In this paper, we introduce a new Glimm functional for general systems of hyperbolic conservation laws. This new functional is consistent with the classical Glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate, so that it can be viewed as optimal in some sense. With this new functional, the consistency of the Glimm scheme is proved clearly for general systems. Moreover, the convergence rate of the Glimm scheme is shown to be the same as the one obtained in Bressan, Marson (Arch Ration Mech Anal 142(2):155176, 1998) for systems with each characteristic field being genuinely nonlinear or linearly degenerate.

Based on the existence theory on the Boltzmann equation with external forces in infinite vacuum, in this paper, we will study the L1 and BV-type stability of the classical solutions for small initial data. The stability results generalize those for the Boltzmann equation without force to the case with external force. In particular, we show that the stability can be established for the soft potentials directly, while the stability for the hard potentials and hard sphere model is obtained through the construction of some nonlinear functionals. The functionals thus constructed generalize those constructed in [S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005) 178205] for the case without force to capture the effect of the force term on the time evolution of the solutions. 2006 Elsevier Inc. All rights reserved.

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to <i>x</i>.

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that ∫_{[0,x]}|∇b(ζ)|e^{b(ζ)}d|ζ|<∞.. In the second part of the paper, we show that the area or volume integral ∫_{B^d}|∇u(w)|p(w,θ)dA(w) for positive harmonic functions u is bounded by the value cu(0) for at least one θ. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

The multi-scale zero relaxation singular limit for gas dynamics in thermal non-equilibrium with multiple non-equilibrium modes in multi-dimensions with physical boundaries from non-equilibrium to thermal equilibrium of compressible Euler flow is proved in this paper for analytical data by establishing the uniform local-in-time estimates. A cancellation mechanism is utilized to deal with the nonlinear singular terms that cause the increase in both time and space derivatives in energy estimates. The rates of the relaxations corresponding to different non-equilibrium modes tending to zero discussed in this paper can be arbitrarily different.