Let M be any 3-dimensional (nonabelian) compact solvmanifold. We apply the methods of representation theory to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations Df = g in C°°(M). We show that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence...

In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traffic flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual sub-characteristic condition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formati...

We study the degenerate parabolic equation ut +∇ · f = ∇ · (Q∇u) + g, where (x, t) ∈ R ×R+, the flux ~ f , the viscosity coefficient Q and the source term g depend on (x, t, u) and Q is nonnegative definite. Due to the possible degeneracy, weak solutions are considered. In general, these solutions are not uniquely determined by the initial data and, therefore, additional conditions must be impo...