Entropy and the Stability of Classical Solutions

2009 ◽  
pp. 97-144
Author(s):  
Constantine M. Dafermos
Keyword(s):  
Classical Solutions ◽  
The Stability

scholarly journals L1STABILITY FOR THE VLASOV–POISSON–BOLTZMANN SYSTEM AROUND VACUUM

2006 ◽  
Vol 16 (09) ◽  
pp. 1505-1526 ◽  
Author(s):  
RENJUN DUAN ◽  
MEI ZHANG ◽  
CHANGJIANG ZHU
Keyword(s):  
Existence Theory ◽  
Classical Solutions ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Small Initial Data ◽  
Dimensional Phase Space ◽  
Nonlinear Functionals ◽  
Poisson Boltzmann ◽  
The Stability ◽  
Hard Potentials

Based on the global existence theory of the Vlasov–Poisson–Boltzmann system around vacuum in the N-dimensional phase space, in this paper, we prove the uniform L1stability of classical solutions for small initial data when N ≥ 4. In particular, we show that the stability can be established directly for the soft potentials, while for the hard potentials and hard sphere model it is obtained through the construction of some nonlinear functionals. These functionals thus generalize those constructed by Ha for the case without force to capture the effect of the force term on the time evolution of solutions. In addition, the local-in-time L1stability is also obtained for the case of N = 3.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

KINKS STABILITY IN TWO-HIGGS MODELS

1999 ◽  
Vol 14 (04) ◽  
pp. 247-256
Author(s):  
Y. BRIHAYE ◽  
P. KOSINSKI
Keyword(s):  
The Other ◽  
Classical Solutions ◽  
The Stability ◽  
Higgs Fields

We analyze the stability of two kink-type classical solutions available in a model describing two scalar Higgs fields in (1+1) dimensions. It is shown that one of these solutions is stable while the other is sphaleron. A few algebraic eigenmodes of the problem are exhibited.

UNIFORM L2-STABILITY ESTIMATES FOR THE RELATIVISTIC BOLTZMANN EQUATION

2009 ◽  
Vol 06 (02) ◽  
pp. 295-312 ◽  
Author(s):  
SEUNG-YEAL HA ◽  
HO LEE ◽  
XIONGFENG YANG ◽  
SEOK-BAE YUN
Keyword(s):  
Boltzmann Equation ◽  
Collision Operator ◽  
Stability Estimate ◽  
Classical Solutions ◽  
Stability Estimates ◽  
Type Estimate ◽  
Relativistic Boltzmann Equation ◽  
The Stability ◽  
L2 Stability ◽  
Linearized Collision Operator

In this paper, we derive an a prioriL2-stability estimate for classical solutions to the relativistic Boltzmann equation, when the initial datum is a small perturbation of a global relativistic Maxwellian. For the stability estimate, we use the dissipative property of the linearized collision operator and a Strichartz type estimate for classical solutions. As a direct application of our stability estimates, we establish that classical solutions in Glassey–Strauss and Hsiao–Yu's frameworks satisfy a uniform L2-stability estimate.

scholarly journals Implicit Difference Inequalities Corresponding to First-Order Partial Differential Functional Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-18
Author(s):  
Z. Kamont ◽  
K. Kropielnicka
Keyword(s):  
Functional Equations ◽  
Classical Solutions ◽  
Functional Inequalities ◽  
Partial Differential ◽  
First Order ◽  
Mixed Problems ◽  
Implicit Difference Schemes ◽  
Difference Methods ◽  
The Stability ◽  
Comparison Technique

We give a theorem on implicit difference functional inequalities generated by mixed problems for nonlinear systems of first-order partial differential functional equations. We apply this result in the investigations of the stability of difference methods. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference schemes. The proof of the convergence of difference method is based on comparison technique, and the result on difference functional inequalities is used. Numerical examples are presented.

scholarly journals Recent Progress in Fighting Ghosts in Quantum Gravity

Universe ◽  
2018 ◽  
Vol 4 (9) ◽  
pp. 91 ◽  
Author(s):  
Filipe Salles ◽  
Ilya Shapiro
Keyword(s):  
Quantum Gravity ◽  
High Energy ◽  
Effective Theory ◽  
Classical Solutions ◽  
Fundamental Theory ◽  
Physical Conditions ◽  
Higher Derivative ◽  
Low Energies ◽  
Gravitational Vacuum ◽  
The Stability

We review some of the recent results which can be useful for better understanding of the problem of stability of vacuum and in general classical solutions in higher derivative quantum gravity. The fourth derivative terms in the purely gravitational vacuum sector are requested by renormalizability already in both semiclassical and complete quantum gravity theories. However, because of these terms, the spectrum of the theory has unphysical ghost states which jeopardize the stability of classical solutions. At the quantum level, ghosts violate unitarity, and thus ghosts look incompatible with the consistency of the theory. The “dominating” or “standard” approach is to treat higher derivative terms as small perturbations at low energies. Such an effective theory is supposed to glue with an unknown fundamental theory in the high energy limit. We argue that the perspectives for such a scenario are not clear, to say the least. On the other hand, recently, there was certain progress in understanding physical conditions which can make ghosts not offensive. We survey these results and discuss the properties of the unknown fundamental theory which can provide these conditions satisfied.

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