Repository: Freie Universit├Ąt Berlin, Math Department

GLOBAL WELL-POSEDNESS FOR PASSIVELY TRANSPORTED NONLINEAR MOISTURE DYNAMICS WITH PHASE CHANGES

Hittmeir, S. and Klein, R. and Li, J. and Titi, E. (2017) GLOBAL WELL-POSEDNESS FOR PASSIVELY TRANSPORTED NONLINEAR MOISTURE DYNAMICS WITH PHASE CHANGES. Nonlinearity, 30 (10). pp. 3676-3718. ISSN 0951-7715

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Official URL: http://dx.doi.org/10.1088/1361-6544/aa82f1

Abstract

We study a moisture model for warm clouds that has been used by Klein and Majda in [21] as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler [20] and Grabowski and Smolarkiewicz [15], in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation. In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos [1] to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms. SABINE HITTMEIR, RUPERT KLEIN, JINKAI LI, AND EDRISS S. TITI

Item Type:Article
Additional Information:SFB1114 Preprint 10/2016
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group
ID Code:1969
Deposited By: Ulrike Eickers
Deposited On:10 Oct 2016 10:17
Last Modified:24 Oct 2017 15:32

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