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Remarks on non controllability of the heat equation with memory

Sergio Guerrero, Oleg Yurievich Imanuvilov (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we deal with the null controllability problem for the heat equation with a memory term by means of boundary controls. For each positive final time T and when the control is acting on the whole boundary, we prove that there exists a set of initial conditions such that the null controllability property fails.

Remarks on positive solutions to a semilinear Neumann problem

Anna Maria Candela, Monica Lazzo (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we study the influence of the domain topology on the multiplicity of solutions to a semilinear Neumann problem. In particular, we show that the number of positive solutions is stable under small perturbations of the domain.

Remarks on regularity criteria for the Navier-Stokes equations with axisymmetric data

Zujin Zhang (2016)

Annales Polonici Mathematici

We consider the axisymmetric Navier-Stokes equations with non-zero swirl component. By invoking the Hardy-Sobolev interpolation inequality, Hardy inequality and the theory of * A β (1 < β < ∞) weights, we establish regularity criteria involving u r , ω z or ω θ in some weighted Lebesgue spaces. This improves many previous results.

Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Zujin Zhang, Chenxuan Tong (2022)

Applications of Mathematics

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that | ω r ( x , t ) | + | ω z ( r , t ) | ≤ C r 10 , 0 < r ≤ 1 2 . By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ω r , ω z and ω θ / r on different hollow cylinders, we are able to improve it and obtain | ω r ( x , t ) | + | ω z ( r , t ) | ≤ C | ln r | r 17 / 2 , 0 < r ≤ 1 2 .

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