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An algebraic upper bound for the decay rate of solutions to the Navier–Stokes and Navier–Stokes–Coriolis equations in the critical space H ˙ 1 2 (R 3) is derived using the Fourier splitting method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts. The proof is carried on purely in the critical space, as no L 2 (R 3) estimates are available for the solution. This is the first instance in which such a method is used for obtaining decay bounds in a critical space for a nonlinear equation. [ABSTRACT FROM AUTHOR]

We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded H ∞ -calculus with angle smaller than π / 2 . As an application, we consider critical spaces for the Navier–Stokes equations on the surface Σ . In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L 2 (Σ , T Σ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field. [ABSTRACT FROM AUTHOR]

This paper considers the large time properties of global solutions to the 3D incompressible magnetohydrodynamic system in critical spaces H ˙ 1 2 (R 3) and L 3 (R 3) . More precisely, we obtain the temporal decay rate (1 + t) - 1 4 for a small global solution in the space H ˙ 1 2 (R 3) , and a small global solution in the space L 3 (R 3) is non-increasing and decays to zero at infinity. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the optimal time decay rate of solutions for the compressible flow of nematic liquid crystals in Lp type critical framework and any dimensionN≥2. Concretely, if the low frequencies of the initial data are in e.g. Lp/2(RN), we could obtain that the Lp norm of the critical global solution (ρ-1,u) decays like t-Np+N4, while the Lp norm of d-d^ decays like t-N2p for t→∞, exactly as shown in the work by Gao et al. (J Differ Equ 261:2334-2383, 2016) when p=2 and N=3, for solutions with high Sobolev regularity. As a by-product, we derive an accurate description of the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The main tools used here are the Littlewood-Paley theory and refined time weighted inequalities in Fourier space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the large time behavior of global solutions to the initial value problem for the compressible Navier–Stokes–Poisson system in the L p critical framework and in any dimension N ≥ 3 . We obtain the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents , which improves the decay results in high Sobolev regularity. The proof is mainly based on the Littlewood–Paley theory and refined time weighted inequalities in Fourier space. [ABSTRACT FROM AUTHOR]