Well-posedness, stability and determining modes to 3D Burgers equation in Gevrey class
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Well-posedness, stability and determining modes to 3D Burgers equation in Gevrey class Ridha Selmi and Abdelkerim Chˆ aabani Abstract. This paper aims to prove that the three-dimensional periodic Burgers equation has a unique global in time solution, in the Lebesgue–Gevrey space. In particular, the initial data that belong to L2a,σ (T3 ) give rise to a solution in 1 (T3 )), where L2 ˙r C(R+ ; L2a,σ (T3 )) ∩ L2 (R+ ; Ha,σ a,σ is identified with the homogeneous Sobolev–Gevrey space Ha,σ when r = 0 with parameters a ∈ (0, 1) and σ ≥ 1. We also prove that the solution is stable under perturbation and that the long-time behavior of Burgers system is determined by a finite number of degrees of freedom in L2a,σ . Energy methods, compactness methods and Fourier analysis are the main tools. Mathematics Subject Classification. Primary 35A01, 35A02; Secondary 35B05, 35B10, 35B35. Keywords. Burgers equation, Turbulence, Gevrey class, Sobolev space, existence, stability, determining modes.
1. Introduction We consider the three-dimensional periodic Burgers system ∂t u − νΔu + (u · ∇)u = 0, u|t=0 = u0 (x), x ∈ T , 3
(t, x) ∈ R+ × T3
(1.1) (1.2)
where ν > 0 is the viscosity, the system is subject to periodic boundary conditions with basic domain T3 = [0, 2πL]3 , the initial data u0 belong to Lebesgue–Gevrey space L2a,σ . The well-posedness of (1.1– 1.2) in critical Sobolev space H 1/2 (T3 ) was proved in [10]. In [12], we have addressed the problem in 1/2 critical Sobolev–Gevrey space Ha,σ (T3 ). In both above-mentioned papers, the aim was a global in time existence and uniqueness result. Several difficulties were encountered, and authors dealt with effectively. In the present paper, we establish a global in time well-posedness result in L2a,σ as well as a smallness theory result that we will use to prove that the long-time behavior of Burgers system is determined by a finite number of degrees of freedom. The result states that if the L2a,σ (T3 )-norm of the initial data is less than the viscosity ν multiplied by a certain constant, then the solution u(t) to Burgers equation (1.1) 2 3 2 3 ˙1 is uniformly bounded with respect to time t in L∞ R+ (La,σ (T )) ∩ LR+ (Ha,σ (T )). It is worth emphasizing that the global well-posedness holds in both cases, namely for arbitrary large initial data and small as required initial data, so it should be said that independent of time refers to the upper bounds not the norms themselves. We also prove a robustness result, i.e., the persistence of regularity under perturbation of the initial condition. It is worthwhile to note that the lack of the divergence free condition prevents us from making the usual L2 energy estimates that would give existence of weak solutions as said in [10]. Here, the situation differs from its counterpart in L2 and the constraint can be overcome by means of the properties of our functional spaces framework choice. In fact, the regularity provided by Gevrey class analyticity is stronger even than that provided by C ∞ r
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