This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals$S$^{$α$}, 1 <$α$< ∞. Alternatively, for every 1 <$α$< ∞, there is a constant$c$_{$α$}> 0 such that $$ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $$ where$f$is a Lipschitz function with $$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$ $$ {\left\| \cdot \right\|_{\alpha }} $$ is the norm is$S$^{$α$}, and$a$and$b$are self-adjoint linear operators such that $$ a - b \in {S^{\alpha }} $$ .

The singular parabolic problem $u_t-\triangle u=\l{\frac{1+\d|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\O$ of $\Rn$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\l_\d^*>0$ such that if $0<\l<\l_\d^*$, all solutions exist globally; while for $\l>\l_\d^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $\l$ is investigated. Furthermore, the quenching set is a compact subset of $\O$, provided $\O$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $\O$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.

In this paper, we consider the compressible NavierStokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support. The viscosity coefficient is assumed to be a power function of the density so that the Cauchy problem is well-posed. New global existence results are established when the density function connects to the vacuum states continuously. For this, some new <i>a priori</i> estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum. We will also give a non-global existence theorem of regular solutions when the initial data are of compact support in Eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum.

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.

This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. We also introduce the free boundary problem for the compressible primitive equations.