Paulo R. Zingano

Nonlinear L under large disturbances

Abstract We derive time-asymptotic decay rates in L2 for large disturbances to some important classes of solutions of the Cauchy problem for a number of uniformly parabolic equations, provided only that the disturbances belong to appropriate Lp spaces at initial time. Examples considered include the scalar nonlinear advection-diffusion equation ut + f(u)x = (b(u)ux)x and the parabolic system u t + (ϕ(¦ u ¦)) x = (B( u ) u x ) x , where u (x,t)∈ R m , ϕ is a given scalar function and B( u ) is a uniformly positive-definite diagonal matrix.

On the supnorm form of Leray’s problem for the incompressible Navier-Stokes equations

We show that t3/4  u(⋅,t) L∞(R3)→0 as t → ∞ for all Leray-Hopf’s global weak solutions u(⋅, t) of the incompressible Navier-Stokes equations in ℝ3. It is also shown that t u(⋅,t)−eΔtu0 L∞(R3)→0 as t → ∞, where eΔt is the heat semigroup, as well as other fundamental new results. In spite of the complexity of the questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.

General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

We derive general bounds for the large time size of supnorm values of solutions to one-dimensional advection-diffusion equations with initial data for some and arbitrary bounded advection speeds , introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

On nilpotent singular systems

Abstract We derive some results for linear differential-algebraic equations (DAEs) of the form N(t) z ′(t)+ z (t)= h (t) , where N(t) is a smooth nilpotent matrix for all t concerned and such that the system is uniquely solvable, i.e., has exactly one solution for each smooth h . Such systems play a fundamental role in the investigation of more general DAEs, but their theory is still incomplete. We give some sufficient conditions for unique solvability, and a global representation for the solution operator constructed in terms of a finite set of special solutions.

On the Hardy--Littlewood Theorem for Functions of Bounded Variation

We derive here an improved version of the well-known Hardy--Littlewood theorem for functions of bounded variation in the classical sense of Jordan. Results for the positive and negative variations are also discussed.