scholarly journals Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2995
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Kinetic Equation ◽  
Linear Partial Differential Equation ◽  
Metal Vapor ◽  
Problem Solution ◽  
Semiclassical Approach ◽  
Active Media ◽  
The Cauchy Problem

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

scholarly journals Continuity equations and ODE flows with non-smooth velocity

2014 ◽  
Vol 144 (6) ◽  
pp. 1191-1244 ◽  
Author(s):  
Luigi Ambrosio ◽  
Gianluca Crippa
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Bounded Variation ◽  
Velocity Fields ◽  
Well Posedness ◽  
Continuity Equations ◽  
Quantitative Estimates ◽  
The Cauchy Problem

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

Estimates for Solutions of Wave Equations with Vanishing Curvature

1985 ◽  
Vol 37 (6) ◽  
pp. 1176-1200 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Linear Combination ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

scholarly journals The probabilistic approach to the analysis of the limiting behavior of an integro-diffebential equation depending on a small parameter, and its application to stochastic processes

1994 ◽  
Vol 7 (1) ◽  
pp. 25-31
Author(s):  
O. V. Borisenko ◽  
A. D. Borisenko ◽  
I. G. Malyshev
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Stochastic Differential Equation ◽  
Small Parameter ◽  
Stochastic Processes ◽  
Probabilistic Approach ◽  
Problem Solution ◽  
Poisson Measure ◽  
Limiting Behavior ◽  
The Cauchy Problem

Using connection between stochastic differential equation with Poisson measure term and its Kolmogorov's equation, we investigate the limiting behavior of the Cauchy problem solution of the integro differential equation with coefficients depending on a small parameter. We also study the dependence of the limiting equation on the order of the parameter.

scholarly journals On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient non-linearity

2020 ◽  
Author(s):  
Maxim Korpusov ◽  
Alexandra Matveeva
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Critical Exponents ◽  
Weak Sense ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

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