arXiv subjects
Publications and source records attributed to Wilberclay G. Melo.
We study the fractional Navier-Stokes equations with cubic damping in the specific Lei-Lin-Gevrey space $\mathcal{X}^0_{a,σ}(\mathbb{R}^3)$ (with $a\geq0$ and $σ\geq1$). We establish local and small-data global well-posedness, quantitative blow-up criteria, Gevrey loss-of-radius estimates, and perturbative stability in the critical endpoint case $γ=\frac12$ of the usual Navier-Stokes system.
This paper studies the local well-posedness and the behavior at potential finite blow-up times of mild solutions to the three-dimensional fractional Boussinesq equations in Lei-Lin and Lei-Lin-Gevrey spaces $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$. By combining fixed point arguments with Fourier estimates adapted to these spaces, we obtain local existence and uniqueness results for data in $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$, including the usual Lei-Lin case $a=0$. We also establish several blow-up criteria for maximal mild solutions and derive lower bounds for the growth of the corresponding norms as the maximal time is approached. In the strict Lei-Lin-Gevrey regime, and when the dissipative exponents coincide, these estimates yield an exponential type blow-up criterion.
This work establishes conditions for the existence and uniqueness of local mild solutions to the Boussinesq equations with fractional dissipations in Sobolev-Gevrey spaces. We prove that a unique mild solution exists in an appropriate Sobolev-Gevrey class and analyze its behavior up to the maximal time of existence. In particular, we derive quantitative lower bounds describing how the norm of the solution must blow up as it approaches a finite maximal time. As a corollary, we deduce that the solution exhibits exponential growth.
In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $α\in (0,\frac{5}{4})$ in $L^2(\mathbb{R}^3)$ and $\dot{H}^s(\mathbb{R}^3)$ ($s\leq0$). More precisely, we prove that $u$ satisfies the following upper bound: $$ \|u(t)\|_{2}^2\leq C(1+t)^{-\frac{3-2p}{2α}}, \quad\forall t>0.$$ This estimate leads us to show the next inequality: $$ \|u(t)\|_{\dot{H}^{-δ}}^2\leq C(1+t)^{-\frac{3-2δ-2p}{2α}}, \quad\forall t>0.$$ These results are obtained by applying standard Fourier Analysis and they hold for $α\in(0,\frac{5}{4}),$ $p\in[-1,\frac{3}{2})$, $δ\in [0, \frac{3-2p}{2})$ and $u_0\in L^2(\mathbb{R}^3)\cap \mathcal{Y}^p(\mathbb{R}^3)$ (and also $u_0\in L^1(\mathbb{R}^3)$ for $p=-1$ and a certain finite set of values of $α$).
In this paper, we obtain new lower bounds on the Homogeneous Sobolev--norms of the maximal solution of the Magnetohydrodynamics Equations. This gives us some insight on the blow-up behavior of the solution. We utilize standard techniques from the Navier--Stokes Equations.