2,932 research outputs found
Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy
This work is devoted to the study of the initial boundary value problem for a
general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin
(1985), which can be used as a phase transition model. We will prove the
existence of strong solutions in finite time with discontinuous initial
density, more precisely is in . Our
analysis improves the results of \cite{fDD} and \cite{fH1}, \cite{fH2} by
working in space of infinite energy. In passing our result allow to consider
initial data with discontinuous interfaces, whereas in all the literature the
results of existence of strong solutions consider always initial density that
are continuous. More precisely we investigate the existence of strong solution
for Korteweg's system when we authorize jump in the pressure across some
hypersurface. We obtain also a result of ill-posedness for Korteweg system and
we derive a new blow-up criterion which is the main result of this paper. More
precisely we show that if we control the vacuum (i.e \frac{1}{\rho}\in
L^{\infty}_{T}(\dot{B}^{0}_{N+\e,1}(\R^{N})) with \e>0) then we can extend
the strong solutions in finite time. It extends substantially previous results
obtained for compressible equations
Asymptotic Behavior of a Viscous Liquid-Gas Model with Mass-Dependent Viscosity and Vacuum
In this paper, we consider two classes of free boundary value problems of a
viscous two-phase liquid-gas model relevant to the flow in wells and pipelines
with mass-dependent viscosity coefficient. The liquid is treated as an
incompressible fluid whereas the gas is assumed to be polytropic. We obtain the
asymptotic behavior and decay rates of the mass functions ,\
when the initial masses are assumed to be connected to vacuum both
discontinuously and continuously, which improves the corresponding result about
Navier-Stokes equations in \cite{Zhu}.Comment: 24 page
- …