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Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions. (English) Zbl 1301.35099

Summary: In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [S. C. Selvaduray and H. F. Yashima, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. (5) 35, 37–69 (2012; Zbl 1279.35073)] as a model for air motion in \({\mathbb{R}^3}\) including water phase transitions. Unknown functions are: the densities \(\rho\) of dry air, \(\pi\) of water vapor, \(\sigma\) and \(\nu\) of water in the liquid and solid state, dependent also on the mass \(m\) of the droplets or ice particles. Air velocity \(v\) and temperature \(T\) are assumed to be known. Solutions (\(\rho\), \(\pi\), \(\sigma\), \(\nu\)) lie in \({L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}\), where \({\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbb{R}^3}\) is open and bounded, and \(\tau^*\) is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to \({\partial\Omega}\); they lie also in \({W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}\), where \({q \in [1, \infty]}\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35L60 First-order nonlinear hyperbolic equations
76T30 Three or more component flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L45 Initial value problems for first-order hyperbolic systems
76T10 Liquid-gas two-phase flows, bubbly flows
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 1279.35073
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References:

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