Kinds Of Initial Data Research Articles

Riemann problems for generalized gas dynamics

In this paper, a few classes of exact solutions are obtained using the differential constraints method for generalized gas dynamics equations. The solutions to Riemann problems for two different kinds of initial data are determined with a complete characterization of the solutions through shock waves and/or rarefaction waves.

On some large global solutions to the incompressible inhomogeneous nematic liquid crystal flows

ABSTRACTIn this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.

Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term

In this paper, we investigate an initial boundary value problem to a class of pseudo-parabolic partial differential equations with Newtonian nonlocal term. First, the local existence and uniqueness of a weak solution is established. In virtue of the energy functional and the related Nehari manifold, we also describe the exponent decay behavior and the blow up phenomenon of weak solutions with different kinds of initial data. Our second conclusion states that some solutions starting in a potential well exist globally, whereas solutions with suitable initial data outside the potential well must blow up. Furthermore, the instability of a ground state equilibrium solution is studied.

Quenching phenomenon for a non-Newtonian filtration equation with singular boundary flux

This paper is concerned with the quenching phenomenon for the one-dimensional non-Newtonian filtration equation with both source term and Neumann boundary condition. With two different kinds of initial data, we prove that the solution must quench in a finite time and the time derivative blows up at a quenching point. The corresponding quenching rate and a lower bound for the quenching time are also obtained.

Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential

We consider the three dimensional Gross-Pitaevskii equation (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confi ned in vertical z direction. The highly confi ned potential induces high oscillations in time. If the confi nement in the z direction is a harmonic trap (which is widely used in physical experiments), the very special structure of the spectrum of the confi nement operator will imply that the oscillations are periodic in time. Based on this observation, it can be proved that the GPE can be averaged out with an error of order of epsilon, which is the typical period of the oscillations. In this article, we construct a more accurate averaged model, which approximates the GPE up to errors of order epsilon squared. Then, expansions of this model over the eigenfunctions (modes) of the vertical Hamiltonian Hz are given in convenience of numerical application. Effi cient numerical methods are constructed for solving the GPE with cylindrical symmetry in 3D and the approximation model with radial symmetry in 2D, and numerical results are presented for various kinds of initial data.

Global Classical Solutions to the Equations of Two–dimensional MHD Transverse Flow

A new method for constructing the symmetric form for the hyperbolic systems is introduced. Then a new symmetric form of the equations of MHD transverse flow is constructed by adding an additional equation. With this new form, we obtain the local existence of smooth solution including the case that the initial density may tend to the vacuum state at infinity. Furthermore, the uniformly a priori estimation for the classical solutions is established and the global smooth solutions for a kind of initial data are obtained.

Stochastic Perturbations of Line Solitons of KP

We consider the properties of localized solutions of the KP equation coupled to a stochastic noise. Corresponding to white noise, we find that the traveling waves are destroyed asymptotically, and we determine the distribution of the wave position and the arrival time. For generalized Ornstein–Uhlenbeck processes, we show that the only effect of noise is to render the asymptotic position random; in particular, when the noise has a sufficiently strong attenuation mechanism, the random wave coincides asymptotically with the unperturbed one. We also consider linearization of the corresponding Cauchy problem in the plane corresponding to this kind of initial data.

A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation

We study the solution of the focusing nonlinear Schrödinger equation in the semiclassical limit. Numerical solutions are presented for four different kinds of initial data, of which three are analytic and one is nonanalytic. We verify numerically the weak convergence of the oscillatory solution by examining the strong convergence of the spatial average of the solution.

LIFE-SPAN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH SLOW DECAY INITIAL DATA

The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1–1.3 in [2] are still valid for this kind of initial data.

Uniqueness of electromagnetic inversion by local surface measurements

Assume that a bounded body has been embedded in a homogeneous half-space. Our aim is to reconstruct the electromagnetic material parameters, namely the electric permittivity, conductivity and magnetic permeability, of the body from local field measurements on the surface of the half-space. All fields are supposed to be governed by time-harmonic Maxwell equations. We consider two kinds of initial data. The first data set consists of tangential electric and magnetic field vectors of a tangential magnetic dipole for all field and source points in an open subset of the surface. The second data set is one that we know locally on the surface - the admittance map - which, by definition, associates the tangential magnetic field with the tangential electric field. The purpose of this paper is to show that both of the data sets determine the material parameters uniquely.

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.