scholarly journals OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS

2012 ◽  
Vol 22 (07) ◽  
pp. 1250012 ◽  
Author(s):  
RENJUN DUAN ◽  
LIZHI RUAN ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Source Term ◽  
Decay Rates ◽  
Heat Diffusion ◽  
Time Behavior ◽  
Time Decay ◽  
Diffusion Type ◽  
Linear Solution ◽  
Optimal Decay Rates

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.

STABILITY OF SHOCK PROFILES FOR NONCONVEX SCALAR VISCOUS CONSERVATION LAWS

1995 ◽  
Vol 05 (03) ◽  
pp. 279-296 ◽  
Author(s):  
MING MEI
Keyword(s):  
Conservation Laws ◽  
Decay Rates ◽  
Asymptotically Stable ◽  
Time Decay ◽  
Shock Condition ◽  
Viscous Conservation Laws ◽  
Time Decay Rates ◽  
Shock Profiles ◽  
Degenerate Shock ◽  
The Stability

This paper is to study the stability of shock profiles for nonconvex scalar viscous conservation laws with the nondegenerate and the degenerate shock conditions by means of an elementary energy method. In both cases, the shock profiles are proved to be asymptotically stable for suitably small initial disturbances. Moreover, in the case of nondegenerate shock condition, time decay rates of asymptotics are also obtained.

THE BOLTZMANN EQUATION IN THE SPACE $L^2\cap L^\infty_\beta$: GLOBAL AND TIME-PERIODIC SOLUTIONS

2006 ◽  
Vol 04 (03) ◽  
pp. 263-310 ◽  
Author(s):  
SEIJI UKAI ◽  
TONG YANG
Keyword(s):  
Boltzmann Equation ◽  
Source Term ◽  
Decay Rates ◽  
Regularity Conditions ◽  
Unique Existence ◽  
The Boltzmann Equation ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Derivatives Of ◽  
Time Periodic

We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator.

scholarly journals A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ruy Coimbra Charão ◽  
Alessandra Piske ◽  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Decay Rates ◽  
Plate Equation ◽  
New Model ◽  
Asymptotic Profile ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Logarithmic Type

<p style='text-indent:20px;'>We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in <inline-formula><tex-math id="M1">\begin{document}$ {{\bf R}}^{n} $\end{document}</tex-math></inline-formula>, and study the asymptotic profile and optimal decay rates of solutions as <inline-formula><tex-math id="M2">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-sense. The operator <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [<xref ref-type="bibr" rid="b7">7</xref>]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.</p>

scholarly journals Global strong solutions and large time behavior of 2D tropical climate model with zero thermal diffusion

2021 ◽  
Author(s):  
Dongjuan Niu ◽  
Huiru Wu
Keyword(s):  
Thermal Diffusion ◽  
Large Time ◽  
Climate Model ◽  
Low Frequency ◽  
Tropical Climate ◽  
Decay Rates ◽  
Thermal Dissipation ◽  
Time Behavior ◽  
Time Decay ◽  
Small Initial Data

In this article, we study the global well-posedness and large-time behaviors of solutions to the two-dimensional tropical climate system with zero thermal diffusion for a small initial data in the whole space. The main approaches include high and low frequency decomposition method and exploiting the structure of system (1) to obtain the estimates of thermal dissipation. We utilize the time decay properties of the kernels to a linear differential equation to obtain the decay rates of solutions of the low frequency part and the decay property of exponential operator for the high frequency part. The key ingredient here is the explicit large-time decay rate of solutions.

scholarly journals Cauchy problem for general time fractional diffusion equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1545-1559
Author(s):  
Chung-Sik Sin
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Behavior ◽  
Analysis Technique ◽  
The Cauchy Problem ◽  
Long Time ◽  
Time Fractional Diffusion Equation

Abstract In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.

scholarly journals A new stability result for a thermoelastic Bresse system with viscoelastic damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Soh Edwin Mukiawa ◽  
Cyril Dennis Enyi ◽  
Tijani Abdulaziz Apalara
Keyword(s):  
Stability Result ◽  
Bending Moment ◽  
Decay Rates ◽  
Viscoelastic Damping ◽  
Physical Parameters ◽  
Bresse System ◽  
Solution Energy ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Weaker Conditions

AbstractWe investigate a thermoelastic Bresse system with viscoelastic damping acting on the shear force and heat conduction acting on the bending moment. We show that with weaker conditions on the relaxation function and physical parameters, the solution energy has general and optimal decay rates. Some examples are given to illustrate the findings.

Sign in / Sign up

Export Citation Format

Share Document