In this paper, we establish the optimal error estimate of the particle method for a family of nonlinear evolutionary partial differential equations, or the so-called b-equation. The b-equation, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. The particle method is an approximation of the b-equation in Lagrangian representation. We also prove short-time existence, uniqueness and regularity of the Lagrangian representation of the b-equation.

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.

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 study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

In this paper, we prove that the celebrated Arnold--Beltrami--Childress (ABC) flow with parameters A=B=C=1, given by $ \dot x =\sin z+\cos y,\ \dot y = \sin x+\cos z,\ \dot z =\sin y + \cos x $, has periodic orbits on (2\pi\mathbb T)^3 with rotation vectors parallel to (1,0,0), (0,1,0), and (0,0,1). Despite ABC flows being studied since the 1960s, this seems to be the first time that existence of nonperturbative periodic orbits has been established for them. The main difficulty here is the lack of a variational structure for these flows, and our proof instead relies on their symmetry properties. As an application of our result, we show that the well-known G-equation model of turbulent combustion with this ABC flow on (2\pi\mathbb T)^3 has a linear (i.e., maximal possible) flame speed enhancement rate as the flow amplitude grows to infinity. To the best of our knowledge, this is the first time an asymptotic flame speed growth law has been established for a