We study the Maxwell system with localized conductivity σ and the boundary conditions of a perfect conductor on a simply connected domain Ω, assuming that there are no electric charges off the support of σ. For matrix-valued permittivity ε and permeability µ, we show strong stability of the underlying semigroup by checking the spectral criteria of the Arendt–Batty–Lyubich–V˜u Theorem. If ε = µ = 1, Ω is the cube (0, π)³ and supp σ contains a strip, the semigroup is polynomially stable of rate 1 2. To derive this result, we establish the resolvent estimate of the Borichev–Tomilov Theorem using an orthonormal basis of eigenfunctions of the Maxwell operator for σ = 0.

Given a compact connected Lie group G, we estimated the constant for the Sobolev inequality for an arbitrary Hörmander sub-Laplacian on G. The constant was estimated in terms of quantities depending on the intrinsic geometry of the group and on the Lebesgue parameters. We also provided applications of this result to the estimation of the operator norm for L^p-L^q-bounded subelliptic pseudo-differential operators in subelliptic Hörmander classes.

This paper studies the stability of a weak mild solution of the Navier–Stokes–Oseen equations in the solenoidal Lorentz space L³ σ,w. Our approach relies on dual space pair and suitable estimates in our setting for the Oseen semigroup. Therefore, we get a new result for the stability of a weak mild solution following the initial datum and external force.

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin’s maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.

Motivated by the study of the possible breakdown of regularity for solutions of the incompressible Navier-Stokes system in the whole three-dimensional space $\mathbb{R}^{3}$, we prove that the norm $\|u\|_{L_{T}^{\infty}\left(B_{2 \infty}^{\frac{1}{2}}\right)}$ and the quantity $\sup _{I \subset[0, T[ } \frac{1}{\sqrt{|I|}} \int_{I}\|\nabla u(t)\|_{L^{2}}^{2}$ dt are non linearly equivalent. The proofs rely on the Duhamel formula, dyadic localization in the Fourier space, and for one of them, on the energy inequality.

The aim of this note is to examine Liouville-type theorems for p-Laplacian-type operators. Guided by the Laplacian case, analogous results are established for the p-Laplacian and sums of operators of this type.

This paper deals with the null controllability of a coupled nonlinear parabolic system. The coefficients of the system operators depend on the states, and the control acts through the first equation. To solve the control problem of the linearized system, we use maximum regularity results in the spaces L^q(0,T,L^p(Ω)), and we use Liusternick’s inverse function theorem for the nonlinear controllability problem. In addition, as an appendix, we prove the well-posedness of the system.