Convergence Of Initial Conditions Research Articles

Global Weak Besov Solutions of the Navier–Stokes Equations and Applications

We introduce a notion of global weak solution to the Navier–Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $${\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}$$ , p > 3. These solutions satisfy a certain stability property with respect to the weak- $${\ast}$$ convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.

Uniform global existence and convergence of Euler-Maxwell systems with small parameters

The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.