Related papers: On Morawetz estimates for the elastic wave equatio…
The Whitham equation is a nonlocal, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at…
We study the implications of time-varying wave mechanics, and show how the standard wave equation is modified if the speed of a wave is not constant in time. In particular, waves which experience longitudinal acceleration are shown to have…
This paper investigates the inverse random source problem for elastic waves in three dimensions, where the source is assumed to be driven by an additive white noise. A novel computational method is proposed for reconstructing the variance…
We prove quantitative scattering for the three-dimensional defocusing energy-critical quintic wave equation on a class of asymptotically flat, possibly non-stationary perturbations of Minkowski space, by establishing the first explicit…
In this paper we derive $W^{1,\infty}$ and piecewise $C^{1,\alpha}$ estimates for solutions, and their $t-$derivatives, of divergence form parabolic systems with coefficients piecewise H\"older continuous in space variables $x$ and smooth…
It is well recognized that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear partial differential equations in hand. In this…
In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space $\mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on…
The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. The motivation for this part of my thesis…
This paper focuses on stability estimates of the inverse random source problems for the polyharmonic, electromagnetic, and elastic wave equations. The source is represented as a microlocally isotropic Gaussian random field, which is defined…
For the solution of the free Schr\"odinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the…
Two methods are explained to exactly solve Maxwell's equations where permittivity, permeability and conductivity may vary in space. In the constitutive relations, retardation is regarded. If the material properties depend but on one…
We demonstrate that a sufficiently smooth solution of the relativistic Euler equations that represents a dynamical compact liquid body, when expressed in Lagrangian coordinates, determines a solution to a system of non-linear wave equations…
In this paper, we are concerned with the study of stabilization problem for the following strongly degenerate wave equation in one space dimension $$w_{tt}(x,t)-\left(x^\alpha w_x(x,t)\right)_x=0$$ where ${\bf\alpha\in [1,2)}$. Thus, using…
New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.
Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical…
We study the two-dimensional gravity water waves with a one-dimensional interface with small initial data. Our main contributions include the development of two novel localization lemmas and a Transition-of-Derivatives method, which enable…
We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type I singularities of solutions with $$\limsup_{t \nearrow T}|{\rm div} u(t, x)|(T - t) \leq \kappa,$$ can never happen at time…
We consider a semilinear wave equation involving a time-dependent structural damping term of the form $\displaystyle\frac{1}{{(1+t)}^{\beta}}(-\Delta)^{\sigma/2} u_t$. Our results show the influence of the parameters $\beta,\sigma$ on the…
Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as…
In this paper, we study the endpoint reversed Strichartz estimates along general time-like trajectories for wave equations in $\mathbb{R}^{3}$. We also discuss some applications of the reversed Strichartz estimates and the structure of wave…