arXiv subjects

Elek Csobo

Publications and source records attributed to Elek Csobo.

5 recordsLinked to original sources

arXiv2021

On blowup for the supercritical quadratic wave equation

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$. In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

math.AP↗

arXiv2020

Existence and orbital stability of standing waves to a nonlinear Schrödinger equation with inverse square potential on the half-line

We investigate the properties of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse square potential). Finally, we establish orbital instability by a blow-up argument for mass super-critical nonlinearity.

math.AP↗

arXiv2019

Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known 'slope condition' for orbital stability.

math.AP↗

arXiv2018

Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential

We study minimal mass blow-up solutions of the focusing $L^2$ critical nonlinear Schrödinger equation with inverse-square potential, \[ i\partial_t u + Δu + \frac{c}{|x|^2}u+|u|^{\frac{4}{N}}u = 0, \] with $N\geqslant 3$ and $0<c<\frac{(N-2)^2}{4}$. We first prove a sharp global well-posedness result: all $H^1$ solutions with a mass (i.e. $L^2$ norm) strictly below that of the ground states are global. Note that, unlike the equation in free space, we do not know if the ground state is unique in the presence of the inverse-square potential. Nevertheless, all ground states have the same, minimal, mass. We then construct and classify finite time blow-up solutions at the minimal mass threshold. Up to the symmetries of the equation, every such solution is a pseudo-conformal transformation of a ground state solution.

math.AP↗

arXiv2015

Numerical reconstruction of pulsatile blood flow from 4D computer tomography angiography data

We present a novel numerical algorithm developed to reconstuct pulsatile blood flow from ECG-gated CT angiography data. A block-based optimization method was constructed to solve the inverse problem corresponding to the Riccati-type ordinary differential equation that can be deduced from conservation principles and Hooke's law. Local flow rate for 5 patients was computed in 10cm long aorta segments that are located 1cm below the heart. The wave form of the local flow rate curves seems to be realistic. Our approach is suitable for estimating characteristics of pulsatile blood flow in aorta based on ECG gated CT scan thereby contributing to more accurate description of several cardiovascular lesions.

math.NA↗