Global Existence and Temporal Decay of Large Solutions for the Poisson--Nernst--Planck Equations in Low Regularity Spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.

The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p ( R 2 ) ∩ L 2 ( R 2 ). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.