Browsing Linear and NonLinear Waves by Title
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The dynamics of vortex filaments with corners
(Communications on Pure and Applied Analysis (CPAA), 20150701)This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the socalled binormal flow. The case of a regular polygon ... 
The initial value problem for the binormal flow with rough data
(Annales Scientifiques de l'Ecole Normale Superieure, 20151231)In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the ... 
The Vortex Filament Equation as a Pseudorandom Generator
(Acta Applicandae Mathematicae, 20150801)In this paper, we consider the evolution of the socalled vortex filament equation (VFE), $$ X_t = X_s \wedge X_{ss},$$ taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel ... 
Threedimensional coarsening dynamics of a conserved, nematic liquid crystalisotropic fluid mixture
(Journal of NonNewtonian Fluid Mechanics, 201709)We present a numerical investigation of the threedimensional coarsening dynamics of a nematic liquid crystalisotropic fluid mixture using a conserved phase field model. The model is a coupled system for a generalized ... 
Uniqueness and Properties of Distributional Solutions of Nonlocal Equations of Porous Medium Type
(Advances in Mathematics, 20160901)We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation $\partial_tu\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ ... 
Uniqueness properties for discrete equations and Carleman estimates
(Journal of Functional Analysis, 20170325)Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of ... 
Variable coefficient Wolfftype inequalities and sharp local smoothing estimates for wave equations on manifolds
(2018)The sharp Wolfftype decoupling estimates of BourgainDemeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian ... 
Variable Lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities
(Nonlinear Analysis, 20180215)We prove global Calder\'onZygmund type estimate in Lorentz spaces for variable power of the gradients to weak solution of nonlinear elliptic equations in a nonsmooth domain. We mainly assume that the nonlinearities are ... 
Vortex filament equation for a regular polygon
(Nonlinearity, 20141231)In this paper, we study the evolution of the vortex filament equation,$$ X_t = X_s \wedge X_{ss},$$with $X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give ... 
Weghted Lorentz and LorentzMorrey estimates to viscosity solutions of fully nonlinear elliptic equations
(Complex Variables and Elliptic Equations, 2018)We prove a global weighted Lorentz and LorentzMorrey estimates of the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equation $F(D^{2}u,x)=f(x)$ defined in a bounded $C^{1,1}$ domain. The ...