm

In this paper we study the theory of operators on complex Hilbert spaces, which achieve the norm in the unit sphere. We prove important results concerning the characterization of the AN operators, see Definition 1.2. The class of AN operators contains the algebra of the compact ones.

math.FA↗
arXiv2010

On uniqueness and decay of solution for Hirota equation

We address the question of the uniqueness of solution to the initial value problem associated to the equation \partial_{t}u+iα\partial^{2}_{x}u+β \partial^{3}_{x}u+iγ|u|^{2}u+δ|u|^{2}\partial_{x}u+εu^{2}\partial_{x}\bar{u} = 0, \quad x,t \in \R, and prove that a certain decay property of the difference $u_1-u_2$ of two solutions $u_1$ and $u_2$ at two different instants of times $t=0$ and $t=1$, is sufficient to ensure that $u_1=u_2$ for all the time.

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