SEMINAR @ SISSSA, dos Santos 21 Sep. 3.00 p.m.

[Barbara Corzani]

********************************************************************
FUNCTIONAL ANALYSIS SECTOR'S ACTIVITIES
*********************************************************************
SEMINAR 

Dott. 
Marcelo Martins dos Santos
(University of Campinas - Brasile)

Title:
"Lagrangean Structure and Propagation of Singularities in
Multidimensional Compressible Flow"


Thursday 21 September  @  03.00 p.m.

SISSA - Main Building - ground floor - room B

********************************************************************
ABSTRACT:

We study the propagation of singularities in solutions of the Navier-Stokes equations of compressible, barotropic fluid flow in two and three space dimensions. The solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded, initial energies are small, and initial velocities
are in certain fractional Sobolev spaces.  We show that, if the initial density is bounded below away from zero in an open set V, then each point of V determines a unique integral curve of the velocity field, and that this system of integral curves defines a homeomorphism of V onto its
image, which is locally H\"older continuous, at each positive time. 
This "Lagrangean structure" is then applied to show
that, if the initial density has a limit at a point of such a set V from a given side of a continuous hypersurface in V, then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface, which is also a continuous manifold. 
If the limits from both sides exist, then the Rankine Hugoniot conditions hold in a strict pointwise sense, showing that the jump in the divergence of the velocity
is proportional to the jump in the pressure. This allows for the derivation of an explicit representation for the strength of the jump in the logarithm of the density, from which it follows that discontinuities persist for all time, convecting along fluid particle paths, and in the case that the pressure is strictly increasing in density, having strengths which decay exponentially in time.