相关论文: Optimal constants of smoothing estimates for Dirac…
We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schr\"odinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal…
We establish sharp weighted smoothing estimates for limit solutions to the Cauchy-Dirichlet problem for the fast diffusion equation on smooth bounded domains. We demonstrate that the critical exponent governing these estimates coincides…
We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…
We look for the optimal range of Lebesque exponents for which inhomogeneous Strichartz estimates are valid. We show that it is larger than the one given by admissible exponents for homogeneous estimates. We prove inhomogeneous estimates…
We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Dirac equation with generic potential. The decay extends the results obtained by Jensen, Kato and Murata for the Schr\"odinger equations.
Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics…
The existence of multidimensional lattice compactons in the discrete nonlinear Schr\"odinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast…
In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM…
In this paper we study sharp estimates for the Schr\"odinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schr\"odinger equation that…
We establish the full asymptotic stability of solitary waves for the focusing cubic Schr\"odinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances…
We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials
We expound in detail a method frequently used to reduce the Dirac equation in D-dimensional (D >= 4) spherically symmetric spacetimes to a pair of coupled partial differential equations in two variables. As a simple application of these…
By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schr\"odinger equation. These commutative relations correspond to the intrinsic symmetry of the…
We establish an observation inequality for the Schr\"odinger equation on $\mathbf{R}^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223…
Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed…
Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for…
For $f \in \mathscr{S}^2(\mathcal S)_{o}$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal S$, we consider the Schr\"odinger maximal function, \begin{equation*} S^* f(x):=…
The initial value problem for the homogeneous Schr\"odinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which…
We consider the Schr\"odinger equation with a (matrix) Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the…
This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper, where dispersive equations were treated.…