The ellipsoidal BGK model was introduced in\cite {Ho} to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In the paper we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation in\cite {DHWY}.

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the l2 operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical Bakry–Émery curvature criterion implies subgaussian concentration for “matrix Lipschitz” functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron–Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.

We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein–Gordon equation (NKGE) with weak power-type nonlinearity and O(1) initial data, while the nonlinearity strength is characterized by εp with a constant p ∈ N+ and a dimensionless parameter ε ∈ (0, 1], for the long-time dynamics up to the time at O(ε^{−β}) with 0 ≤ β ≤ p. In fact, when 0 < ε ≪ 1, the problem is equivalent to the long-time dynamics of NKGE with small initial data and O(1) nonlinearity strength, while the amplitude of the initial data (and the solution) is at O(ε). By reformulating the NKGE into a relativistic nonlinear Schr ̈odinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at O(h^m + ε^{p−β} τ^2) of the TSFP method with h mesh size, τ time step and m ≥ 2 depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at O(ε^{−β}) and uniformly valid for ε ∈ (0, 1]. Especially, the error bounds are uniformly at the second order rate for the large time step τ = O(ε^{−(p−β)/2}) in the parameter regime 0 ≤ β < p. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at O(1) in space and O(ε^β) in time and wave velocity at O(ε^{−β}).

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