Linear and Non-Linear Waves
http://hdl.handle.net/20.500.11824/4
2022-09-26T04:13:53ZSharp local smoothing estimates for Fourier integral operators
http://hdl.handle.net/20.500.11824/1513
Sharp local smoothing estimates for Fourier integral operators
Beltran D.; Hickman J.; Sogge C.D.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory
integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions
arising from the structure of the Fourier integrals.
2019-01-01T00:00:00ZENERGY CONSERVATION FOR 2D EULER WITH VORTICITY IN L(log L)α*
http://hdl.handle.net/20.500.11824/1486
ENERGY CONSERVATION FOR 2D EULER WITH VORTICITY IN L(log L)α*
Ciampa, G.
In these notes we discuss the conservation of the energy for weak solutions of the twodimensional incompressible Euler equations. Weak solutions with vorticity in (Formula presented) with p > 3/2 are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class L(logL)α with α > 1/2.
2022-01-01T00:00:00ZOn the Hausdorff dimension of Riemann's non-differentiable function
http://hdl.handle.net/20.500.11824/1454
On the Hausdorff dimension of Riemann's non-differentiable function
Eceizabarrena, D.
Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.
2021-01-01T00:00:00ZA pseudospectral method for the one-dimensional fractional Laplacian on R
http://hdl.handle.net/20.500.11824/1453
A pseudospectral method for the one-dimensional fractional Laplacian on R
Cayama, J.; Cuesta, C.M.; De la Hoz, F.
In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric series. Therefore, the central point of this paper is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we also do the simulation of Fisher's equation with fractional Laplacian in the monostable case.
2021-01-15T00:00:00Z