An Initial and boundary value problem on a strip for a large class of quasilinear hyperbolic systems arising from an atmospheric model

TL;DR

This paper proves well-posedness for a broad class of quasilinear hyperbolic systems modeling atmospheric water phase transitions, using linear transport analysis and fixed point methods to establish local existence and uniqueness.

Contribution

It introduces a novel approach to analyze IBVPs for complex hyperbolic systems arising from atmospheric models, extending existing methods to more general conditions.

Findings

Established well-posedness for a large class of hyperbolic systems

Derived estimates for solutions of linear transport equations

Applied results to atmospheric water phase transition models

Abstract

In this paper well-posedness is proved for an initial and boundary value problem (IBVP) relative to a large class of quasilinear hyperbolic systems, in $p+q$ equations, on a strip, arising from a model of $H_2O$-phase transitions in the atmosphere. To obtain this result, first, we extensively study an IBVP for the generic linear transport equation on $S_{t_1 } = (0,t_1 ) \times G$ with uniformly locally Lipschitz data and associated vector field in $L_{x_d }^{-} (S_{t_1 } \times \mathbb{R}^p )$ (this cone of $L_t^\infty (0,t_1 ;L_{(x,y_{loc} )}^\infty (G \times \mathbb{R}^p ))^d $ is not cointained in $W^{1,\infty }_{\left( {t,x,y_{loc} } \right)} \left( {S_{t_1} \times \mathbb{R}^p} \right)^d$), that involves parametric vector functions in $ L^\infty \left( {0,t_1;W^{1,\infty } \left( G \right)} \right)^p$, by the method of characteristics. We obtain that the solution belongs to $W_{(t,x,y_{loc} )}^{1,\infty } (S_{t_1 } \times \mathbb{R}^p )$ and interesting estimates about it. Afterwards, using fixed point arguments, we establish the local existence, in time, and uniqueness of the solution in $W^{1,\infty } \left( {S_{t^* } } \right)^{p+q}$ for our class of quasilinear hyperbolic systems. Finally, we apply this result to study an IBVP for the hyperbolic part of an atmospheric model on the transition of water in the three states, introduced in \cite{[SF]}, such that rain and ice fall from it.