scholarly journals LOCAL WELL-POSEDNESS BELOW THE CHARGE NORM FOR THE DIRAC–KLEIN–GORDON SYSTEM IN TWO SPACE DIMENSIONS

2007 ◽  
Vol 04 (02) ◽  
pp. 295-330 ◽  
Author(s):  
PIERO D'ANCONA ◽  
DAMIANO FOSCHI ◽  
SIGMUND SELBERG
Keyword(s):  
Dirac Spinor ◽  
Existence Problem ◽  
Time Estimates ◽  
Homogeneous Wave ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
Two Space Dimensions ◽  
Positive Index ◽  
Well Posed ◽  
Homogeneous Wave Equation

We prove that the Cauchy problem for the Dirac–Klein–Gordon equations in two space dimensions is locally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor, and an associated range of spaces of positive index for the meson field. In particular, we can go below the charge norm, that is, the L2 norm of the spinor. We hope that this can have implications for the global existence problem, since the charge is conserved. Our result relies on the null structure of the system, and bilinear space-time estimates for the homogeneous wave equation.

scholarly journals GLOBAL WELL-POSEDNESS OF THE 1D DIRAC–KLEIN–GORDON SYSTEM IN SOBOLEV SPACES OF NEGATIVE INDEX

2009 ◽  
Vol 06 (03) ◽  
pp. 631-661 ◽  
Author(s):  
ACHENEF TESFAHUN
Keyword(s):  
Sobolev Spaces ◽  
Well Posedness ◽  
Dirac Spinor ◽  
Negative Index ◽  
Main Ingredient ◽  
Time Estimates ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
Positive Index ◽  
Well Posed

We prove that the Cauchy problem for the Dirac–Klein–Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of "almost conservation law" and "I-method" introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. Our proof also relies on the null structure in the system, and bilinear space–time estimates of Klainerman–Machedon type.

scholarly journals ON THE ASYMPTOTIC ANALYSIS OF THE DIRAC–MAXWELL SYSTEM IN THE NONRELATIVISTIC LIMIT

2005 ◽  
Vol 02 (01) ◽  
pp. 129-182 ◽  
Author(s):  
PHILIPPE BECHOUCHE ◽  
NORBERT J. MAUSER ◽  
SIGMUND SELBERG
Keyword(s):  
Convergence Result ◽  
Nonrelativistic Limit ◽  
Dirac Spinor ◽  
Maxwell System ◽  
Poisson System ◽  
Pauli Equation ◽  
Existence Time ◽  
Self Consistent Field ◽  
Klein Gordon ◽  
Well Posed

We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H1 data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H1. Moreover, if the datum for the Dirac spinor converges in H1, then the solution of DM converges, modulo a phase correction, in C([0,T];H1) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit. The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

Designing localized waves

1993 ◽  
Vol 440 (1910) ◽  
pp. 541-565 ◽  
Keyword(s):  
Wave Equation ◽  
Fourier Transforms ◽  
Transform Domain ◽  
Localized Solutions ◽  
Homogeneous Wave ◽  
Klein Gordon ◽  
Domain Phase ◽  
Localized Waves ◽  
Homogeneous Wave Equation ◽  
Geometrical Consideration

In this paper we re-interpret a recently introduced method for obtaining non-separable, localized solutions of homogeneous partial differential equations. This reinterpretation is in the form of a geometrical consideration of the algebraic constraint that the Fourier transforms of such solutions must satisfy in the transform domain (phase space). With this approach we link two classes of localized, non-separable solutions of the homogeneous wave equation, and examine the transform domain characteristic that determines the space-time localization properties of these classes. This characterization allows us to design classes of solutions with better localization properties. In particular, we design and discuss the properties of several novel subluminal and superluminal solutions of the homogeneous wave equation. We also design families of non-separable, localized, subluminal and superluminal solutions of the Klein-Gordon equation by using the same technique.

scholarly journals Global well-posedness in energy space for the Chern–Simons–Higgs system in temporal gauge

2016 ◽  
Vol 13 (02) ◽  
pp. 331-351 ◽  
Author(s):  
Hartmut Pecher
Keyword(s):  
Cauchy Problem ◽  
Energy Space ◽  
Well Posedness ◽  
Null Condition ◽  
Temporal Gauge ◽  
Time Estimates ◽  
Dimensional Minkowski Space ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

The Cauchy problem for the Chern–Simons–Higgs system in the [Formula: see text]-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an [Formula: see text]-estimate for solutions of the wave equation, and also takes advantage of a null condition.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

scholarly journals Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

2020 ◽  
Vol 23 (6) ◽  
pp. 1663-1677
Author(s):  
Michael Ruzhansky ◽  
Berikbol T. Torebek
Keyword(s):  
Cauchy Problem ◽  
Exponential Function ◽  
Evolution Equations ◽  
Oscillatory Integrals ◽  
Schrödinger Equations ◽  
Fractional Evolution Equations ◽  
Type Function ◽  
Fractional Schrödinger Equations ◽  
The Cauchy Problem ◽  
Klein Gordon

Abstract The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

scholarly journals The cauchy problem for non-linear Klein-Gordon equations

1993 ◽  
Vol 152 (3) ◽  
pp. 433-478 ◽  
Author(s):  
Jacques C. H. Simon ◽  
Erik Taflin
Keyword(s):  
Cauchy Problem ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Klein Gordon
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