Papers by lorenzo di ruvo
Milan Journal of Mathematics
The dynamics of aeolian sand ripples is described by a 1D non-linear evolutive fourth order equat... more The dynamics of aeolian sand ripples is described by a 1D non-linear evolutive fourth order equation. In this paper, we prove the well-posedness of the classical solutions of the Cauchy problem, associated with this equation.
Journal of Hyperbolic Differential Equations, Jun 1, 2022
Mathematical Methods in The Applied Sciences, Mar 25, 2021
The fifth‐order short pulse equation models the nonlinear propagation of optical pulses of a few ... more The fifth‐order short pulse equation models the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media. In particular, it models the propagation of circularly and elliptically polarized few‐cycle solitons in a Kerr medium. In this paper, we prove the well‐posedness of the classical solutions for the Cauchy problem associated with this equation.
Springer proceedings in mathematics & statistics, Aug 28, 2013
The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid... more 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 welposedness of the Cauchy problem and of an initial boundary value problem associated to this equation is studied.
Symmetry, Jul 27, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Acta Applicandae Mathematicae, Dec 15, 2020
Parabolic equations with degenerate diffusivity describe several phenomena like vehicular dynamic... more Parabolic equations with degenerate diffusivity describe several phenomena like vehicular dynamics or flow in porous media. In this paper, we give some conditions on the singular diffusion that guarantee the well-posedness of continuous solutions for the Cauchy problem.
Mediterranean Journal of Mathematics, Feb 15, 2023
Kuramoto-Velarde equation describes the spatiotemporal evolution of the morphology of steps on cr... more Kuramoto-Velarde equation describes the spatiotemporal evolution of the morphology of steps on crystal surfaces, or the evolution of the spinoidal decomposition of phase separating systems in an external field. We prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation for each choice of the terminal time T .
Ricerche Di Matematica, Jul 27, 2021
The Kuramoto-Sinelshchikov-Cahn-Hilliard equation models the spinodal decomposition of phase sepa... more The Kuramoto-Sinelshchikov-Cahn-Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.
Journal of Mathematical Analysis and Applications
Ricerche di Matematica
The Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase sepa... more The Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.
Communications on Pure and Applied Analysis
Mediterranean Journal of Mathematics
Kuramoto–Velarde equation describes the spatiotemporal evolution of the morphology of steps on cr... more Kuramoto–Velarde equation describes the spatiotemporal evolution of the morphology of steps on crystal surfaces, or the evolution of the spinoidal decomposition of phase separating systems in an external field. We prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation for each choice of the terminal time T.
Partial Differential Equations and Applications
In this paper, we prove the well-posedness of the initial-boundary value problem for a non-local ... more In this paper, we prove the well-posedness of the initial-boundary value problem for a non-local elliptic-hyperbolic system related to the short pulse equation. Our arguments are based on energy estimates and passing to the limit in a vanishing viscosity approximation of the problem.
Contemporary Mathematics
The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface ... more The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface morphology during Molecular Beam Epitaxy growth, provoked by an interplay between deposition of atoms onto the surface and the relaxation of the surface profile through surface diffusion. It consists of a nonlinear fourth order partial differential equation. Using energy methods we prove the well-posedness (i.e., existence, uniqueness and stability with respect to the initial data) of the classical solutions for the Cauchy problem, associated with this equation.
arXiv (Cornell University), Jan 5, 2018
We consider a scalar, possibly degenerate parabolic equation with a source term, in several space... more We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these solutions converge, in the vanishing-viscosity limit, to the Kruzhkov entropy solution of the corresponding hyperbolic equation. The proof exploits the H-measure compactness in several space dimensions.
Discrete and Continuous Dynamical Systems - S
We study a porous medium equation with dissipative and hyperdiffusive effects and no sign restric... more We study a porous medium equation with dissipative and hyperdiffusive effects and no sign restrictions on the diffusion coefficient.
Symmetry
The fifth order Kudryashov–Sinelshchikov equation models the evolution of the nonlinear waves in ... more The fifth order Kudryashov–Sinelshchikov equation models the evolution of the nonlinear waves in a gas–liquid mixture, taking into account an interphase heat transfer, surface tension, and weak liquid compressibility simultaneously at the derivation of the equations for non-linear-waves. We prove the well-posedness of the solutions for the Cauchy problem associated with this equation for each choice of the terminal time T.
Contemporary Mathematics, 2020
The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In thi... more The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.
The Ostrovsky-Hunter equation provides a model for small-amplitude long waves in a rotating fluid... more 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 we study the well-posedness for the Cauchy problem associated to this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities by an Oleinik-type estimate and prove uniqueness via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation is admissible only if it jumps down in value (like the inviscid Burgers equation).
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Papers by lorenzo di ruvo