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In this paper we address the decay of solutions to the four-dimen\-sional energy-critical nonlinear heat equation in the critical space $\dot{H}^1$. Recently, it was proven that the $\dot{H}^1$ norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2206.09445

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter $\kappa$ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large $\kappa$ regardless of the size of initial data., Comment: 27 pages

In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term $A \mathcal{R}_{1} \vartheta$, in the domain $\Omega =\mathbb{T}^{1} \times \mathbb{R}$. In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described by a linear equation. On the other hand, performing the combined fast rotation and inviscid limits, we show that the initial data $\overline{\vartheta}_{0}$ is transported along the motion. The proof of the convergence is based on an application of the Aubin-Lions lemma., Comment: Submitted. arXiv admin note: text overlap with arXiv:2005.04279, arXiv:2106.13699, arXiv:2109.09841, arXiv:2205.11889

An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ 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., Comment: 20 pages. Proof of Theorem 1.1 reformulated. Coauthor added