arXiv subjects

Pablo Braz e Silva

Publications and source records attributed to Pablo Braz e Silva.

7 recordsLinked to original sources

arXiv2026

Asymptotic profiles and large-time behavior for 3D micropolar fluid equations with possibly vanishing spin viscosity

We consider 3D micropolar flows with possible vanishing spin viscosity and investigate the decay of the energy for large times. We compute first the exact $L^2$-asymptotic profile, as $t\to+\infty$, for solutions to the linear 3D micropolar equations, up to the second order. For the nonlinear micropolar system, we first establish the existence of restricted Leray solutions. This new notion of solutions is required because it is not known whether the weak finite energy solutions verify a strong energy inequality. Next, we study the large-time behavior of restricted Leray solutions, and prove that they behave asymptotically in $L^2$ like their linear counterpart, up to the critical algebraic decay rate $O(t^{-5/2})$ for the energy. Applying a remarkable linear enstrophy identity, we show that the microrotation field exhibits faster decay in $L^2$ than the velocity field, allowing us to impose our hypothesis on the velocity field only and not on the angular velocity.

math.AP↗

arXiv2026

Control of the Schrödinger equation in $\mathbb{R}^3$: The critical case

This article deals with the $H^{1}$--level local null controllability for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$. Firstly, we demonstrate that the problem under consideration is well-posed using Strichartz estimates. Moreover, through the Hilbert uniqueness method, we prove the linear Schrödinger equation to be controllable. Finally, we use a perturbation argument and show local controllability for the critical nonlinear Schrödinger equation.

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arXiv2024

Stabilization of a perturbed quintic defocusing Schrödinger equation in $\mathbb{R}^{3}$

This article addresses the stabilizability of a perturbed quintic defocusing Schrödinger equation in $\mathbb{R}^{3}$ at the $H^1$--energy level, considering the influence of a damping mechanism. More specifically, we establish a profile decomposition for both linear and nonlinear systems and use them to show that, under certain conditions, the sequence of nonlinear solutions can be effectively linearized. Lastly, through microlocal analysis techniques, we prove the local exponential stabilization of the solution to the perturbed Schrödinger equation in $\mathbb{R}^{3}$ showing an observability inequality for the solution of the system under consideration, which is the key result of this work.

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arXiv2024

Stability and eigenvalue bounds for micropolar shear flows

We prove eigenvalue bounds for two-dimensional linearized disturbances of parallel flows of micropolar fluids, deriving the Orr-Sommerfeld equations and providing a sufficient condition for linear stability of such flows. We also derive wave speed bounds.

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arXiv2017

Some remarks on the regularity time of Leray solutions to the Navier-Stokes equations

In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4} if n = 3, and t* <= K_{4} nu^{-3} || u(.,0) ||_{L2}^{2} if n = 4 (in particular, by reducing the current best known values for the constants K_{3}, K_{4}). Some related results of clear interest are also included (derived) in our discussion.

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arXiv2003

On the resolvent technique for stability of plane Couette flow

We discuss the application of the resolvent technique to prove stability of plane Couette flow. Using this technique, we derive a threshold amplitude for perturbations that can lead to turbulence in terms of the Reynolds number. Our main objective is to show exactly how much control one should have over the perturbation to assure stability via this technique.

math.AP↗

arXiv2002

Resolvent estimates for 2 dimensional perturbations of plane Couette Flow

We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing perturbations of the Couette flow. We prove estimates on the L_2 norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the unstable half plane. For the remaining region, we show that the problem can be reduced to estimating the solution of a homogeneous ordinary differential equation with non-homogeneous boundary conditions. Numerical approximations indicate that the norm of the resolvent is proportional to R in the whole region of interest.

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