arXiv subjects
Publications and source records attributed to Simão Correia.
We consider the Benjamin-Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a consequence, unconditional uniqueness for the Benjamin-Ono equation is proved.
We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a general instability criterion and then apply it to some well-known models arising in several physical contexts. In particular, this mass-transfer instability allows us to exhibit $L^2$-subcritical unstable ground-states.
In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called nonlinear smoothing. We employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing for dispersive PDEs, contingent only on establishing two particular bounds. We then apply this theorem to show that the nonlinear smoothing property holds, depending on the regularity of the initial data, for five classical dispersive equations: in $\mathbb{R}$, the cubic nonlinear Schrödinger, the Korteweg-de Vries, the modified Korteweg-de Vries and the derivative Schrödinger equations; in $\mathbb{R}^2$, the modified Zakharov-Kuznetsov equation. For the aforementioned one-dimensional equations, this unifying methodology matches or improves the existing nonlinear smoothing results, while enlarging the ranges of Sobolev regularities where the property holds.
We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and construct solutions with a prescribed blow up behavior as t $\rightarrow$ 0.
Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $$ \partial_t u = (a + iα) Δu - (b + i β) |u|^σu + k u, \,\, t > 0,\,\, x\in Ω, $$ with $a>0, α, b, β, k \in \mathbb{R}$, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when $Ω=\mathbb{R}$ and $-k$ large enough.
We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W^{1,\infty} around a carefully chosen, two term ansatz. Such knowledge is crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. In the defocusing case, the self-similar profiles are solutions to the Painlevé II equation. Although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. We are able to relate the constants involved in the description in Fourier space with those involved in the description in physical space.
We study the Schrödinger-Debye system over $\mathbb{R}^d$ iu_t+\frac 12Δu=uv,\quad μv_t+v=λ|u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in dimensions $d=2,3,4$. Moreover, in dimension $d=1$, we prove a Hayashi-Naumkin modified scattering result.
We consider the quadratic Schrödinger system $$iu_t+Δ_{γ_1}u+\overline{u}v=0$$ $$2iv_t+Δ_{γ_2}v-βv+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $γ_1,γ_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.
We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data $u_0, v_0$, if $|y|$ is large, then the concatenated initial data $u_0+v_0(\cdot -y)$ gives rise to globally decaying solutions.
We consider the Cauchy problem for the nonlinear Schrödinger equation on $\mathbb{R}^d$, where the initial data is in $\dot{H}^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$. We prove local well-posedness for large ranges of $p$ and discuss some global well-posedness results.
Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over $\mathbb{R}^d$ $$ iu_t + u_{xx} - Δ_{\textbf{y}} u + λ|u|^σu=0. $$ We deduce the conservation laws associated with (HNLS) and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including \textit{spatial plane waves} and \textit{spatial standing waves}, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.
We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional equation. We then proceed to prove local well-posedness over a space which includes $L^3(\mathbb{R}^3)$ and these solutions. Finally, we prove $L^3$-stability of spatial plane waves, with no condition on their size.
We consider the Cauchy problem for the nonlinear Schrödinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + λ|u|^σu =0$, $λ\in \mathbb{R}$, $σ>0$. We introduce new functional spaces over which the initial value problem is well-posed. Their construction is based on \textit{spatial plane waves} (cf. arXiv:1510.08745). These spaces contain $H^1(\mathbb{R}^2)$ and do not lie within $L^2(\mathbb{R}^2)$. We prove several global well-posedness and stability results over these new spaces, including a new global well-posedness result of $H^1$ solutions with indefinitely large $H^1$ and $L^2$ norms. Some of these results are proved using a new functional transform, the \textit{plane wave transform}. We develop a suitable theory for this transform, prove several properties and solve classical linear PDE's with it, highlighting its wide range of application.
We focus on the study of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. We extend the characterization result from a previous work (arXiv:1410.7993) to the case where both attraction and repulsion are present and cannot be studied separately. Furthermore, we derive some perturbation and classification results to study the general system where components may be out of phase. In particular, we present several conditions to the existence of nontrivial ground-states.
In this work we consider the weakly coupled Schrödinger cubic system \[ \begin{cases} \displaystyle -Δu_i+λ_i u_i= μ_i u_i^{3}+ u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quad i=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $λ_i,μ_i >0$ and $b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, that is solutions $\mathbf{u}=(u_1,\ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $λ_i,μ_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $d\geq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $λ_i\equiv λ$ and $b_{ij}\equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.
In this paper we consider the nonlinear Schrö\-din\-ger equation $i u_t +Δu +κ|u|^αu=0$. We prove that if $α<\frac {2} {N}$ and $\Im κ<0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $α>\frac {2} {N}$ and $κ\in {\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} < α\le \frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$.
We focus on the study of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we study global existence, $L^2$-concentration phenomena and blowup profile for the evolution system in the $L^2$-critical power case.
We focus on the study of the stability properties of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. Our results are generalizations of the theory for the single equation and the technique for this kind of results is simplified. Depending on the power of the nonlinearity, we may observe stability, instability and weak instability. We also obtain results for three distinct classes of bound-states, which is a special feature of the $M\ge2$ case.