We compute the sharp time decay rates of the solutions of the IVP for quasi-geostrophic equation and the Boussinesq model, subject to fractional dissipation. Moreover, we explicitly identify the asymptotic profiles, the kernel of the α stable processes, which are analogues of the Oseen vortices.
This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation (α,β) : α + β = 1. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as α > 2/3. This improves upon some recent works [13] and [23].The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by new set of commutator estimates (in both negative and positive index Sobolev spaces!), which may be of independent interest.1 Regularity of the critical 2D Boussinesq EquationsMathematically, the problem for global regularity of 1.1 is an interesting and a subtle one. Intuitively, the lower the values of α,β, the harder it is to prove that solutions emanating from sufficiently smooth and localized data persist globally. In particular, the problem with no dissipation (i.e. ν = κ = 0) remains open. This is very similar to the Euler equation in two and three spatial dimensions and in fact numerous studies explore the possibility of finite time blow up, [22].Next, we take on the difficult task of reviewing the recent results regarding wellposedness issues for 1.1. Indeed, there has been tremendous interest in this problem in the last fifteen years. In the classical case, when the diffusion is given by the regular Laplacian (i.e. α = β = 2), the global regularity follows just as it does for the 2D Navier-Stokes model, [6,18]. In the works [1,12], global regularity was proved in the presence of one full Laplacian, that is in the cases α = 2,β = 0 or α = 0,β = 2. In more recent years, the full two parameter range of α,β was explored in detail. Based on the currently available results, it is natural to draw the conclusion that one expects global regularity in the cases α + β ≥ 1, while the case α + β < 1 generally remains open 1 . We thus adopt the notion of criticality -namely, we say that a pair (α,β) is subcritical if α + β > 1, critical if α + β = 1 and supercritical if α + β < 1.As it was alluded above, in the supercritical regime the behavior of the solutions remains a mystery. Apart from some numerical simulations, the only rigorous results that we are aware of is the eventual regularity of the solutions, [27], for appropriate supercritical regime of the diffusivity parameters. To be sure, such statement does not, per se exclude a finite time blow up of some solutions. It remains to discuss the critical and subcritical cases. This is probably a good place to observe that if global regularity holds for critical pair (α 0 ,β 0 ) : α 0 + β 0 = 1, then it must hold for all subcritical pairs in the form 2 (α,β 0 ),α > α 0 and (α 0 ,β) : β > β 0 . Thus, clearly global regularity results on the critical line are superior, in the sense described above, to subcritical ones. That being said, the subcritical theory is far from obvious or well-understood. Many res...
We investigate a coordinate-free model of flame fronts introduced by Frankel and Sivashinsky; this model has a parameter α which relates to how unstable the front might be. We first prove short-time well-posedness of the coordinate-free model, for any value of α > 0. We then argue that near the threshold α ≈ 1, the solution stays arbitrarily close to the solution of the weakly nonlinear Kuramoto-Sivashinsky (KS) equation, as long as the initial values are close.
We consider the mass-in-mass (MiM) lattice when the internal resonators are very small. When there are no internal resonators the lattice reduces to a standard Fermi-Pasta-Ulam-Tsingou (FPUT) system. We show that the solution of the MiM system, with suitable initial data, shadows the FPUT system for long periods of time. Using some classical oscillatory integral estimates we can conclude that the error of the approximation is (in some settings) higher than one may expect.
In this paper, a class of finite difference numerical technique is presented for the solution of the second-order linear inhomogeneous damped wave equation. The consistency, stability and convergences of these numerical schemes are discussed. The results obtained are compared to the exact solution as well as ordinary explicit, implicit finite difference methods, and the fourth-order compact method (FOCM) of [6]. The general idea of these methods is developed by using C 0 -semigroups operator theory. We also showed that the stability region for the explicit finite difference scheme depends on the damping coefficient.
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