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On the implosion of a three dimensional compressible fluid
We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of the analysis are the existence of Cβ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow which are addressed in the companion papers \cite{MRRSprofile, MRRSdefoc}. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar
Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation
We study the nonlinear inhomogeneous wave equation in one space dimension:
. By constructing some "decoupled" Riccati type
equations for smooth solutions, we provide a singularity formation result
without restrictions on the total variation of unknown, which generalize
earlier singularity results of Lax and the first author. These results are
applied to several one-dimensional hyperbolic models, such as compressible
Euler flows with a general pressure law, elasticity in an inhomogeneous medium,
transverse MHD flow, and compressible flow in a variable area duct
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