In this work, we study the stability of the expanding configurations for radiation gaseous stars. Such expanding configurations exist for the isentropic model and the thermodynamic model. In particular, we focus on the effect of the monatomic gas viscosity tensor on the expanding configurations. With respect to small perturbation, it is shown the self-similarly expanding homogeneous solutions of the isentropic model is unstable, while the linearly expanding homogeneous solutions of the isentropic model and the thermodynamic model are stable. This is an extensive study of the result in arXiv: 1605.08083 by Hadzic and Jang.

We prove the existence of the global weak entropy solution to the Doi–Saintillan–Shelley model for active and passive rod-like particle suspensions, which couples a Fokker–Planck equation with the incompressible Navier–Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd611))L2(Ω;L1(Sd611)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd611)L2(Ω×Sd611) weak solution in two and three dimensions and the uniqueness in two dimension.

In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [<i>Comm. Pure Appl. Math.</i>, 48 (1995), pp. 555--563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [<i>Comm. Math. Phys.</i>, 108 (1987), pp. 153--175]. The travelling wave solutions with strong shock profile are shown to be asymptotically stable under small disturbances with integral zero using an elementary but technical energy method. Proofs involve detailed study of the error equation for disturbances using the same weight function introduced in [<i>Comm. Math. Phys.</i>, 165 (1994), pp. 83--96].

We study the large time asymptotic speeds (turbulent flame speeds sT) of the simplified HamiltonJacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L1 type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L2 type) inviscid HJ model of MajdaSouganidis derived from the KolmogorovPetrovskyPiskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995)[10], we compare the turbulent flame speeds sTs from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models. 2013 Elsevier Masson SAS. All rights reserved.

Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).