Existence and Smoothing for a Nondivergence-Form Degenerate Diffusion from Plasma-Wave Theory

Abstract

We prove existence, positive-time smoothing, and physical admissibility of weak solutions to the degenerate parabolic Cauchy-Dirichlet problem $\partial_t u = \rho_\lambda(x) u \partial_x^2 u + \rho_\lambda(x) g(x) u$ on the half-line, where $\rho_\lambda$ vanishes at the boundary and grows at infinity. This scalar problem arises by formally reducing the system of equations given by the quasilinear theory of plasma waves in the one-dimensional case. This theory models a background distribution of electrons $f$ coupled to a spectral energy density $W$ through wave-particle resonance. For the scalar problem we construct weak solutions from weighted $L^p$ initial data and bounded reaction, admitting the unbounded, discontinuous data that the physical model demands, and lying beyond the reach of the continuous-data theories developed for nearby problems. We identify a parabolic smoothing effect for the constructed solution, namely one-sided bounds on $\partial_t u$: from merely integrable data, the solution becomes locally H\"older in space and time and locally Lipschitz in space at positive times. This spatial regularity is shown to be sharp by explicit examples. Finally, we address the quasilinear system itself, whose well-posedness remains open: we prove that the scalar solution induces a particle-wave pair $(f^{\ast}, W^{\ast})$ which is a weak solution of the system. Under nonnegativity and finite-moment hypotheses on the initial data, both components remain nonnegative and the initial mass is conserved. Moreover, the pair inherits positive-time regularity, with $W^{\ast}$ decaying quantitatively at large wavenumber and $f^{\ast}$ smoothing to a locally bounded function even when initially a measure.