ON THE TIME-DEPENDENT HARTREE–FOCK EQUATIONS COUPLED WITH A CLASSICAL NUCLEAR DYNAMICS

1999 ◽  
Vol 09 (07) ◽  
pp. 963-990 ◽  
Author(s):  
ERIC CANCÈS ◽  
CLAUDE LE BRIS
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Hartree Fock ◽  
Newtonian Dynamics ◽  
The Cauchy Problem ◽  
Time Existence

We prove a global-in-time existence and uniqueness result for the Cauchy problem in the setting of some model of Molecular Quantum Chemistry. The model we are concerned with consists of a coupling between the time-dependent Hartree–Fock equations (for the electrons) and the classical Newtonian dynamics (for the nuclei). The proof combines semigroup techniques and the Schauder fixed-point theorem. We also extend our result in order to treat the case of a molecule subjected to a time-dependent electric field.

The relativistic Enskog equation near the vacuum in the Robertson–Walker space-time

2019 ◽  
Vol 10 (3) ◽  
pp. 273-284
Author(s):  
Fidele Lavenir Ciake Ciake ◽  
Etienne Takou
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Hard Sphere ◽  
Weighted Space ◽  
Space Time ◽  
Uniqueness Result ◽  
Enskog Equation ◽  
The Cauchy Problem

Abstract In this paper, we consider the Cauchy problem for the relativistic Enskog equation with near vacuum data for a hard sphere gas in the Robertson–Walker space-time. We prove an existence and uniqueness result of the global (in time) mild solution in a suitable weighted space. We also study the asymptotic behavior of the solution as well as the {L^{\infty}} -stability.

LOCAL EXISTENCE AND UNIQUENESS THEORY FOR THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

2012 ◽  
Vol 09 (01) ◽  
pp. 177-193 ◽  
Author(s):  
KEIICHI KATO ◽  
YUUSUKE SUGIYAMA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Local Existence ◽  
Second Sound ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Uniqueness Theory

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.

Cauchy problem for the time-dependent Ginzburg–Landau model of superconductivity

2000 ◽  
Vol 130 (6) ◽  
pp. 1383-1404 ◽  
Author(s):  
A. Rodriguez-Bernal ◽  
B. Wang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Result ◽  
Uniqueness Result ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Landau Model ◽  
The Cauchy Problem ◽  
Ginzburg Landau Equations ◽  
Well Posed

The Cauchy problem for the time-dependent Ginzburg–Landau equations of superconductivity in Rd (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L2. When d = 3, we establish the existence result of solutions for L3 initial data and the uniqueness result for L4 initial data.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

scholarly journals A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Irregular Meshes ◽  
Complicated Geometry

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

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