Lorenzo di Ruvo

Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

Well-posedness results for the short pulse equation

The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. It is a nonlinear evolution equation. In this paper, the well-posedness of bounded solutions for the homogeneous initial boundary value problem and the Cauchy problem associated with this equation are studied.

Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky–Hunter equation

The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper the welposed- ness of bounded solutions for a non-homogeneous initial boundary value problem associated to this equation is studied.

A Singular Limit Problem for the Rosenau–Korteweg-de Vries-Regularized Long Wave and Rosenau-regularized Long Wave Equations

Abstract We consider the Rosenau–Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations, which contain nonlinear dispersive effects. We prove that by adding small diffusion to the equations, as the diffusion and dispersion parameters tends to zero, the solutions of the duffusive/dispersive equations converge to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p

On the convergence of the modified Rosenau and the modified Benjamin–Bona–Mahony equations

{L^{p}}

Convergence of the Solutions on the Generalized Korteweg–de Vries Equation*

setting.

Convergence of the Ostrovsky equation to the Ostrovsky–Hunter one ☆

Abstract We consider the modified Rosenau and the modified Benjamin–Bona–Mahony equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equations converge to entropy solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.

Oleinik type estimates for the Ostrovsky–Hunter equation

We consider the generalized Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting.