1,721,032 research outputs found
The visco-capillarity kinetic condition for sonic phase transitions
No full textWe consider a model for phase transitions, in one space dimension. In a recent
paper, [5], the author and M. Sable-Tougeron have studied sonic phase boundaries for this
model, by taking into account a kinetic relation. We prove here that both in the piecewise
linear case and in a cubic case the assumptions that were made in [5] hold if the kinetic
relation is provided by a visco-capillarity approach
Subsonic and sonic phase transitions
No full textWe consider a hyperbolic system of two
conservation laws, in one space dimension, modelling phase
transitions; these are understood as discontinuous solutions with
values in two disjoint open sets of the state space. For states close
to a subsonic phase transition the Riemann problem is underdetermined,
and we select the phase boundary by an entropy equality. In the limit
sonic case, we still use such a criterion either for subsonic and
sonic phase boundaries; some conditions are needed to solve the
Riemann problems. These assumptions are satisfied by some significant
models if the phase boundary is chosen according to a
viscosity-capillarity criterion. In both cases we provide results of
global existence of solutions to the Cauchy problem if the initial
data have suitable small total variation
The problem of local solvability of the linear partial differential equations
No full textThe author provide a comprehensive survey on the problem of the local solvability of linear partial differential operator, with a special emphasis to the solvability Gevrey classes. More than one hundred papers are taken into consideration
Asymptotic analysis of contact discontinuities
No full textWe study the perturbation of a contact discontinuity by a small amplitude, rapidly oscillating wave train. Under a suitable stability assumption the perturbed solution is still a contact discontinuity, and we give its asymptotic development, as well as that of the contact curve, in terms of the wavelength of the perturbation
Weakly nonlinear geometric optics for hyperbolic systems of conservation laws with shock waves
No full textWe consider in this paper a strictly hyperbolic system of conservation laws, in one space dimension. We suppose that a shock wave solution to this system is given and superimpose to it small amplitude fast oscillations. These oscillations are described by sets of phase functions, and resonances are taken into account. We justify the asymptotic expansions given by the weakly non-linear geometric optics: the oscillating part of the perturbed solution is approximated by an almost periodic function (a profile) which is solution to a non-linear integro-differential mixed problem. We give at the same time an asymptotics to the shock front
On local solvability in Gevrey classes of linear partial differential operators with multiple characteristics
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