REGULARITY OF SOLUTIONS TO LINEAR STOCHASTIC SCHRÖDINGER EQUATIONS

2007 ◽  
Vol 10 (02) ◽  
pp. 237-259 ◽  
Author(s):  
CARLOS M. MORA ◽  
ROLANDO REBOLLEDO
Keyword(s):  
Strong Solution ◽  
Existence And Uniqueness ◽  
Mean Value ◽  
Quantum Measurements ◽  
Regularity Of Solutions ◽  
Schrodinger Equations ◽  
Quantum Trajectories ◽  
Initial Condition ◽  
Schrödinger Equations ◽  
The Mean

We develop linear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (LSSs) that unravel quantum master equations in Lindblad form into quantum trajectories. More precisely, this paper establishes the existence and uniqueness of the smooth strong solution Xt to a LSS with regular initial condition. Moreover, we obtain that the mean value of the square norm of Xt is constant. We also treat the approximation of LSSs by ordinary stochastic differential equations. We apply our results to: (i) models of quantum measurements of position and momentum; and (ii) a system formed by fermions.

scholarly journals The Navier–Stokes regularity problem

2020 ◽  
Vol 378 (2174) ◽  
pp. 20190526
Author(s):  
James C. Robinson
Keyword(s):  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Initial Condition ◽  
Theme Issue ◽  
Uniqueness Of Solutions ◽  
Rigorous Results ◽  
Navier Stokes Equations

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

scholarly journals INFINITELY MANY BROWNIAN GLOBULES WITH BROWNIAN RADII

2010 ◽  
Vol 10 (04) ◽  
pp. 591-612
Author(s):  
MYRIAM FRADON ◽  
SYLVIE RŒLLY
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Strong Solution ◽  
Local Time ◽  
Existence And Uniqueness ◽  
Infinite System ◽  
Time Dependent ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
Time Existence

We consider an infinite system of non-overlapping globules undergoing Brownian motions in ℝ3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional stochastic differential equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.

scholarly journals NONLINEAR MEAN-VALUE FORMULAS ON FRACTAL SETS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850091
Author(s):  
J. C. NAVARRO ◽  
J. D. ROSSI
Keyword(s):  
Comparison Principle ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Lipschitz Continuity ◽  
Continuous Solution ◽  
Sierpinski Gasket ◽  
Mean Value ◽  
Fractal Sets ◽  
Continuity Of Solutions ◽  
The Mean

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem [Formula: see text] in the Sierpiński gasket with prescribed values [Formula: see text], [Formula: see text] and [Formula: see text] at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.

scholarly journals Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations

2015 ◽  
Vol 145 (6) ◽  
pp. 1251-1282 ◽  
Author(s):  
Stefan Le Coz ◽  
Dong Li ◽  
Tai-Peng Tsai
Keyword(s):  
Solitary Waves ◽  
Fast Moving ◽  
Existence And Uniqueness ◽  
Large Time ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
New Construction

We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

scholarly journals Coupled local/nonlocal models in thin domains

2021 ◽  
pp. 1-31
Author(s):  
Bruna C. dos Santos ◽  
Sergio M. Oliva ◽  
Julio D. Rossi
Keyword(s):  
Mean Value ◽  
Limit Problem ◽  
Nonlocal Diffusion ◽  
Initial Condition ◽  
Thin Domains ◽  
Qualitative Properties ◽  
Uniqueness Of Solutions ◽  
Nonlocal Equations ◽  
Nonlocal Models ◽  
The Mean

In this paper, we analyze a model composed by coupled local and nonlocal diffusion equations acting in different subdomains. We consider the limit case when one of the subdomains is thin in one direction (it is concentrated to a domain of smaller dimension) and as a limit problem we obtain coupling between local and nonlocal equations acting in domains of different dimension. We find existence and uniqueness of solutions and we prove several qualitative properties (like conservation of mass and convergence to the mean value of the initial condition as time goes to infinity).

Unitary propagators for N-body Schrödinger equations in external field

2020 ◽  
pp. 2060002
Author(s):  
Kenji Yajima
Keyword(s):  
External Field ◽  
Existence And Uniqueness ◽  
Schrodinger Equations ◽  
Schrödinger Equations

We report our recent results on the existence and uniqueness of unitary propagators for [Formula: see text]-particle Schrödinger equations which may be applied to most interesting problems in physics.

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