The Boltzmann equation with frictional force

The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro-micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3-4) (2004) 178-192] and Liu and Yu [Boltzmann equation: micro-macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133-179] through energy estimates.     

Uniqueness and perturbation bounds for sparse non-negative tensor equations

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the efficiency of the theoretical results.     

Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II)

This is a continuation of the paper (Comm. Math. Phys. 230 (2002) 329) on the study of the compressible Navier-Stokes equations for isentropic flow when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. This assumption comes from physical consideration and it also gives the well-posedness of the Cauchy problem. A new global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to .     

Tensor complementarity problems: the GUS-property and an algorithm

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.     

Commutators with maximal Frobenius norm

It is known that for any nonzero complex n×n matrices X and Y the quotient of Frobenius norms