Large Time Behavior Of Global Solutions Research Articles

Large-time behavior of global solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas

This paper is concerned with the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas when the viscosity is a positive constant or depends on the density. According to Saha's ionization equation, the equations of state for the basic thermodynamic variables of an ionized gas depend also on the degree of ionization, which leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas together with the unboundedness of the domain under our consideration makes it hard to derive the desired dissipative estimates on the first-order spatial derivative of both the bulk velocity and the absolute temperature, thus making the problem more challenging.In this paper, for a class of constant non-vacuum equilibrium states with positive temperature, we succeed in deducing the desired uniform-in-time bounds on the above mentioned dissipative estimates on both the bulk velocity and the absolute temperature. Based on such an estimate, we can then establish the global existence of solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas with positive constant or density-dependent viscosity and show that such a class of constant non-vacuum equilibrium states is time asymptotically nonlinear stable.

On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field

The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global-in-time solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in finite time. Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology.