Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun and Zhu (Calc Var Partial Differ Equ 60:1–26, 2021), we propose a heat flow for the mean field equation on a connected finite graph G=(V,E). Namely ∂_tϕ(u)=Δu−Q+ρ\frac{e^u}{∫_V e^u dμ} u(⋅,0)=u_0, where Δ is the standard graph Laplacian, ρ is a real number, Q:V→R is a function satisfying ∫_V Qdμ=ρ, and ϕ:R→R is one of certain smooth functions including ϕ(s)=es. We prove that for any initial data u_0 and any ρ∈R, there exists a unique solution u:V×[0,+∞)→R of the above heat flow; moreover, u(x, t) converges to some function u_∞:V→R uniformly in x∈V as t→+∞, and u_∞ is a solution of the mean field equation Δu_∞−Q+ρ\frac{e^{u_∞}}{∫_V e^{u_∞}dμ}=0. Though G is a finite graph, this result is still unexpected, even in the special case Q≡0. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz–Simon type inequality and use it to conclude the convergence of the heat flow.

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In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the onedimensional heat conductive compressible Navier--Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier--Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

The inflow problem of full compressible NavierStokes equations is considered on the half-line (0,+\infty). First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state (0,+\infty) belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds.