arXiv subjects

Márcio Cavalcante

Publications and source records attributed to Márcio Cavalcante.

At least 19 recordsLinked to original sources

arXiv2026

Local well-posedness for the KdV equation on the half-line at the critical regularity $H^{-\frac34}$

We prove local well-posedness for the initial-boundary-value problem for the Korteweg--de Vries equation on the right half-line at the critical regularity $s=-3/4$, with initial and boundary data in $H^{-3/4}(\mathbb R_+)$ and $H^{1/12}(\mathbb R_+)$, respectively. This reaches the endpoint left open in the previous half-line results of Holmer (2006) and Bona, Sun, and Zhang (2006), who established local well-posedness for $s>-3/4$. The proof is based on a contraction argument in the Besov-type Bourgain space introduced by Kishimoto. The main difficulty is to construct a boundary forcing operator compatible with the endpoint $b=1/2$ and the required $\ell^1$-summability in the modulation variable. We construct such an operator using a Laplace-transform representation of the linear half-line problem and a carefully designed extension to $x<0$. The extension combines the required regularity across the boundary with prescribed vanishing spatial moments, yielding favorable low-frequency behavior of the associated kernel. A pointwise kernel estimate then provides the crucial dyadic summability needed for the $X^{-3/4,1/2,1}$ estimate. Together with a time-trace estimate for the Duhamel term at $b=1/2$, this yields the critical local well-posedness result. The ideas used here have prospects to be used in other nonlinear dispersive equations on the half-lines.

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arXiv2025

The Kawahara equation on star graphs

In this paper, we establish local well-posedness for the Cauchy problem associated with the Kawahara equation on a general metric star graph. Initially, we identify suitable boundary conditions that produce a well-behaved dynamics for the linear equation. Subsequently, we derive the integral formula using the forcing operator method, previously applied to the Kawahara equation on the half-line by Cavalcante and Kwak (NoDEA 2020), and the Fourier restriction method of Bourgain (GAFA 1993). This work has the potential to be extended to other fifth-order nonlinear dispersive equations on star graphs.

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arXiv2025

On the Propagation of Regularity of Solutions to the KdV Equation on the positive Half-line

We study special regularity properties of solutions to the initial-boundary value problem associated with the Korteweg-de Vries equations posed on the positive half-line. In particular, for initial data $u_0 \in H^{\frac{3}{4}^{+}}(\mathbb{R}^+)$ and boundary data $f\in H^{\frac32^+}(\R^+)$, where the restriction of $u_0$ to some subset of $(b,\infty)$ has an extra regularity for any $b>0$, we prove that the regularity of solutions $u$ moves with infinite speed to its left as time evolves until a certain time $T^*$. The existence of a stopping time $T^{*}$ appears because of the effect of the boundary function $f$. Also, as a consequence of our proof, we prove a gain in the regularity of the trace derivatives of the solutions for the Korteweg-de Vries on the half-line.

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arXiv2025

The Korteweg-de Vries Equation on general star graphs

In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.

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arXiv2024

Dynamics of the Korteweg-de Vries equation on a balanced metric graph

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of $δ$-type at the {unique} graph-vertex. Also, we extend the linear instability result of Angulo and Cavalcante (2021) to one of nonlinear instability. For the proof of local well posedness theory the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class $C^2$.

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arXiv2022

Stability of mKdV breathers on the half-line

In this paper we study the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise.

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arXiv2021

Controllability for Schrödinger type system with mixed dispersion on compact star graphs

In this work we are concerned with solutions to the linear Schrödinger type system with mixed dispersion, the so-called biharmonic Schrödinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure $\mathcal{G}$ for $T>T_{min}$, with $$T_{min}=\sqrt{ \frac{ \overline{L} (L^2+π^2)}{π^2\varepsilon(1- \overline{L} \varepsilon)}},$$ when the couplings and the controls appear only on the Neumann boundary conditions.

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arXiv2021

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

In this work we study the initial boundary value problem associated with the coupled Schrödinger equations {with quadratic nonlinearities, that appears in nonlinear optics}, on the half-line. We obtain local well-posedness for data {in Sobolev spaces} with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces $X^{s,b}$ with $b<\frac12$, jointly with bilinear estimates in adapted Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the constant $a$ measures the scaling-diffraction magnitude indices.

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arXiv2020

Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane

This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic cubic nonlinear Schrödinger equation on star graphs are also discussed.

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arXiv2020

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

In a recent article \textit{"Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane", to appear on Pacific Journal of Mathematics [15]}, the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [15] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure $\mathcal{G}$. Precisely, consider $\mathcal{G}$ composed by $N$ edges parameterized by half-lines $(0,+\infty)$ attached with a common vertex $ν$. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the \textit{boundary forcing operator approach}. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

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arXiv2020

Linear instability of stationary solutions for the Korteweg-de Vries equation on a star graph

The aim of this work is to establish a linear instability criterium of stationary solutions for the Korteweg-de Vries model on a star graph with a structure represented by a finite collections of semi-infinite edges. By considering a boundary condition of $δ$-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable in a balanced star graph. The use of the analytic perturbation theory of operators and the extension theory of symmetric operators is a piece fundamental in our stability analysis. The arguments presented in this investigation has prospects for the study of the instability of stationary waves solutions of other nonlinear evolution equations on star graphs.

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arXiv2019

The cubic nonlinear fractional Schrödinger equation on the half-line

We study the cubic nonlinear fractional Schrödinger equation with Lévy indices $\frac{4}{3}<α< 2$ posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp with respect for known results on the full real line $\mathbb R$. Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data. To get our results we use the Colliander and Kenig approach based in the Riemann--Liouville fractional operator combined with Fourier restriction method of Bourgain \cite{Bourgain3} and some ideas of the recent work of Erdogan, Gurel and Tzirakis \cite{tzirakis2}. The method applies to both focusing and defocusing nonlinearities. As the consequence of our analysis we prove a smothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line $\mathbb R$ for the case of the low regularity assumption, which was point out at the recent work \cite{tzirakis2}.

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arXiv2019

Local well-posedness of the fifth-order KdV-type equations on the half-line

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

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arXiv2018

The initial-boundary value problem for Schrödinger-Korteweg-de Vries system on the half-line

We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger-Korteweg de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, being one of these family developed by Holmer to study the IBVP associated to the Korteweg-de Vries equation (Communications in Partial Differential Equations, 31 (2006)) and the other family one was recently introduced by Cavalcante (Differential and Integral Equations (2017)) in the context of nonlinear Schrödinger with quadratic nonlinearities.

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arXiv2018

Stability of KdV solitons on the half-line

In this paper we study the stability problem for KdV solitons on the left and right half-line. Unlike standard KdV, these are not exact solutions to the equations posed in the half-line. However, we are able to show that solitons placed initially far away from the origin are strongly stable for the problem posed on the right half-line, assuming homogeneous boundary conditions. For the problem posed on the left half-line, the positive infinitetime stability problem makes no sense for the case of KdV solitons, but in this setting we prove a result of stability for all negative times. The proof involves the use of almost conserved quantities adapted to the evolution of the KdV soliton on the particular case of the half-line. Adaptations to other boundary conditions or star graphs are also discussed.

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arXiv2018

Well-posedness and long time behavior for the Schrödinger-Korteweg-de Vries interactions on the half-Line

The initial-boundary value problem for the Schrödinger-Korteweg-de Vries system is considered on the left and right half-line for a wide class of initial-boundary data, including the energy regularity $H^1(\R^{\pm})\times H^1(\R^{\pm})$ for initial data. Assuming homogeneous boundary conditions it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space; furthermore, for negative coupling interactions it was proved, for a certain class of regular initial data, the following result: if the respective solution does not exhibits finite time blow-up in $H^1(\R^-)\times H^1(\R^-)$, then the norm of the weighted space $L^2\big(\R^-,\, |x|dx\big)\times L^2\big(\R^-,\, |x|dx\big)$ blows-up at infinity time with \textit{super-linear rate}, this is obtained by using a satisfactory algebraic manipulation of a new global virial type identity associated to the system .

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arXiv2018

The Korteweg-de Vries equation on a metric star graph

We prove local well-posedness for the Cauchy problem associated to Korteweg-de Vries equation on a metric star graph with three semi-infinite edges given by one negative half-line and two positives half-lines attached to a common vertex, for two classes of boundary conditions. The results are obtained in the low regularity setting by using the Duhamel Boundary Forcing Operator, in context of half-lines, introduced by Colliander, Kenig (2002),and extended by Holmer (2006) and Cavalcante (2017).

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arXiv2018

The initial-boundary value problem for the Kawahara equation on the half-line

This paper concerns the initial-boundary value problem (IBVP) of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander - Kenig \cite{CK} in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in $X^{s,b}$ space for $b < \frac12$, which is almost sharp compared to IVP of Kawahara equation \cite{CLMW2009, JH2009}.

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