scholarly journals Asymptotic behaviors of the semigroup of the linearized Landau operator for the very soft and Coulomb potentials

2021 ◽  
Author(s):  
Jiawei SUN ◽  
Yakui WU
Keyword(s):  
Spectral Gap ◽  
Collision Operator ◽  
Accretive Operator ◽  
Decay Estimates ◽  
Time Decay ◽  
Complete Spectrum ◽  
Asymptotic Behaviors ◽  
Hard Potentials ◽  
Spectrum Structure ◽  
Coulomb Potentials

We study the asymptotic behaviors of the semigroup generated by the linearized Landau operator in the case of the very soft potentials and Coulomb potential. Compared with the hard potentials, Maxwellian molecules and moderately soft potentials, there is no spectral gap for the linearized Landau operator with the very soft and Coulomb potentials. By introducing a new decomposition of the linear Landau collision operator $L$ including an accretive operator and a relatively compact operator, we establish the complete spectrum structure for the linearized Landau operator with the very soft and Coulomb potentials and furthermore derive the time decay estimates of the corresponding semigroup in a weighted velocity space.

scholarly journals Global small data solutions for semilinear waves with two dissipative terms

2021 ◽  
Author(s):  
Wenhui Chen ◽  
Marcello D’Abbicco ◽  
Giovanni Girardi
Keyword(s):  
Wave Equations ◽  
Space Dimension ◽  
Full Range ◽  
Small Data ◽  
Decay Estimates ◽  
Time Decay ◽  
Nonexistence Result ◽  
Derivative Type ◽  
Long Time ◽  
Dissipative Terms

AbstractIn this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity $$|u|^p$$ | u | p or nonlinearity of derivative type $$|u_t|^p$$ | u t | p , in any space dimension $$n\geqslant 1$$ n ⩾ 1 , for supercritical powers $$p>{\bar{p}}$$ p > p ¯ . The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive $$L^r-L^q$$ L r - L q long time decay estimates for the solution in the full range $$1\leqslant r\leqslant q\leqslant \infty $$ 1 ⩽ r ⩽ q ⩽ ∞ . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers $$p<{\bar{p}}$$ p < p ¯ .

scholarly journals Velocity decay estimates for Boltzmann equation with hard potentials

Nonlinearity ◽  
2020 ◽  
Vol 33 (6) ◽  
pp. 2941-2958
Author(s):  
Stephen Cameron ◽  
Stanley Snelson
Keyword(s):  
Boltzmann Equation ◽  
Decay Estimates ◽  
Velocity Decay ◽  
Hard Potentials

LANDAU–GINZBURG TYPE EQUATIONS IN THE SUBCRITICAL CASE

2003 ◽  
Vol 05 (01) ◽  
pp. 127-145 ◽  
Author(s):  
NAKAO HAYASHI ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Decay Rate ◽  
Unique Solution ◽  
Linear Case ◽  
Decay Estimates ◽  
Time Decay ◽  
Subcritical Case ◽  
Small Norm ◽  
The Cauchy Problem

We study the Cauchy problem for the nonlinear Landau–Ginzburg equation [Formula: see text] where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case [Formula: see text]. We assume that θ = | ∫ u0(x) dx| ≠ 0 and ℜδ (α, β) > 0, where [Formula: see text] Furthermore we suppose that the initial data u0 ∈ L1 are such that (1+|x|)au0 ∈ L1, with sufficiently small norm ε = ‖(1 + |x|)a u0 ‖1, where a ∈ (0,1). Also we assume that σ is sufficiently close to [Formula: see text]. Then there exists a unique solution of the Cauchy problem (*) such that [Formula: see text] satisfying the following time decay estimates for large t > 0[Formula: see text] Note that in comparison with the corresponding linear case the decay rate of the solutions of (*) is more rapid.

Sign in / Sign up

Export Citation Format

Share Document