Decay rates for the 4D energy-critical nonlinear heat equation

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:2206.0944

Algebraic decay rates for 3D Navier-Stokes and Navier-Stokes-Coriolis equations in H˙12 \dot{H}^{\frac{1}{2}}

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. Title changed. Emphasis placed on new estimates in critical spac

Global well-posedness for a family of regularized Benjamin-type equations

In this work we prove local and global well-posedness results for the Cauchy problem of a family of regularized nonlinear Benjamin-type equations in both periodic and nonperiodic Sobolev spaces.Comment: 22 page

Decay of weak solutions to the 2D dissipative quasi-geostrophic equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data is in $L^2$ only, we prove that the $L^2$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For the initial data in $L^p \cap L^2$, with $1 \leq p < 2$, we are able to obtain a uniform decay rate in $L^2$. We also prove that when the $L^{\frac{2}{2 \alpha -1}}$ norm of the initial data is small enough, the $L^q$ norms, for $q > \frac{2}{2 \alpha -1}$ have uniform decay rates. This result allows us to prove decay for the $L^q$ norms, for $q \geq \frac{2}{2 \alpha -1}$, when the initial data is in $L^2 \cap L^{\frac{2}{2 \alpha -1}}$.Comment: A paragraph describing work by Carrillo and Ferreira proving results directly related to the ones in this paper is added in the Introduction. Rest of the article remains unchange

Large Time Behavior of the Navier-Stokes Flow

International audienceDifferent results related to the asymptotic behavior of incompressible fluid equations are analyzed as time tends to infinity. The main focus is on the solutions to the Navier-Stokes equations, but in the final section a brief discussion is added on solutions to Magneto-Hydrodynamics, Liquid crystals, Quasi-Geostrophic and Boussinesq equations. Consideration is given to results on decay, asymptotic profiles, and stability for finite and nonfinite energy solutions