Initial Data Problem Research Articles

Conservative solutions to a nonlinear variational sine-Gordon equation

This paper is concerned with a nonlinear variational sine-Gordon equation u t t − c ( u ) [ c ( u ) u x ] x + λ 2 2 s i n ( 2 u ) = 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. To deal with the possible blowup of solutions, we introduce a new set of variables depending on the energy, whereby all singularities are resolved.

Pathologies in asymptotically Lifshitz spacetimes

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.

On geometrical and physical consequences of using certain special classes of comoving physical frames of reference II

This paper develops the study of the geometrical and physical characteristics of the class of the frames of reference associated to isotropic coordinates in the case of spherical symmetry. In particular, through the separate invariant formulation of the initial data problem and the restricted evolution problem, we study motions comoving with isotropic coordinates: i.e. shear free motions in which p = p(μ) or p = 0. All the possible solutions of the restricted evolution problem are determined by specifying all the admissible choice of the data in the case of shear free. Furthermore, if the characteristics of the motions are initially valid, they are valid as time evolves.

A semilinear parabolic system with coupling variable exponents

This paper deals with semilinear parabolic equations coupled via variable sources, subject to the homogeneous Dirichlet condition in a bounded domain. Since the variable exponents in the sources are just assumed to be positive, the non-linearities may be non-Lipschitz. We establish the existence–uniqueness with comparison principle of local solutions to the regularized problem at first, and then consider the maximal solutions of the original problem as the limits of the solutions of the regularized problem. Some criteria are established for distinguishing global and non-global solutions of the problem, dependent or independent of initial data. Especially, we prove a Fujita type conclusion that the solutions blow up for any non-trivial initial data under certain assumptions on the variable sources and the size of the domain.

On the restricted evolution problem of the Einstein relativistic mechanics

This work is focused on the interior Cauchy problem for the Einstein’s field equations. Precisely, in the relativistic study of the evolution of a continuum reversible system, the Cauchy problem is broken into two separate problems: the initial data problem and the restricted problem of evolution.

The Scattering Problem for a Noncommutative Nonlinear Schrödinger Equation

We investigate scattering properties of a Moyal deformed version of the nonlinear Schrodinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory

We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

On geometrical and physical consequences of using certain special classes of comoving physical frames of reference

This paper develops the study of the geometrical characteristics of the class of frames of reference associated with the two well known sets of coordinates: Levi-Civita’s curvature coordinates and gaussian polar coordinates. In particular, through the separate invariant formulation of the initial data problem and of the restricted evolution problem, the analysis is applied to a spherically symmetric perfect fluid, whose stream lines coincide with those of the above mentioned class of frames.

Einstein-Maxwell system in3+1form and initial data for multiple charged black holes

We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell's equations in curved spacetimes are derived in detail using a $3+1$ formalism and their hyperbolic properties are analyzed, showing that the resulting system is symmetric hyperbolic. We also focus on the problem of finding initial data for multiple charged black holes assuming time-symmetric initial data and using a puncturelike method to solve the Hamiltonian and the Gauss constraints. We study the behavior of the resulting initial data families, and show that previous results in this direction can be obtained as particular cases of our approach.

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchywith complex-valued initial data and prove unique solvability globally intime for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inversealgebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

Riemann problem and elementary wave interactions in isentropic magnetogasdynamics

In this paper, we consider the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady simple wave flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. This class of equations includes, as a special case, the equations of isentropic gasdynamics. We study the shock and rarefaction waves and their properties, and discuss the geometry of shock curves using the Riemann invariant coordinates. Under certain conditions, we show the existence and uniqueness of the solution to the Riemann problem for arbitrary initial data, and then discuss the vacuum state in isentropic magnetogasdynamics. Finally, we discuss numerical results for different initial data, and discuss all possible interactions of elementary waves. It is noticed that although the magnetogasdynamic system is more complex than the corresponding gasdynamic system, all the parallel results remain identical. However, unlike the ordinary gasdynamic case, the solution inside rarefaction waves in magnetogasdynamics cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. It is also observed that the presence of a magnetic field makes both the shock and rarefaction stronger compared to what they would have been in the absence of a magnetic field.

L1 Stability of Spatially Periodic Solutions in Relativistic Gas Dynamics

This paper proves the well posedness of spatially periodic solutions of the relativistic isentropic gas dynamics equations. The pressure is given by a γ-law with initial data of large amplitude, provided γ − 1 is sufficiently small. As a byproduct of our techniques, we obtain the same results for the classical case. At the limit c → + ∞, the solutions of the relativistic system converge to the solutions of the classical one, the convergence rate being 1/c2. We also construct the semigroup of solutions of the Cauchy problem for initial data with bounded total variation, which can be large, as long as γ − 1 is small.

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial–boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial–boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.

An asymptotic analysis of initial-value problems for thin elastic plates

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.

Review Article

The isolated horizon formalism recently introduced by Ashtekar et al. aims at providing a quasi-local concept of a black hole in equilibrium in an otherwise possibly dynamical spacetime. In this formalism, a hierarchy of geometrical structures is constructed on a null hypersurface. On the other side, the 3+1 formulation of general relativity provides a powerful setting for studying the spacetime dynamics, in particular gravitational radiation from black hole systems. Here we revisit the kinematics and dynamics of null hypersurfaces by making use of some 3+1 slicing of spacetime. In particular, the additional structures induced on null hypersurfaces by the 3+1 slicing permit a natural extension to the full spacetime of geometrical quantities defined on the null hypersurface. This four-dimensional point of view facilitates the link between the null and spatial geometries. We proceed by reformulating the isolated horizon structure in this framework. We also reformulate previous works, such as Damour's black hole mechanics, and make the link with a previous 3+1 approach of black hole horizon, namely the membrane paradigm. We explicit all geometrical objects in terms of 3+1 quantities, putting a special emphasis on the conformal 3+1 formulation. This is in particular relevant for the initial data problem of black hole spacetimes for numerical relativity. Illustrative examples are provided by considering various slicings of Schwarzschild and Kerr spacetimes.

$L_p$-$L_q$ maximal regularity and viscous incompressible flows with free surface

We prove the $L_p$-$L_q$ maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. And as an application, we consider a free boundary problem for the Navier-Stokes equation. We prove a locally in time unique existence of solutions to this problem for any initial data and a globally in time unique existence of solutions to this problem for some small initial data.

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u t = Δ u + u p + u q ( x * , t ) in B ( R ) where p , q > 0 , B ( R ) = { x ∈ R N : | x | < R } and x * ≠ 0 . If max ( p , q ) > 1 , there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution u ( x , t ) = u ( r , t ) ( r = | x | ) of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q. (i) If q < p ( p > 1 ) or p = q > 2 , then single point blow-up occurs whenever solutions blow up. (ii) If 1 < p < q , both phenomena, total blow-up and single point blow-up, occur depending on the initial data. (iii) If p ⩽ 1 < q , total blow-up occurs whenever solutions blow up. (iv) If max ( p , q ) ⩽ 1 , every solution exists globally in time.

EQUATION-FREE, EFFECTIVE COMPUTATION FOR DISCRETE SYSTEMS: A TIME STEPPER BASED APPROACH

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff &amp; Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, Lp–Lq type estimates are obtained. By use of the Lp–Lq estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto-micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.