arXiv subjects
Publications and source records attributed to Luis Vega.
The motion of incompressible and ideal fluids is studied in the plane. The stability in $L^1$ of circular vortex patches is established among the class of all bounded vortex patches of equal strength without any restriction on the size of the initial perturbation.
We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a non trapping condition, which are expressed as smallness of suitable components of the potentials. However, the potentials themselves can be large, and we avoid completely any a priori spectral assumption on the operator. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.
We present a numerical study of the self-similar solutions of the Localized Induction Approximation of a vortex filament. These self-similar solutions, which constitute a one-parameter family, develop a singularity at finite time. We study a number of boundary conditions that allow us reproduce the mechanism of singularity formation. Some related questions are also considered.
In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski equation with time-variable coefficients.
We prove bilinear virial identities for the nonlinear Schrodinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.
We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension $n\geq3$, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.
We study some qualitative properties of global solutions to the following focusing and defocusing critical $NLW$: \begin{equation*} \Box u+ λu|u|^{2^*-2}=0, \hbox{} λ\in {\mathbf R} \end{equation*} $$\hspace{2cm} u(0)=f\in \dot H^1({\mathbf R}^n), \partial_t u(0)=g\in L^2({\mathbf R}^n)$$ on ${\mathbf R}\times {\mathbf R}^n$ for $n\geq 3$, where $2^*\equiv \frac{2n}{n-2}$. We will consider the global solutions of the defocusing $NLW$ whose existence and scattering property is shown in \cite{shst}, \cite{sb} and \cite{bg}, without any restriction on the initial data $(f,g)\in \dot H^1({\mathbf R}^n) \times L^2({\mathbf R}^n)$. As well as the solutions constructed in \cite{pecher} to the focusing $NLW$ for small initial data and to the ones obtained in \cite{mk}, where a sharp condition on the smallness of the initial data is given. We prove that the solution $u(t, x)$ satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see \cite{mor1}). As a by--product we deduce that any global solution to critical $NLW$ belonging to a natural functional space satisfies: $$\lim_{R\to \infty}\frac 1R \int_{\mathbf R} \int_{|x|<R} |\nabla_{x} u(t,x)|^2 \hbox{} dxdt $$ $$=\lim_{R\to \infty} \frac 1{2R} \int_{\mathbf R} \int_{|x|<R} (|\nabla_{t,x} u(t,x)|^2 + \frac{2 λ}{2^*} |u(t,x)|^{2^*}) \hbox{} dxdt$$ $$=\int_{{\mathbf R}^n} (|\nabla_{t, x} u(0, x)|^2+ \frac{2 λ}{2^*} |u(0, x)|^{2^*}) \hbox{} dx.$$
We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.
We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - Δu + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.
In this article we will study the initial value problem for some Schrödinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so called Hasimoto transformation.
We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<ξ>^s u^_0||_{L^r'_ξ}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.
In this paper we establish the equivalence of solutions between Schrödinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
We consider the Helmholtz equation with a variable index of refraction $n(x)$, which is not necessarily constant at infinity but can have an angular dependency like $n(x)\to n\_\infty(x/|x |)$ as $|x |\to \infty$. Under some appropriate assumptions on this convergence and on $n\_\infty$ we prove that the Sommerfeld condition at infinity still holds true under the explicit form $$ \int\_{\R^d} | \nabla u -i n\_\infty^{1/2} u \xox |^2 \f{dx}{|x |}<+\infty. $$ It is a very striking and unexpected feature that the index $n\_{\infty}$ appears in this formula and not the gradient of the phase as established by Saito in \cite {S} and broadly used numerically. This apparent contradiction is clarified by the existence of some extra estimates on the energy decay. In particular we prove that $$ \int\_{\R^d} | \nabla\_ωn\_\infty(\xox)|^2 \f{| u |^2}{|x |} dx < +\infty. $$ In fact our main contribution is to show that this can be interpreted as a concentration of the energy along the critical lines of $n\_\infty$. In other words, the Sommerfeld condition hides the main physical effect arising for a variable $n$ at infinity; energy concentration on lines rather than dispersion in all directions.
In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz estimates.
In this paper we study uniqueness properties of solutions of the k-generalized Korteweg-de Vries equation. Our goal is to obtain sufficient conditions on the behavior of the difference $u_1-u_2$ of two solutions $u_1, u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee that $u_1\equiv u_2$.
We investigate the formation of singularities in a self-similar form of regular solutions of the Localized Induction Approximation (also referred as to the binormal flow). This equation appears as an approximation model for the self-induced motion of a vortex filament in an inviscid incompressible fluid. The solutions behave as 3d-logarithmic spirals at infinity. The proofs of the results are strongly based on the existing connection between the binormal flow and certain Schrödinger equations.
We discuss the question whether solutions of the initial value problem for the generalized KdV equation can have compact support at two different times.