scholarly journals Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior

2015 ◽  
Vol 32 (3) ◽  
pp. 687-713 ◽  
Author(s):  
Tsukasa Iwabuchi
Keyword(s):  
Besov Spaces ◽  
Large Time ◽  
Burgers Equation ◽  
Global Solutions ◽  
Large Time Behavior ◽  
Time Behavior

Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

2018 ◽  
Vol 15 (02) ◽  
pp. 259-290 ◽  
Author(s):  
Weixuan Shi ◽  
Jiang Xu
Keyword(s):  
Large Time ◽  
Characteristic Root ◽  
Global Solutions ◽  
Strong Solutions ◽  
Parabolic Systems ◽  
Large Time Behavior ◽  
Optimal Time ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Mhd System

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.

scholarly journals The Asymptotic Properties of Turbulent Solutions to the Navier-Stokes Equations

10.14311/1688 ◽  
2012 ◽  
Vol 52 (6) ◽  
Author(s):  
Zdenek Skalák
Keyword(s):  
Besov Spaces ◽  
Large Time ◽  
Stokes Equations ◽  
Asymptotic Properties ◽  
Large Time Behavior ◽  
Energy Concentration ◽  
Navier Stokes ◽  
Related Phenomenon ◽  
Time Behavior ◽  
Navier Stokes Equations

In this paper we study the large time behavior of solutions to the Navier-Stokes equations. We present a brief survey of results concerning energy decay, and discuss a related phenomenon of the large time energy concentration in the frequency space occurring in any turbulent solution. This leads us to the study of solutions in the Besov spaces and to proof that if we choose a suitable initial condition then in some Besov spaces the energy of the associated solution does not decrease asymptotically to zero.

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