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G. Ponce

Publications and source records attributed to G. Ponce.

15 recordsLinked to original sources
arXiv2024

On the fractional Sch\"odinger equation with variable coefficients

We study the initial value problem (IVP) associated to the semi-linear fractional Sch\"odinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

math.AP↗
arXiv2015

Hardy Uncertainty Principle, Convexity and Parabolic Evolutions

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.

math.AP↗
arXiv2014

On persistence properties in fractional weighted spaces

In this work we derive a point-wise formula that will allows us to study the well-posedness of initial value problem associated to nonlinear dispersive equations in fractional weighted Sobolev spaces $H^s(\R)\cap L^2(|x|^{2r}dx)$, $s, r \in \R$. As an application of this formula we will study local and global well posedness of the $k$-generalized Korteweg-de Vries equation in these weighted Sobolev spaces.

math.AP↗
arXiv2010

The Hardy Uncertainty Principle Revisited

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.

math.AP↗
arXiv2009

Uncertainty Principle of Morgan type and Schrödinger Evolutions

We prove unique continuation properties for solutions of evolution Schrödinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an application of our method we also obtain results concerning the possible concentration profiles of solutions of semi-linear Schrödinger equations.

math.AP↗
arXiv2009

The sharp Hardy Uncertainty Principle for Sch\"odinger evolutions

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schr\"odinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schr\"odinger equations.

math.AP↗
arXiv2008

Hardy's Uncertainty Principle, Convexity and Schrödinger Evolutions

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

math.AP↗
arXiv2007

On the decay properties of solutions to a class of Schrödinger equations

We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections P+ and P- onto the positive and negative frequencies.

math.AP↗
arXiv2005

On unique continuation of solutions of Schr\"odinger equations

We study uniqueness properties of solutions of Schr\"odinger equations. The aim is to obtain sufficient conditions on the decay behavior of the difference of two solution $u_1-u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee the uniqueness of the solution, i.e. that $u_1\equiv u_2$.

math.AP↗
arXiv2005

Variable coefficient Schrödinger flows for ultrahyperbolic operators

In this paper we study the local solvability of nonlinear Schrödinger equations of the form $$\p_t u = i {\cal L}(x) u + \vec b_1(x)\cdot \nabla_x u + \vec b_2(x)\cdot \nabla_x \bar u + c_1(x)u+c_2(x)\bar u +P(u,\bar u,\nabla_x u, \nabla_x\bar u), where $x\in\mathbb R^n$, $t>0$, $\displaystyle{\cal L}(x) = -\sum_{j,k=1}^n\p_{x_j}(a_{jk}(x)\p_{x_k})$, $A(x)=(a_{jk}(x))_{j,k=1,..,n}$ is a real, symmetric and nondegenerate variable coefficient matrix, and $P$ is a polynomial with no linear or constant terms. Equations of the form described in with $A(x)$ merely invertible as opposed to positive definite arise in connection with water wave problems, and in higher dimensions as completely integrable models. Under appropriate assumptions on the coefficients we shall show that the associated initial value problem is local well posed.

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