The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent γ lies in the stability regime (4/3, 2), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent γ lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.

It is well-known that in Banach spaces with finite cotype, the $R$ -bounded and $\gamma $ -bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$ -boundedness implies $\gamma $ -boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$ -boundedness is stable under taking adjoints if and only if the underlying space is $K$ -convex.

The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.

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In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.