Knudsen type group for time in ℝ and related Boltzmann type equations

2021 ◽  
pp. 2150072
Author(s):  
Jörg-Uwe Löbus
Keyword(s):  
Boundary Conditions ◽  
Global Solution ◽  
Physical Space ◽  
Type Equation ◽  
Collision Kernel ◽  
Unique Global Solution ◽  
Initial Value ◽  
Type Group ◽  
Diffusive Boundary ◽  
Diffusive Part

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [Formula: see text]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [Formula: see text] or for time [Formula: see text] for some [Formula: see text] which is independent of the initial value at time 0. Depending on the collision kernel, [Formula: see text] can be arbitrarily small.

DIFFUSION APPROXIMATION FOR TRANSPORT PROCESSES WITH GENERAL REFLECTION BOUNDARY CONDITIONS

2006 ◽  
Vol 16 (05) ◽  
pp. 717-762 ◽  
Author(s):  
CRISTINA COSTANTINI ◽  
THOMAS G. KURTZ
Keyword(s):  
Boundary Conditions ◽  
Stochastic Averaging ◽  
Transport Processes ◽  
Collision Kernel ◽  
Diffusion Approximations ◽  
Transport Models ◽  
Scattering Kernel ◽  
Reflection Boundary ◽  
Averaging Techniques ◽  
Number Of Particles

Diffusion approximations are obtained for space inhomogeneous linear transport models with reflection boundary conditions. The collision kernel is not required to satisfy any balance condition and the scattering kernel on the boundary is general enough to include all examples of boundary conditions known to the authors (with conservation of the number of particles) and, in addition, to model the Debye sheath. The mathematical approach does not rely on Hilbert expansions, but rather on martingale and stochastic averaging techniques.

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

scholarly journals A Paneitz–Branson type equation with Neumann boundary conditions

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Bonheure ◽  
Hussein Cheikh Ali ◽  
Robson Nascimento
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Type Equation ◽  
Neumann Boundary ◽  
Rayleigh Quotient ◽  
Neumann Boundary Conditions ◽  
Asymptotic Estimates ◽  
Best Constant ◽  
Order Elliptic Problem ◽  
Nonconstant Solution

AbstractWe consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

scholarly journals Global Attractors for the Higher-Order Evolution Equation

2020 ◽  
Vol 5 (1) ◽  
pp. 195-210
Author(s):  
Erhan Pişkin ◽  
Hazal Yüksekkaya
Keyword(s):  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Global Solution ◽  
Global Attractor ◽  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Evolution Type

AbstractIn this paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.

scholarly journals Dissipationof Energyofan AC Electric Fieldina Semi-Infinit Electron Plasma with Diffusive Boundary Conditions

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Electric Field ◽  
Boundary Conditions ◽  
Electron Plasma ◽  
Surface Absorption ◽  
Ac Electric Field ◽  
Plasma Response ◽  
Diffusive Boundary

Let us consider the electron plasma response with an arbitary degree of degeneracy to an external ac electric field. Surface absorption of the energy of an electric field is calculated.

scholarly journals Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Equilibrium Solution ◽  
Global Solutions ◽  
Initial Time ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Unique Global Solution ◽  
Initial Value ◽  
The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

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