Global Well-Posedness for the Thermodynamically Refined Passively Transported Nonlinear Moisture Dynamics with Phase Changes

Abstract

In this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies (Hittmeir et al. in Nonlinearity 30:3676–3718, 2017) take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential, e.g. in the case of deep convective cloud columns in Hittmeir and Klein (Theoret Comput Fluid Dyn 32(2):137–164, 2017). The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of solutions.

1 Introduction

Precipitation causes one of the major uncertainties in weather forecast and climate modelling and thus also the incorporation of moisture and phase changes into atmospheric flow models is still actively debated, see, e.g., Bannon (2002). In the framework of systematically derived reduced mathematical models for atmospheric dynamics, it has been shown that not only the inclusion of moist processes alone, but also the detailed structure of the moist-air submodels can decisively affect the overall flow dynamics, see, e.g., Smith and Stechmann (2017), Khouider (2019), Stechmann and Hottovy (2020). Often the difference of the gas constants for water vapor and dry air is neglected and further the simple form of the dry ideal gas law is assumed to hold. Even more typically also the dependence of the internal energy on the moisture components is neglected, which results in a much simpler form of the thermodynamic equation. So far global well-posedness of solutions to moisture models has only been proven based upon these assumptions, see also (Bousquet et al 2014; Coti Zelati et al. 2013; Coti Zelati and Temam 2012; Coti Zelati et al. 2015; Hittmeir et al. 2017, 2020). As demonstrated, e.g. in the asymptotical analysis in Hittmeir and Klein (2017) for deep convective cloud columns, exactly these refined thermodynamics lead to a much stronger coupling of the thermodynamic equation (see below) to the moisture components and thereby even change the force balances to leading order. The aim here is to also incorporate them into the analysis, where the refined thermodynamical setting in comparison to Hittmeir et al. (2017) requires a different approach for proving a priori nonnegativity and uniform boundedness of the solution components, since the antidissipative term in the equation for temperature does not vanish anymore when rewriting it in terms of the potential temperature. We thus employ an iterative method similar to the one used by Coti Zelati et al. (2015) to derive an upper bound on the temperature.

Bousquet et al (2014), Coti Zelati et al. (2013, 2015, 2012) analysed a basic moisture model consisting of one moisture quantity coupled to temperature and containing only the process of condensation during upward motion, see, e.g., Haltiner and Williams (1980). Since the source term there is modeled via a Heaviside function as a switching term between saturated and undersaturated regions, the analysis requires elaborate techniques. The approach based on differential inclusions and variational techniques has then further been applied to the moisture model coupled to the primitive equations in Coti Zelati et al. (2015).

In preceding works (Hittmeir et al. 2017, 2020) we studied a moisture model consisting of three moisture quantities for water vapor, cloud water, and rain water, which contains besides the phase changes condensation and evaporation also the autoconversion of cloud water to rain water after a certain threshold is reached, as well as the collection of cloud water by the falling rain droplets. It corresponds to a basic form of a bulk microphysics model in the spirit of Kessler (1969) and Grabowski and Smolarkiewicz (1996). In Hittmeir et al. (2017) we assumed the velocity field to be given and studied the moisture balances coupled to the thermodynamic equation through the latent heat. In Hittmeir et al. (2020) this moisture model has been successfully coupled to the primitive equations by taking over the ideas of Cao and Titi (2007) for their breakthrough on the global solvability of the latter system.

In this work we extend this moisture model for warm clouds consisting of three moisture balances and the thermodynamic equation by the refined thermodynamic setting as explained above, which leads in particular to a much stronger coupling of the model equations.

In the remainder of this section we introduce the moisture model. In Sect. 2 we then formulate the full problem with boundary and side conditions and state the main result on the global existence and uniqueness of bounded solutions. In Sect. 3 we carry out the proof for the existence and uniqueness of strong solutions.

1.1 Governing Equations

When modelling atmospheric flows in general, the full compressible governing equations need to be considered. However, under the assumption of hydrostatic balance, which in particular guarantees the pressure to decrease monotonically in height, the pressure p can be used as the vertical coordinate, which have the main advantage that the continuity equation takes the form of the incompressibility condition (see also (15) below). We therefore work in the following with the governing equations in the pressure coordinates (x, y, p) and as in Hittmeir et al. (2017) assume the velocity field to be given

$$\begin{aligned} \overline{\textbf{v}}=(\overline{\textbf{v}}_h,{\overline{\omega }})=({\overline{u}}, {\overline{v}}, {{\overline{\omega }}}), \end{aligned}$$

where we note that the vertical velocity $\omega =\tfrac{\hbox {d}p}{\hbox {d}t}$ in pressure coordinates takes the inverse sign in comparison to cartesian coordinates for upward and downward motion. Also the horizontal and the vertical derivatives and accordingly the velocity components in pressure coordinates have different units. The total derivative in pressure coordinates reads

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} = \partial _t + \overline{\textbf{v}}_h\cdot \nabla _h + {\overline{\omega }}\partial _p , \qquad \text {with} \quad \nabla _h =(\partial _x,\partial _y). \end{aligned}$$

(1)

For the closure of the turbulent and molecular transport we use

$$\begin{aligned} {\mathcal D}^* = \mu _*\Delta _h+ \nu _*\partial _p\left( \left( \frac{g p }{R_d \bar{T}}\right) ^2\partial _p\right) ,\qquad \text {with} \quad \Delta _h=\partial _x^2+\partial _y^2, \end{aligned}$$

(2)

where $\bar{T}=\bar{T}(p)$ corresponds to some background distribution being uniformly bounded from above and below and $R_d$ is the individual gas constant for dry air. The operator ${\mathcal D}^*$ thereby provides a close approximation to the full Laplacian in cartesian coordinates, see also (Lions et al. 1992; Petcu et al.). The thermodynamic quantities are related via the ideal gas law

$$\begin{aligned} p=p_d + p_v = R_d \rho _d T + R_v \rho _v T, \end{aligned}$$

(3)

where $R_v$ is the individual gas constant for water vapor and $p_d, p_v, \rho _d, \rho _v$ denote the partial pressures and densities of dry air and water vapor, where we note that liquid water does not exert any pressure on the volume of air, see, e.g. also (Bannon 2002; Cotton et al. 2011).

Before going more into details with the ideal gas law (see (7) below), we need to introduce the moisture quantities. In the case of moisture being present typically the water vapor mixing ratio, defined as the ratio of the density of $\rho _v$ over the density of dry air $\rho _d$ ,

$$\begin{aligned} q_{v}=\frac{\rho _v}{\rho _d}, \end{aligned}$$

is used for a measure of quantification. If saturation effects occur, then water is also present in liquid form as cloud water and rain water represented by the additional moisture quantities

$$\begin{aligned} q_{c}=\frac{\rho _c}{\rho _d},\qquad q_{r}=\frac{\rho _r}{\rho _d}. \end{aligned}$$

We focus here on warm clouds, where water is present only in gaseous and liquid form, i.e. no ice and snow phases occur. The total water content is therefore given by

$$\begin{aligned} q_T=q_v+q_c+q_r. \end{aligned}$$

For these mixing ratios for water vapor, cloud water and rain water we have the following moisture balances

$$\begin{aligned} \frac{\hbox {d}q_v}{\hbox {d}t}= & {} S_{ev} - S_{cd}+\mathcal {D}^{q_v} q_v, \end{aligned}$$

(4)

$$\begin{aligned} \frac{\hbox {d}q_c}{\hbox {d}t}= & {} S_{cd} - S_{ac}- S_{cr}+\mathcal {D}^{q_c} q_c, \end{aligned}$$

(5)

$$\begin{aligned} \frac{\hbox {d}q_r}{\hbox {d}t}+V\partial _p\left( \frac{p}{R_d \bar{T}} q_r\right)= & {} S_{ac} + S_{cr}- S_{ev}+\mathcal {D}^{q_r} q_r\, \end{aligned}$$

(6)

with $\frac{\hbox {d}}{\hbox {d}t}$ as in (1) and $\mathcal {D}^{q_j}$ as in (2). Here $S_{ev}, S_{cd}, S_{ac}, S_{cr}$ are the rates of evaporation of rain water, the condensation of water vapor to cloud water and the inverse evaporation process, the auto-conversion of cloud water into rainwater by accumulation of microscopic droplets, and the collection of cloud water by falling rain. Moreover, V denotes the terminal velocity of falling rain and is assumed to be constant.

Having introduced the mixing ratios, we can now reformulate the ideal gas law (3) as

$$\begin{aligned} p=\rho \widetilde{R} T, \end{aligned}$$

(7)

where $\rho =\rho _d+\rho _v+\rho _c+\rho _r$ is the total density and $\widetilde{R}$ depends on the moisture content

$$\begin{aligned} \widetilde{R}=R_d \frac{1+\frac{q_v}{E}}{1+q_T},\qquad \text {where} \qquad E=\frac{R_d}{R_v}, \end{aligned}$$

see, e.g. also (Bannon 2002; Cotton et al. 2011; Hittmeir and Klein 2017). The thermodynamic equation accounts for the diabatic source and sink terms, such as latent heating, radiation effects, etc., but we will in the following only focus on the effect of latent heat in association with phase changes [see e.g. also (Klein and Majda 2006; Coti Zelati et al. 2013, 2015; Hittmeir et al. 2017)]. The temperature equation in pressure coordinates then reads, see, e.g. Hittmeir and Klein (2017); Cotton et al. (2011),

$$\begin{aligned} \frac{\hbox {d}T}{\hbox {d}t}-{\widetilde{\kappa }}\frac{T}{p} {\overline{\omega }} + \frac{c_l}{\widetilde{C}}q_r V \partial _p T=\widetilde{L}(S_{cd}-S_{ev})+ {\mathcal D}^TT, \end{aligned}$$

(8)

where

$$\begin{aligned} {\widetilde{\kappa }}=\frac{\widetilde{R}}{\widetilde{C}},\quad \text {and} \quad \widetilde{C}=c_{pd} + c_{pv} q_v + c_l(q_c+q_r) \end{aligned}$$

with the heat capacities $c_{pd},c_{pv}$ at constant pressure for dry air and water vapor and the heat capacity for liquid water $c_l$, respectively. For the latent heat term, we denote

$$\begin{aligned} \widetilde{L}=\frac{L}{\widetilde{C}},\quad \text {where}\quad L(T)= L_0 - (c_{l}-c_{pv})(T-T_0) \quad \text {and} \quad L_0=L(T_0),\nonumber \\ \end{aligned}$$

(9)

where $T_0$ is the reference temperature which is typically chosen as $T_0=273.15K$. We emphasize here once more that so far only models with $L, \widetilde{R}, {\widetilde{\kappa }}, \widetilde{C}$ constant and $c_l=0$ have been considered in mathematical analysis studies. Thus, this physically more refined setting has several extensions in comparison to existing studies.

Remark 1

To describe the state of the atmosphere a common thermodynamic quantity used instead of the temperature is the potential temperature

$$\begin{aligned} \theta = T \left( \frac{p_0}{p}\right) ^\kappa , \qquad \text {where} \qquad \kappa = \frac{R_d}{c_{pd}}. \end{aligned}$$

(10)

In case of the typical simplification ${\widetilde{\kappa }}=\kappa$ and $c_l=0$, the left-hand side of (8) simply reduces to $\frac{T}{\theta }\frac{\hbox {d}}{\hbox {d}t}\theta$. This property was essential in the preceding works (Hittmeir et al. 2017) and also (Coti Zelati and Temam 2012) to derive a priori nonnegativity of the moisture quantities and temperature.

1.2 Explicit Expressions for the Source Terms

The saturation mixing ratio

$$\begin{aligned}q_{vs}=\frac{\rho _{vs}}{\rho _d},\end{aligned}$$

gives the threshold for saturation, i.e. $q_v<q_{vs}$ for undersaturation, $q_v = q_{vs}$ corresponds to saturation, and $q_v>q_{vs}$ accordingly holds in oversaturated regions. The saturation vapor mixing ratio satisfies

$$\begin{aligned} q_{vs}(p, T)=\frac{E e_s(T)}{p-e_s(T)}, \end{aligned}$$

with the saturation vapor pressure $e_s$ as a function of T being defined by the Clausius-Clapeyron equation:

$$\begin{aligned} \frac{\hbox {d} \ln e_s}{\hbox {d}T}=\frac{L(T)}{R_v T^2}. \end{aligned}$$

From this formula it is obvious that $e_s$ increases in T (as long as L(T) is positive). Since the temperature T is given in Kelvin K, only positive values are physical. It should be noted that the Clausius-Clapeyron equation is only meaningful for temperature ranges appearing in the troposphere, thus in particular we shall pose in the following the natural assumption

$$\begin{aligned} e_s(T)=0 \, \textit{Pa}\quad \text {and} \quad q_{vs}(p,T)=0 \quad \text {for} \ T \le \underline{T}, \end{aligned}$$

for some $\underline{T}\ge 0\, \textit{K}$, which will also be helpful for proving nonnegativity of the moisture quantities and the temperature, see also (Hittmeir et al. 2017).

Recalling the fact that $c_l-c_{pv}>0$, we see from (9) that L(T) decreases in T. In particular, there exists the critical temperature

$$\begin{aligned} T_{crit}=\frac{L_0}{(c_l-c_{pv})}-T_0 \end{aligned}$$

(11)

at which the latent heat of evaporation vanishes, i.e. $L(T_{crit})=0$. At such high temperatures of about 700K, the gaseous and liquid state become indistinguishable. Such temperatures however clearly exceed by far the ones present in the relevant atmospheric layers. Therefore, we in the following pose the natural assumption that

$$\begin{aligned} e_s(T)=0 \, \textit{Pa}\quad \text {and} \quad q_{vs}(p,T)=0 \quad \text {for} \ T \ge T_{crit}. \end{aligned}$$

(12)

For deriving the uniqueness of the solutions, we need additionally the uniform Lipschitz continuity of $q_{vs}\ge 0$ in T, i.e. we assume

$$\begin{aligned} |q_{vs}(p, T_1)-q_{vs}(p, T_2)|\le C|T_1-T_2|, \end{aligned}$$

for a positive constant C independent of p.

For the source terms of the mixing ratios, we take over the setting of Klein and Majda (2006) corresponding to a basic form of the bulk microphysics closure in the spirit of Kessler (1969) and Grabowski and Smolarkiewicz (1996), which has also been used in the preceding work (Hittmeir et al. 2017):

$$\begin{aligned} S_{ev}= & {} C_{ev}\widetilde{R}T (q_r^+)^\beta (q_{vs}-q_v)^+,\qquad \\ S_{cr}= & {} C_{cr} q_c q_r,\qquad \\ S_{ac}= & {} C_{ac} (q_c-q_{ac}^*)^+ \end{aligned}$$

where $C_{ev},C_{cr},C_{ac}$ are dimensionless rate constants. Moreover, $(g)^+=\max \{0,g\}$ and $q_{ac}^*\ge 0$ denotes the threshold for cloud water mixing ratio beyond which autoconversion of cloud water into precipitation become active. The cutoff of the negative part in $q_r$ is only technical since clearly only nonnegative values for T and $q_j$ for $j\in \{v,c,r\}$ are meaningful.

The exponent $\beta$ in the evaporation term $S_{ev}$ in the literature typically appears to be chosen as $\beta \approx 0.5$, see e.g.  Grabowski and Smolarkiewicz (1996); Klein and Majda (2006) and the references therein. Exponent $\beta \in (0,1)$ causes difficulties in the analysis for the uniqueness of the solutions. In the case that both ${\widetilde{C}}$ and ${\widetilde{R}}$ are constants, this problem, however, was overcome in Hittmeir et al. (2017) by introducing new unknowns, which allow for certain cancellation properties of the source terms and reveal advantageous monotonicity properties. Here, however, we need to generalise the setting to incorporate the more complicated structure of the thermodynamic equation and in particular the nonconstant ${\widetilde{C}}, \widetilde{L}$.

We shall use the closure of the condensation term in a similar fashion to Klein and Majda (2006)

$$\begin{aligned} S_{cd}=C_{cd}(q_v-q_{vs})q_c +C_{cn}(q_v-q_{vs})^+, \end{aligned}$$

which is in the literature often defined implicitly via the equation of water vapor at saturation, see, e.g.  Grabowski and Smolarkiewicz (1996).

2 Formulation of the Problem and Main Result

We analyse the moisture model consisting of the moisture equations (4)–(6) coupled to the thermodynamic equation (8). As in Hittmeir et al. (2017), we assume the velocity field $\bar{\textbf{v}}=(\overline{\textbf{v}}_h,{{\overline{\omega }}})$ to be given and to satisfy

$$\begin{aligned}{} & {} \overline{\textbf{v}}_h \in (L^2_{\text {loc}}([0,\infty );H^1({\mathcal M})))^2\cap (L_{\text {loc}}^\infty ([0,\infty );\nonumber \\{} & {} L^2({\mathcal M})))^2\cap (L^r_{\text {loc}}([0,\infty );L^q({\mathcal M})))^2, \end{aligned}$$

(13)

$$\begin{aligned}{} & {} \quad {{\overline{\omega }}} \in L_{\text {loc}}^\infty ([0,\infty );L^2({\mathcal M})) \cap L^r_{\text {loc}}([0,\infty );L^q({\mathcal M})), \end{aligned}$$

(14)

for some $2\le r\le \infty$ and $3\le q \le \infty$ satisfying $\frac{2}{r}+\frac{3}{q}<1$. Moreover, we assume mass conservation, taking in pressure coordinates the form of the incompressibility condition

$$\begin{aligned} \nabla _h\cdot \overline{\textbf{v}}_h+\partial _p{{\overline{\omega }}} =0 \quad \text {in} \ \ {\mathcal M}, \end{aligned}$$

(15)

and the no-penetration boundary condition

$$\begin{aligned} \overline{\textbf{v}}_h\cdot \textbf{n}_h + {{\overline{\omega }}}\, n_p=0\,\quad \text {on} \ \ \partial {\mathcal M}. \end{aligned}$$

(16)

This is motivated from the solution of the viscous primitive equations (without moisture) satisfying these required regularity properties in (13), see (Cao et al. 2020, 2016, 2014a, b; Cao and Titi 2007).

Similar as in Coti Zelati et al. (2013); Hittmeir et al. (2017), we let ${{\mathcal M}}$ be a cylinder of the form

$$\begin{aligned} {{\mathcal M}}=\{(x,y,p): (x,y) \in {{\mathcal M}}', p\in (p_1,p_0)\}, \end{aligned}$$

where ${{\mathcal M}}'$ is a smooth bounded domain in $\mathbb {R}^2$ and $p_0>p_1>0$. The boundary is given by

$$\begin{aligned}{} & {} \Gamma _0=\{(x,y,p)\in \overline{{{\mathcal M}}}: p=p_0\}, \\{} & {} \Gamma _1=\{(x,y,p)\in \overline{{{\mathcal M}}}: p=p_1\},\\{} & {} \Gamma _{{\ell }} = \{(x,y,p)\in \overline{{{\mathcal M}}}: (x,y)\in \partial {{\mathcal M}}', p_0\ge p\ge p_1\}. \end{aligned}$$

The boundary conditions read as

$$\begin{aligned}{} & {} \Gamma _0:\ \partial _pT=\alpha _{0T}(T_{b 0}(x,y,t)-T),\quad \partial _pq_{j}=\alpha _{0j} (q_{b 0 j}(x,y,t)-q_j), \nonumber \\{} & {} \quad j\in \{v,c,r\}, \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \Gamma _1:\ \partial _pT=0,\qquad \partial _pq_j=0,\quad j\in \{v,c,r\}, \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \Gamma _{{\ell }}:\ \partial _nT=\alpha _{{{\ell }}T}(T_{b {\ell }}(p,t)-T),\quad \partial _nq_{j}=\alpha _{{{\ell }} j} (q_{b {{\ell }} j}(p,t)-q_j), \nonumber \\{} & {} \quad j\in \{v,c,r\}, \end{aligned}$$

(19)

where all given functions $\alpha _{0 j }, \alpha _{{\ell }j}, \alpha _{0 T }, \alpha _{{\ell }T}$ and $T_{b 0 }, T_{b {\ell }}, q_{b 0 j }, q_{b {\ell }j }$ are assumed to be nonnegative, sufficiently smooth and uniformly bounded.

Throughout this paper, we use the abbreviation

$$\begin{aligned} \Vert f\Vert =\Vert f\Vert _{L^2({\mathcal M})},\qquad \Vert f\Vert _{L^p}=\Vert f\Vert _{L^p({\mathcal M})}. \end{aligned}$$

According to the weight in the vertical diffusion terms, we also introduce the weighted norms

$$\begin{aligned} \Vert f\Vert _w=\Big \Vert \left( \frac{g p}{R_d\bar{T}}\right) f\Big \Vert ,\qquad \Vert f\Vert ^2_{H_w^1}=\Vert f\Vert ^2+\Vert \nabla _h f\Vert ^2+\Vert \partial _p f\Vert _w^2, \end{aligned}$$

where we emphasize that, since the weight $\frac{gp}{R_d\bar{T}}$ is uniformly bounded from above and below by positive constants, the $H_w^1({\mathcal M})$-norm is equivalent to the $H^1({\mathcal M})$-norm. Moreover, we shall often use for convenience the notation

$$\begin{aligned} \Vert (f_1,\dots ,f_n)\Vert ^2=\sum _{j=1}^n\Vert f_j\Vert ^2. \end{aligned}$$

For the initial data space of functions for the moisture components and the temperature, we introduce

$$\begin{aligned} {\mathcal {X}}=L^\infty ({\mathcal M})\cap H^1({\mathcal M}) \end{aligned}$$

and accordingly for strong solutions

$$\begin{aligned} {\mathcal {Y}}_{\mathcal T}=\{f|f\in L^\infty ({\mathcal M}\times (0,{\mathcal T}))\cap L^2(0,{\mathcal T};H^2({\mathcal M})),\partial _tf\in L^2(\mathcal M\times (0,\mathcal T))\}. \end{aligned}$$

The main result of this paper is the following theorem.

Theorem 1

Let $\beta =1$ and assume that the given velocity field $(\overline{\textbf{v}}_h, {{\overline{\omega }}})$ satisfies (13)–(16) and the initial data $(T_0,q_{v0},q_{c0},$ $q_{r0}) \in {\mathcal {X}}^4$ is nonnegative. Then, for any ${\mathcal T}>0$ there exists a unique global strong solution $( T,q_{v},q_{c},q_{r}) \in {\mathcal {Y}}_{\mathcal T}^4$ to system (4)–(8), subject to (17)–(19), on $\mathcal M\times (0,\mathcal T)$, and the solution components $(T,q_{v},q_{c},q_{r})$ remain nonnegative and uniformly bounded from above with bounds growing with ${\mathcal T}$.

The proof of this theorem will be presented in the next section.

Remark 2

In Hittmeir et al. (2017), we treated the more complicated case of an evaporation source with a general exponent $\beta \in (0,1]$ of $q_r$ that causes in particular difficulties in the uniqueness. To overcome this problem, we introduced the new unknowns $Q=q_v+q_r$ and $H=T-\widetilde{L} (q_c+q_r)$ in Hittmeir et al. (2017), where we recall that $\widetilde{L}$ was assumed to be constant there. Due to the challenge here of treating the additional terms arising from the refined thermodynamics, we stick to the case corresponding $\beta =1$ here and leave the more general case $\beta \in (0,1]$ for future work.

Throughout this paper, we use C to denote a general positive constant which may be different at different places. For the aim of the future studies on the coupled system of the moisture dynamics investigated in the present paper to the primitive equations with either isotropic or anisotropic dissipations, see (Cao et al. 2020, 2016, 2014a, b; Cao and Titi 2007), the dependence of the constant C on the a priori bounds of the given velocity field will be explicitly pointed out at the relevant places. However, the dependence of C on the initial data or the parameters in the system will not be paid attention to. We will also use $C_k, k\in \mathbb N$, to denote constants having relevant units.

3 Proof of Theorem 1

As a start point, we consider the following modified system, which however is equivalent to the original system (4)–(8) for nonnegative solutions:

$$\begin{aligned}{} & {} \partial _tT+(\overline{\textbf{v}}_h\cdot \nabla _h)T+{{\overline{\omega }}}\partial _pT-{\widetilde{\kappa }}^+\frac{T}{p} {{\overline{\omega }}}+\frac{c_lq_r}{\widetilde{C}^+}V\partial _pT=\widetilde{L}^+(S_{cd}^+-S_{ev}^+)+ {\mathcal D}^TT, \end{aligned}$$

(20)

$$\begin{aligned}{} & {} \partial _tq_v+\overline{\textbf{v}}_h\cdot \nabla _hq_v+{{\overline{\omega }}}\partial _pq_v = S_{ev}^+ - S_{cd}^++\mathcal {D}^{q_v} q_v, \end{aligned}$$

(21)

$$\begin{aligned}{} & {} \partial _tq_c+\overline{\textbf{v}}_h\cdot \nabla _hq_c+{{\overline{\omega }}}\partial _pq_c= S_{cd}^+ -S_{ac}^+- S_{cr}^++\mathcal {D}^{q_c} q_c,\end{aligned}$$

(22)

$$\begin{aligned}{} & {} \partial _tq_r+\overline{\textbf{v}}_h\cdot \nabla _hq_r+{{\overline{\omega }}}\partial _pq_r+V\partial _p \left( \frac{p}{R_d\bar{T}} q_r\right) = S_{ac}^+ + S_{cr}^+- S_{ev}^++\mathcal {D}^{q_r} q_r, \end{aligned}$$

(23)

where

$$\begin{aligned} {\widetilde{\kappa }}^+= & {} \frac{{{\widetilde{R}}}^+}{{{\widetilde{C}}}^+}, \quad {{\widetilde{L}}}^+=\frac{L(T)}{{\widetilde{C}}^+},\quad \widetilde{R}^+=\frac{ R_d+R_vq_v^+}{1+q_v^++q_c^++q_r^+},\\{} & {} \quad \quad {\widetilde{C}}^+=c_{pd}+c_{pv}q_c^++c_l(q_c^++q_r^+),\\ S_{ev}^+= & {} C_{ev} \widetilde{R}^+ T^+q_r^+(q_{vs}(p,T)-q_v)^+,\quad S_{cr}^+=C_{cr}q_c^+q_r^+,\\{} & {} \quad S_{ac}^+=S_{ac}=C_{ac}(q_c-q_{ac}^*)^+,\\ S_{cd}^+= & {} C_{cd}(q_v^+-q_{vs}(p,T))q_c^++C_{cn}(q_v-q_{vs}(p,T))^+, \end{aligned}$$

with L(T) given by (9).

Since all the nonlinear terms $S_{ev}^+, S_{cr}^+, S_{ac}^+, S_{cd}^+$ and all the coefficients $\widetilde{\kappa }^+, \frac{1}{{\widetilde{C}}^+}, {\widetilde{L}}^+$ are Lipschitz with respect to $q_v, q_c, q_r,$ and T, the local existence of strong solutions to the initial boundary value problem of system (20)–(23) follows by the standard fixed point arguments. In fact, by following the proof in Hittmeir et al. (2017), we can prove the following proposition on the local existence and uniqueness.

Proposition 1

Assume that the given velocity field $(\overline{\textbf{v}}_h, {{\overline{\omega }}})$ satisfies (13)–(16) and the initial data $(T_0,q_{v0},q_{c0},$ $q_{r0}) \in {\mathcal {X}}^4$ is nonnegative. Then, there exists a positive time $\mathcal T_0$ depending only on the upper bound of $\Vert (T_0, q_{v0}, q_{c0}, q_{r0})\Vert _{H^1(\mathcal M)}$, such that system (20)–(23), subject to (17)–(19), on $\mathcal M\times (0,\mathcal T_0)$, has a unique strong solution $( T,q_{v},q_{c},q_{r}) \in {\mathcal {Y}}_{{\mathcal T}_0}^4$.

By applying Proposition 1 inductively, one can extend uniquely the solution $( T, q_v, q_c, q_r)$ obtained there to the maximal time interval $(0,\mathcal T_\text {max})$, where $T_{\text {max}}$ is characterized as

$$\begin{aligned} \limsup _{\mathcal T\rightarrow \mathcal T_{\text {max}}^-}\Vert ( T, q_v, q_c, q_r)\Vert _{H^1(\mathcal M)}=\infty ,\quad \text{ if } \mathcal T_{\text {max}}<\infty . \end{aligned}$$

(24)

Observe that if $T_\text {max}=\infty$, then Proposition 1 implies Theorem 1. Therefore, our aim is to show that $T_\text {max}=\infty$. To this end, we assume by contradiction that $T_\text {max}<\infty$. Due to this fact, the following assumption will be made in the subsequent propositions throughout this section.

Assumption 1

Let all the assumptions in Proposition 1 hold, and let the solution $(T, q_v, q_c, q_r)$ obtained in Proposition 1 be extended uniquely to the maximal interval of existence $(0,\mathcal T_{\text {max}})$, where $\mathcal T_{\text {max}}<\infty$.

The main part of this section is to carry out a series of a priori estimates on $( T, q_v, q_c, q_r)$.

First, the following proposition about the nonnegative and uniform boundedness of the moisture components $q_v, q_c,$ and $q_r$ can be proved by slightly modifying the corresponding proof of Proposition 3.2 in Hittmeir et al. (2017).

Proposition 2

Let Assumption 1 hold, then the solution $(T,q_{v},q_{c},q_{r})$ satisfies

$$\begin{aligned} 0\le q_{v}\le q_v^*,\quad 0\le q_{c}\le q_c^*,\quad 0\le q_{r}\le q_r^*, \end{aligned}$$

for any $\mathcal T \in (0,\mathcal T_{max})$, where

$$\begin{aligned}{} & {} q_v^*=\max \big \{\Vert q_{v 0}\Vert _{L^\infty {({\mathcal M})}},\Vert q_{b0v}\Vert _{L^\infty ((0,{\mathcal T})\times {\mathcal M}')},\Vert q_{b{\ell }v}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}, q_{vs}^*\big \},\\{} & {} q_c^*\, =q_c^*\big ({\mathcal T}, \Vert q_{c 0}\Vert _{L^\infty {({\mathcal M})}},\Vert q_{b0c}\Vert _{L^\infty ((0,T)\times {\mathcal M}')},\Vert q_{b{\ell }c}\Vert _{L^\infty ((0,T)\times \Gamma _{\ell })},q_v^*,q_{vs}^*\big ),\\{} & {} q_r^*\, =q_r^*\big ({\mathcal T}, \Vert q_{r 0}\Vert _{L^\infty {({\mathcal M})}},\Vert q_{b0r}\Vert _{L^\infty ((0,T)\times {\mathcal M}')},\Vert q_{b{\ell }r}\Vert _{L^\infty ((0,T)\times \Gamma _{\ell })},q_c^*\big ), \end{aligned}$$

with $q_{vs}^*=\max q_{vs}$ and moreover $q_c^*$ and $q_r^*$ are continuous in $\mathcal T\in (0,\infty )$.

Due to the nonnegativity of $q_v, q_c, q_r$, it is clear that ${\widetilde{R}}^+={\widetilde{R}}, {\widetilde{C}}^+={\widetilde{C}}, \widetilde{\kappa }^+=\widetilde{\kappa }, {\widetilde{L}}^+={\widetilde{L}}, S_{cr}^+=S_{cr}, S_{cd}^+=S_{cd}$, $S_{ac}^+=S_{ac}$, and

$$\begin{aligned} 0< {\widetilde{\kappa }} = \frac{\widetilde{R}}{\widetilde{C}} \le \kappa _1,\quad 0\le \frac{c_l q_r}{\widetilde{C}}\le 1,\quad 0<\frac{1}{{\widetilde{C}}}\le \frac{1}{c_{pd}}, \end{aligned}$$

(25)

for some positive constant $\kappa _1$. Besides, by the uniform boundedness of $q_v, q_c, q_r$, one has

$$\begin{aligned} |S_{cd}|+|S_{cr}|+|S_{ac}|\le C, \end{aligned}$$

(26)

for some positive constant C depending on $q_v^*, q_c^*, q_r^*$.

Proposition 3

Let Assumption 1 hold, then for any $\mathcal T\in (0,T_\text {max})$,

$$\begin{aligned} \sup _{0\le t\le {\mathcal T}} \Vert (q_v,q_c,q_r)\Vert ^2 + \int _0^{\mathcal T} \Vert \nabla (q_v,q_c,q_r)\Vert ^2 \hbox {d}t\le K_0(\mathcal T), \end{aligned}$$

for a continuous bounded function $K_0({\mathcal T})$ determined by $q_v^*, q_c^*,$ and $q_r^*$.

Proof

Testing (21), (22), and (23), respectively, with $q_v, q_c,$ and $q_r$, summing the resultants up, using the uniform boundedness of $q_v, q_c, q_r, S_{ac}, S_{cr}, S_{cd}$ (due to Proposition 2 and (26)), and noticing that $S_{ev}^+q_r\ge 0$, one deduces

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\hbox {d}}{\hbox {d}t}\Vert (q_v, q_c, q_r)\Vert ^2-\sum _{j\in \{v,c,r\}}\int _{\mathcal M}q_j\mathcal D^{q_j}q_j\hbox {d}{\mathcal M}\nonumber \\{} & {} \quad =-\sum _{j\in \{v,c,r\}}\int _{\mathcal M}(\overline{\textbf{v}}_h\cdot \nabla _hq_j+{{\overline{\omega }}}\partial _pq_j)q_jd-V\int _{\mathcal M}q_r\partial _p\left( \frac{p q_r}{R_d\bar{T}}\right) \hbox {d}{\mathcal M}\nonumber \\{} & {} \qquad +\int _{\mathcal M}[ (S_{ev}^+-S_{cd})q_v + (S_{cd}-S_{ac}-S_{cr})q_c + (S_{ac}+S_{cr}-S_{ev}^+)q_r]\hbox {d}{\mathcal M}\nonumber \\{} & {} \quad \le -\sum _{j\in \{v,c,r\}}\int _{\mathcal M}(\overline{\textbf{v}}_h\cdot \nabla _hq_j+{{\overline{\omega }}}\partial _pq_j)q_jd-V\int _{\mathcal M}q_r\partial _p\left( \frac{p q_r}{R_d\bar{T}}\right) \hbox {d}{\mathcal M}\nonumber \\{} & {} \qquad +\int _{\mathcal M}S_{ev}^+ q_v\hbox {d}{\mathcal M}+C. \end{aligned}$$

(27)

The integrals in (27) are estimated as follows. For the diffusion terms, integration by parts and using the boundary conditions (17)–(19), one deduces

$$\begin{aligned}{} & {} -\int _{\mathcal M}q_j{\mathcal D}^{q_j}q_j\hbox {d}{\mathcal M}\\{} & {} \quad =-\int _{\mathcal M}\left[ \mu _{q_j}\Delta _hq_j+\nu _{q_j}\partial _p \left( \left( \frac{gp}{R\bar{T}}\right) ^2\partial _pq_j\right) \right] q_j\hbox {d}{\mathcal M}\\{} & {} \quad =-\mu _{q_j}\int _{\Gamma _{\ell }}(\partial _nq_j)q_j\hbox {d}\Gamma _{\ell }-\nu _{q_j}\left. \int _{{\mathcal M}'} \left( \frac{gp}{R\bar{T}}\right) ^2(\partial _pq_j)q_j\hbox {d}{\mathcal M}'\right| _{p_1}^{p_0}\\{} & {} \qquad +\int _{\mathcal M}\left[ \mu _{q_j}\nabla _hq_j\cdot \nabla _hq_j+\nu _{q_j}\left( \frac{gp}{R\bar{T}}\right) ^2\partial _pq_j\partial _pq_j\right] \hbox {d}{\mathcal M}\\{} & {} \quad =\mu _{q_j}\Vert \nabla _hq_j\Vert ^2+\nu _{q_j}\Vert \partial _pq_j\Vert _w^2-\mu _{q_j} \int _{\Gamma _{\ell }}\alpha _{{\ell }j}(q_{{\ell }j}-q_j)q_j\hbox {d}\Gamma _{\ell }\\{} & {} \qquad -\nu _{q_j}\int _{{\mathcal M}'}\left( \frac{gp_0}{R\bar{T}}\right) ^2\alpha _{b0j}(q_{b0j}-q_j)q_j\hbox {d}{\mathcal M}'\\{} & {} \quad =\mu _{q_j}\Vert \nabla _hq_j\Vert ^2+\nu _{q_j}\Vert \partial _pq_j\Vert _w^2+\mu _{q_j} \int _{\Gamma _{\ell }}\alpha _{{\ell }j}\left[ \left( q_j-\frac{q_{{\ell }j}}{2}\right) ^2-\frac{q_{{\ell }j}^2}{4}\right] \hbox {d}\Gamma _{\ell }\\{} & {} \qquad +\nu _{q_j}\int _{{\mathcal M}'}\left( \frac{gp_0}{R\bar{T}}\right) ^2\alpha _{b0j}\left[ \left( q_j-\frac{q_{b0j}}{2}\right) ^2-\frac{q_{b0j}^2}{4}\right] \hbox {d}{\mathcal M}', \end{aligned}$$

for $j\in \{v,c,r\}$. This implies for $j\in \{v,c,r\}$ that

$$\begin{aligned} -\int _{\mathcal M}q_j{\mathcal D}^{q_j}q_j\hbox {d}{\mathcal M}\ge \mu _{q_j}\Vert \nabla _hq_j^-\Vert ^2+\nu _{q_j}\Vert \partial _pq_j^-\Vert _w^2-C, \end{aligned}$$

(28)

for a constant C depending only on the given inhomogeneous boundary functions $\alpha _{0 j }, \alpha _{{\ell }j}$ and $q_{b 0 j }, q_{b {\ell }j }$. The integral containing the advection term vanishes due to (15) and (16), since

$$\begin{aligned}{} & {} \int _{\mathcal M}(\overline{\textbf{v}}_h\cdot \nabla _h q_j + {{\overline{\omega }}} \partial _p q_j) q_j\hbox {d}{\mathcal M}= -\frac{1}{2}\int _{\mathcal M}(\overline{\textbf{v}}_h\cdot \nabla _h + {{\overline{\omega }}} \partial _p) (q_j)^2 \hbox {d}{\mathcal M}\nonumber \\{} & {} \quad = -\frac{1}{2}\int _{ \partial {\mathcal M}}(\overline{\textbf{v}}_h\cdot \textbf{n}_h + {{\overline{\omega }}} n_p)(q_j)^2d(\partial {\mathcal M})+\frac{1}{2}\int _{ {\mathcal M}}(q_j)^2(\nabla _h\cdot \overline{\textbf{v}}_h+ \partial _p{{\overline{\omega }}} )\hbox {d}{\mathcal M}=0.\qquad \nonumber \\ \end{aligned}$$

(29)

The Young inequality leads to

$$\begin{aligned} -V\int _{\mathcal M}q_r\partial _p\left( \frac{p q_r}{R_d\bar{T}}\right) \hbox {d}{\mathcal M}\le \frac{\nu _{q_r}}{8}\Vert \partial _pq_r\Vert _w^2+C\Vert q_r\Vert ^2. \end{aligned}$$

(30)

To estimate the term $\int _{\mathcal M}S_{ev}^+ q_v\hbox {d}{\mathcal M}$, we decompose the domain ${\mathcal M}$ as ${\mathcal M}={\mathcal M}_+(t)\cup {\mathcal M}_-(t)$, where ${\mathcal M}_+(t)=\{(x,y,p)\in {\mathcal M}|T(x,y,p,t)\ge T_{\text {crit}}\}$ and ${\mathcal M}_-(t)={\mathcal M}\setminus {\mathcal M}_+(t).$ Due to (12), one can check that $S_{ev}^+=0$ on ${\mathcal M}_+(t)$, while on ${\mathcal M}_-(t)$, due to the nonnegativity and uniform boundedness of $q_v, q_c, q_r$ guaranteed by Proposition 2, one has $S_{ev}^+\le C$. Therefore, we always have

$$\begin{aligned} 0\le S_{ev}^+\le C\quad \text{ on } {\mathcal M}\end{aligned}$$

and, as result, it holds that $\int _{\mathcal M}S_{ev}^+ q_v\hbox {d}{\mathcal M}\le C$. Thanks to this and combining (28)–(30), the conclusion follows from (27) by the Grönwall inequality. $\square$

We would like to point out that the a priori estimates obtained in Proposition 2 and Proposition 3 do not depend on the a priori bounds of the given velocity $(\overline{\textbf{v}}_h, {{\overline{\omega }}})$.

The following lemma will be used later.

Lemma 1

Let $f \in L^\infty ( 0,{\mathcal T};L^2({\mathcal M}))$ and $g,h \in L^2(0,{\mathcal T};H^1({\mathcal M}))\cap L^\infty (0,{\mathcal T};L^2({\mathcal M}))$. Then for some arbitrary $\delta _g,\delta _h >0$ at a.e. $t\in (0,{\mathcal T})$ the following estimate holds

$$\begin{aligned} \left| \int _{\mathcal M}f \,g \,h\, \textrm{d}{\mathcal M}\right| \le \delta _g \Vert \nabla g\Vert ^2_{L^2({\mathcal M})} + \delta _h \Vert \nabla h\Vert ^2_{L^2({\mathcal M})} +C \left( \Vert g\Vert ^2_{L^2({\mathcal M})} +\Vert h\Vert ^2_{L^2({\mathcal M})}\right) , \end{aligned}$$

where $C=C(\delta _1,\delta _2,\Vert f\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))})$.

Proof

We first bound the integral by

$$\begin{aligned} \left| \int _{\mathcal M}f \,g \,h\, \hbox {d}{\mathcal M}\right| \le \Vert f\Vert _{L^2({\mathcal M})}\Vert g\,h \Vert _{L^2({\mathcal M})}\le C (\Vert g^2\Vert _{L^2({\mathcal M})}+\Vert h^2\Vert _{L^2({\mathcal M})}) \end{aligned}$$

where C depends on $\Vert f\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))}$. We next employ the Gagliardo Nirenberg inequality, see e.g. [Friedman (1969), Theorem 10.1] and [Zheng (1995), Theorem 1.1.4], to estimate

$$\begin{aligned} \Vert g^2\Vert _{L^2({\mathcal M})}= & {} \Vert g\Vert ^2_{L^4({\mathcal M})}\le C\Vert g\Vert ^{2\vartheta }_{L^2({\mathcal M})}\Vert \nabla g\Vert ^{2(1-\vartheta )}_{L^2({\mathcal M})} + \Vert g\Vert ^2_{L^2({\mathcal M})} \\\le & {} \delta _g \Vert \nabla g\Vert ^2_{L^2({\mathcal M})} + C(\delta _g) \Vert g\Vert ^2_{L^2({\mathcal M})} \end{aligned}$$

which holds for $\vartheta = \frac{1}{4}$. In the last estimate we used Young’s inequality. The same estimate holds for h, which concludes the proof. $\square$

Nonnegativity of the temperature is proved in the following proposition.

Proposition 4

Let Assumption 1 hold, then the temperature T is nonnegative.

Proof

As already mentioned above, for the refined thermodynamic modelling used in this work, the anti-dissipative term ${\widetilde{\kappa }}^+\frac{T}{p} {{\overline{\omega }}}$ in the thermodynamic equation (20) does not vanish anymore, when switching to the potential temperature $\theta$. We therefore perform the estimate directly from the equation for T.

Testing the equation for T with $-T^-$ yields

$$\begin{aligned}{} & {} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\int _{\mathcal M}(T^-)^2\hbox {d}{\mathcal M}+\int _{\mathcal M}T^-{\mathcal D}^{T}T\hbox {d}{\mathcal M}-\int _{\mathcal M}(\overline{\textbf{v}}_h\cdot \nabla T+{{\overline{\omega }}}\partial _pT)T^-\hbox {d}{\mathcal M}\nonumber \\{} & {} \quad = \int _{\mathcal M}\frac{{\widetilde{\kappa }}}{p}(T^-)^2{\overline{\omega }} \hbox {d}{\mathcal M}-\int _{\mathcal M}\frac{c_l q_r}{\widetilde{C}} V T^- \partial _p T^- \hbox {d}{\mathcal M}-\int _{\mathcal M}\widetilde{L}T^-(S_{cd}-S^+_{ev}) \hbox {d}{\mathcal M}.\nonumber \\ \end{aligned}$$

(31)

The various terms in the above equality are estimated as follows. For the diffusion term, integration by parts and using the boundary conditions (17)–(19), one deduces

$$\begin{aligned} \int _{\mathcal M}T^-{\mathcal D}^{T}T\hbox {d}{\mathcal M}= & {} \int _{\mathcal M}\left[ \mu _{T}\Delta _hT+\nu _{T}\partial _p \left( \left( \frac{gp}{R\bar{T}}\right) ^2\partial _pT\right) \right] T^-\hbox {d}{\mathcal M}\\= & {} \mu _{T}\int _{\Gamma _{\ell }}(\partial _nT)T^-\hbox {d}\Gamma _{\ell }+\nu _{T}\left. \int _{{\mathcal M}'} \left( \frac{gp}{R\bar{T}}\right) ^2(\partial _pT)T^-\hbox {d}{\mathcal M}'\right| _{p_1}^{p_0}\\{} & {} -\int _{\mathcal M}\left[ \mu _{T}\nabla _hT\cdot \nabla _hT^-+\nu _{T}\left( \frac{gp}{R\bar{T}}\right) ^2\partial _pT\partial _pT^-\right] \hbox {d}{\mathcal M}\\= & {} \mu _{T}\Vert \nabla _hT^-\Vert ^2+\nu _{T}\Vert \partial _pT^-\Vert _w^2+\mu _{T} \int _{\Gamma _{\ell }}\alpha _{{\ell }T}(T_{{bl}}-T)T^-\hbox {d}\Gamma _{\ell }\\{} & {} +\nu _{T}\int _{{\mathcal M}'}\left( \frac{gp_0}{R\bar{T}}\right) ^2\alpha _{0T}(T_{b0}-T)T^-\hbox {d}{\mathcal M}'. \end{aligned}$$

Since the functions $\alpha _{{\ell }T}, T_{{bl}}, \alpha _{0T}, T_{b0}$ are nonnegative and $T T^- =- ( T^-)^2$, the last two boundary integrals are nonnegative, and we obtain

$$\begin{aligned} \int _{\mathcal M}T^-{\mathcal D}^{T}T\hbox {d}{\mathcal M}\ge \mu _{T}\Vert \nabla _hT^-\Vert ^2+\nu _{T}\Vert \partial _pT^-\Vert _w^2. \end{aligned}$$

(32)

Same as (29), the integral containing the advection terms vanishes due to (15) and (16). To bound the first term on the right-hand side containing the vertical velocity component, we apply Lemma 1 and use (25) as follows

$$\begin{aligned}{} & {} \left| \int _{\mathcal M}\frac{{\widetilde{\kappa }}}{p}(T^-)^2{\overline{\omega }} \hbox {d}{\mathcal M}\right| \le C \int _{\mathcal M}|{\overline{\omega }}|(T^-)^2 \hbox {d}{\mathcal M}\le \frac{\mu _T}{4}\Vert \nabla _h T^-\Vert ^2 \\{} & {} \quad + \frac{\nu _T}{4}\Vert \partial _pT^-\Vert ^2_w + C\Vert T^-\Vert ^2, \end{aligned}$$

where C depends on $\Vert {\overline{\omega }}\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))}$. Using (25) again and by the Young inequality, we can estimate the second term as

$$\begin{aligned} \left| \int _{\mathcal M}\frac{c_l q_r}{\widetilde{C}} V T^- \partial _p T^- \hbox {d}{\mathcal M}\right| \le \frac{\nu _T}{4}\Vert \partial _p T^-\Vert _w^2 + C\Vert T^-\Vert ^2. \end{aligned}$$

It remains to bound the integral with the latent heating terms.

$$\begin{aligned}{} & {} -\int _{\mathcal M}\widetilde{L}T^-S_{cd} \hbox {d}{\mathcal M}\\{} & {} \quad = - \int \frac{L_0 +(c_l-c_{pv} )T_0}{{\widetilde{C}}}[C_{cd} (q_v-q_{vs}) q_c + C_{cn} (q_v-q_{vs})^+]T^- \hbox {d}{\mathcal M}\\{} & {} \qquad - \int _{\mathcal M}\frac{c_l-c_{pv}}{\widetilde{C}} (T^-)^2 [C_{cd} (q_v-q_{vs}) q_c + C_{cn} (q_v-q_{vs})^+] \hbox {d}{\mathcal M}\le 0, \end{aligned}$$

where we used the fact $c_{pv}<c_l$, $q_{vs}=0$ for $T\le 0$, and the nonnegativity of all moisture quantities. The integral term with the evaporation term in (31) vanishes since $S_{ev}^+ T^-=C_{ev}{\widetilde{R}}T^+q_r(q_{rs}(p,T)-q_v)^+T^-=0$.

Combining all bounds above, we thus obtain from (31) that

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t} \Vert T^-\Vert ^2 + \frac{\mu _T}{2}\Vert \nabla _h T^-\Vert ^2 + \frac{\nu _T}{2}\Vert \partial _p T^-\Vert _w^2 \le C \Vert T^-\Vert ^2, \end{aligned}$$

and we can conclude by the Grönwall inequality the nonnegativity of T since $T_0^-=0$. $\square$

From now on, the a priori estimates to be carried out depend on the a priori bounds of the given velocity $(\overline{\textbf{v}}_h,{{\overline{\omega }}})$. The relevant bounds of the velocity on which the solutions depend will be explicitly pointed out in the statements of the propositions.

Proposition 5

Let Assumption 1 hold, then for any ${\mathcal T}\in (0,{\mathcal T}_\text {max})$

$$\begin{aligned} \sup _{0\le t\le {\mathcal T}} \Vert T \Vert ^2(t)+ \int _0^{\mathcal T}\Vert \nabla T \Vert ^2(t) \textrm{d}t \le K_1(\mathcal T), \end{aligned}$$

for a continuous bounded function $K_1({\mathcal T})$ determined by the quantities $\Vert {{\overline{\omega }}}\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))}$, $q_v^*, q_c^*, q_r^*$.

Proof

By testing the temperature equation with T and by the same calculations as (28) and (29) to the diffusion terms and the convection terms, we have the following estimate

$$\begin{aligned}{} & {} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert T\Vert ^2 + \mu _T\Vert \nabla _h T\Vert ^2 +\nu _T\Vert \partial _pT\Vert _w^2\\{} & {} \quad \le \int _{\mathcal M}\frac{{\widetilde{\kappa }}}{p} {\overline{\omega }} T^2 \hbox {d}{\mathcal M}- \int _{\mathcal M}\frac{c_l q_r}{\widetilde{C}}V \partial _p T \, T \hbox {d}{\mathcal M}+ \int _{\mathcal M}\widetilde{L}(S_{cd}-S_{ev}) T\hbox {d}{\mathcal M}+C. \end{aligned}$$

To bound the first term on the right-hand side, we use Lemma 1 and (25) to get

$$\begin{aligned} \int _{\mathcal M}\frac{{\widetilde{\kappa }}}{p} {\overline{\omega }} T^2 \hbox {d}{\mathcal M}\le C \int _{\mathcal M}|{\overline{\omega }}| T^2 \hbox {d}{\mathcal M}\le \frac{\mu _T}{4}\Vert \nabla _h T\Vert ^2 +\frac{\nu _T}{4}\Vert \partial _pT\Vert _w^2 + C\Vert T\Vert ^2, \end{aligned}$$

where C depends on $\Vert {\overline{\omega }}\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))}$. By Young’s inequality and using (25) again, we obtain

$$\begin{aligned} \left| \int _{\mathcal M}\frac{c_l q_r}{\widetilde{C}}V \partial _p T \, T \hbox {d}{\mathcal M}\right| \le \frac{\nu _T}{4}\Vert \partial _p T\Vert ^2_w + C\Vert T\Vert ^2. \end{aligned}$$

Recalling the expression of L(T) given by (9), we obtain due to (25) and (26) that

$$\begin{aligned} \left| \int _{\mathcal M}\widetilde{L}S_{cd} T \hbox {d}{\mathcal M}\right| =\left| \int _{\mathcal M}\frac{L(T)}{{\widetilde{C}}} S_{cd}T\hbox {d}{\mathcal M}\right| \le C (C_1+\Vert T\Vert ^2). \end{aligned}$$

Since (12) implies $S_{ev}=C_{ev}{\widetilde{R}}Tq_r(q_{vs}(p,T)-q_v)^+=C_{ev}{\widetilde{R}}Tq_r(-q_v)^+=0$, for $T>T_{\text {crit}}$, while (9) and (11) lead to $L(T)\ge 0$, for $0\le T\le T_{\text {crit}}$, we therefore have

$$\begin{aligned} {\widetilde{L}} S_{ev}=\frac{L(T)}{{\widetilde{C}}}S_{ev}\ge 0 \end{aligned}$$

(33)

and thus

$$\begin{aligned} -\int _{\mathcal M}\widetilde{L}S_{ev} T \hbox {d}{\mathcal M}\le 0. \end{aligned}$$

Combining all above estimates, we obtain

$$\begin{aligned} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert T\Vert ^2 + \mu _T\Vert \nabla _h T\Vert ^2 +\nu _T\Vert \partial _p T\Vert _w^2 \le C(C_2+\Vert T\Vert ^2), \end{aligned}$$

leading to the conclusion by the Grönwall inequality. $\square$

Uniform boundedness of T is stated and proved in the following proposition.

Proposition 6

Let Assumption 1 hold, then for any ${\mathcal T}\in (0,{\mathcal T}_\text {max})$

$$\begin{aligned} \sup _{0\le t\le {\mathcal T}}\Vert T\Vert _{L^\infty ({\mathcal M})}(t)\le K_2({\mathcal T}), \end{aligned}$$

for a continuous bounded function $K_2({\mathcal T})$ determined by the quantities $\Vert {{\overline{\omega }}}\Vert _{L^\infty (0,{\mathcal T};L^2({\mathcal M}))},$ $\Vert T_0\Vert _{L^\infty {({\mathcal M})}},$ $\Vert T_{b0}\Vert _{L^\infty ((0,T)\times {\mathcal M}')}$, $\Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}$, $q_v^*$, $q_c^*,$ and $q_r^*$.

Proof

In Hittmeir et al. (2017), the upper bound for the temperature was derived by employing the potential temperature equation. Again here this does not alleviate the computations due to the stronger coupling of the thermodynamic equation (8) to the moisture quantities. We thus instead apply here the proof of Coti Zelati et al. (2015) based on the De Giorgi technique.

Let $\lambda _k\ge \max \{\Vert T_0\Vert _{L^\infty ({\mathcal M})},T_{\text {crit}},\Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}, \Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}\}$, and denote $T_{\lambda _k} =(T-\lambda _k)^+.$ We claim that

$$\begin{aligned} {\widetilde{L}}S_{cd}T_{\lambda _k}\le 0. \end{aligned}$$

(34)

In fact, if $T<\lambda _k$, then $T_{\lambda _k}=(T-\lambda _k)^+=0$ and, as a result, ${\widetilde{L}}S_{cd}T_{\lambda _k}=0$, while if $T\ge \lambda _k$, it is clear by the definition of $\lambda _k$ that $T\ge T_{\text {crit}}$, and as a result, noticing that in this case $L(T)\le 0$ (due to (9) and (11)) and $q_{vs}(p,T)=0$ (due to (12)), it holds that

$$\begin{aligned} {\widetilde{L}}S_{cd}T_{\lambda _k}=\frac{L(T)}{{\widetilde{C}}}[C_{cd}(q_v-q_{vs})q_c+C_{cn}(q_v-q_{vs})^+]T_{\lambda _k} \le 0. \end{aligned}$$

(35)

Testing equation (8) with $T_{\lambda _k}$ and by similar calculations as (32) and (29) to the diffusion terms and the convection terms, we have the following estimate

$$\begin{aligned}{} & {} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\Vert T_{\lambda _k}\Vert ^2 + \mu _T\Vert \nabla _h T_{\lambda _k}\Vert ^2 +\nu _T\Vert \partial _p T_{\lambda _k}\Vert _w^2\nonumber \\{} & {} \quad \le \int _{\mathcal M}\frac{{\widetilde{\kappa }}}{p} {\overline{\omega }} TT_{\lambda _k} \hbox {d}{\mathcal M}- \int _{\mathcal M}\frac{c_l q_r}{\widetilde{C}}V \partial _p T_{\lambda _k} \, T_{\lambda _k} \hbox {d}{\mathcal M}+ \int _{\mathcal M}\widetilde{L}(S_{cd}-S_{ev}) T_{\lambda _k}\hbox {d}{\mathcal M}\,\nonumber \\{} & {} \quad \le C\int _{\mathcal M}|{\overline{\omega }}| (T_{\lambda _k}^2 + {\lambda _k} T_{\lambda _k}) \hbox {d}{\mathcal M}+C\int _{\mathcal M}| \partial _p T_{\lambda _k}|T_{\lambda _k} \hbox {d}{\mathcal M}, \end{aligned}$$

(36)

where (25), (33), (34), and (35) were used in the last step. By Lemma 1 and the Young inequality, it follows that

$$\begin{aligned} C\int _{\mathcal M}(|{\bar{\omega }}|T_{\lambda _k}^2+|\partial _pT_{\lambda _k}|T_{\lambda _k})\hbox {d}{\mathcal M}\le \frac{1}{2}\left( \mu _T\Vert \nabla _hT_{\lambda _k}\Vert ^2+\nu _T\Vert \partial _pT_{\lambda _k}\Vert _w^2\right) +C\Vert T_{\lambda _k}\Vert ^2, \end{aligned}$$

where C depends on $\Vert {\bar{\omega }}\Vert _{L^\infty (0,{\mathcal T}; L^2({\mathcal M}))}$, which substituted into (36) yields

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert T_{\lambda _k}\Vert ^2 + \mu _T\Vert \nabla _h T_{\lambda _k}\Vert ^2 +\nu _T\Vert \partial _p T_{\lambda _k}\Vert _w^2 \le C\Vert T_{\lambda _k}\Vert ^2+C{\lambda _k}\int _{\mathcal M}|{\bar{\omega }}| T_{\lambda _k} \hbox {d}{\mathcal M}. \end{aligned}$$

Due to $T_{\lambda _k}|_{t=0}=0$, we obtain by applying the Grönwall inequality to the above

$$\begin{aligned} J_k:= & {} \sup _{t\in (0, {\mathcal T})} \Vert T_{\lambda _k}\Vert ^2(t)+ \int _0^{\mathcal T}(\mu _T\Vert \nabla _h T_{\lambda _k}\Vert ^2 +\nu _T\Vert \partial _p T_{\lambda _k}\Vert _w^2) \hbox {d}t\nonumber \\\le & {} C{\lambda _k}\int _0^{\mathcal T}\int _{\mathcal M}|{\bar{\omega }}| T_{\lambda _k} \hbox {d}{\mathcal M}\hbox {d}t. \end{aligned}$$

(37)

Let $M\ge 2\max \{\Vert T_0\Vert _{L^\infty ({\mathcal M})},T_{\text {crit}},\Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}, \Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}\}$ be a positive constant to be determined later and choose

$$\begin{aligned} \lambda _k:= M(1-2^{-k}), \quad Q_k:=\left\{ (x,t)\in {\mathcal M}\times (0,{\mathcal T})|T(x,t)>\lambda _k\right\} ,\quad \text {for} \quad k\ge 1. \end{aligned}$$

For any $(x,t)\in Q_k$, noticing that $T(x,t)>\lambda _k>\lambda _{k-1}$, one deduces

$$\begin{aligned} T_{\lambda _{k-1}}(x,t)=(T-\lambda _{k-1})^+(x,t)=T(x,t)-\lambda _{k-1}>\lambda _k-\lambda _{k-1}=2^{-k}M \end{aligned}$$

and, thus,

$$\begin{aligned} \chi _{Q_k}(x,t)\le \frac{2^k}{M}T_{\lambda _{k-1}}(x,t). \end{aligned}$$

Thanks to this and noticing that $T_{\lambda _k}\le T_{\lambda _{k-1}}$ and $\lambda _k\le M$, one deduces from (37) and the Gagliardo–Nirenberg inequality that

$$\begin{aligned} J_k\le & {} CM\int _0^{{\mathcal T}}\int _{\mathcal M}|{\bar{\omega }}|T_{\lambda _k}\hbox {d}{\mathcal M}\hbox {d}t=CM\int _0^{\mathcal T}\int _{\mathcal M}|{\bar{\omega }}|T_{\lambda _k}\chi _{Q_k}^\frac{4}{3}\hbox {d}{\mathcal M}\hbox {d}t \\\le & {} \frac{C}{M^\frac{1}{3}}16^{\frac{k}{3}}\int _0^{\mathcal T}\int _{\mathcal M}|{\bar{\omega }}|T_{\lambda _{k-1}}^\frac{7}{3}\hbox {d}{\mathcal M}\hbox {d}t\le \frac{C}{M^\frac{1}{3}}16^{\frac{k}{3}}\int _0^{\mathcal T}\Vert {\bar{\omega }}\Vert \Vert {\mathcal T}_{\lambda _{k-1}}\Vert _{L^\frac{14}{3}({\mathcal M})}^\frac{7}{3} \hbox {d}t \\\le & {} \frac{C}{M^\frac{1}{3}}16^{\frac{k}{3}}\int _0^{\mathcal T}\Vert {\mathcal T}_{\lambda _{k-1}}\Vert ^\frac{1}{3}\left( \Vert {\mathcal T}_{\lambda _{k-1}}\Vert ^2+\Vert \nabla T_{\lambda _{k-1}}\Vert ^2\right) \hbox {d}t\\\le & {} \frac{C_*}{M^\frac{1}{3}}16^\frac{k}{3}J_{k-1}^\frac{7}{6}, \qquad k=2,3,4,\ldots , \end{aligned}$$

that is

$$\begin{aligned} J_k\le \frac{C_*}{M^\frac{1}{3}}16^\frac{k}{3}J_{k-1}^\frac{7}{6}, \qquad k=2,3,4,\cdots , \end{aligned}$$

(38)

where C depends on $\Vert {\bar{\omega }}\Vert _{L^\infty (0,{\mathcal T}; L^2({\mathcal M}))}$. Setting

$$\begin{aligned} a=64^7C_*^6M^{-2},\quad b=64,\quad S_k=ab^kJ_k, \end{aligned}$$

and by simple calculations, one can check from (38) that

$$\begin{aligned} S_{k+1}=ab^{k+1}J_{k+1}\le (ab^kJ_k)^\frac{7}{6}=S_k^{\frac{7}{6}} \end{aligned}$$

and, thus,

$$\begin{aligned} 64^{k+8}C_*^6M^{-2}J_{k+1}=S_{k+1}\le S_1^{\left( \frac{7}{6}\right) ^k}=\left[ 64^8C_*^6M^{-2}J_1\right] ^{\left( \frac{7}{6}\right) ^k},\quad k=1,2,\cdots . \qquad \end{aligned}$$

(39)

Recalling the definition of $J_k$ and applying Proposition 5 lead to

$$\begin{aligned} J_1= & {} \sup _{t\in (0, {\mathcal T})} \Vert T_{\lambda _1}\Vert ^2(t)+ \int _0^{\mathcal T}(\mu _T\Vert \nabla _h T_{\lambda _1}\Vert ^2 +\nu _T\Vert \partial _p T_{\lambda _1}\Vert _w^2) \hbox {d}t\nonumber \\\le & {} \sup _{t\in (0, {\mathcal T})} \Vert T\Vert ^2(t)+ \int _0^{\mathcal T}(\mu _T\Vert \nabla _h T\Vert ^2 +\nu _T\Vert \partial _p T\Vert _w^2) \hbox {d}t\le C_{**}. \end{aligned}$$

(40)

Choose M large enough such that $M\ge 2\max \{\Vert T_0\Vert _{L^\infty ({\mathcal M})},T_{\text {crit}},\Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}, \Vert T_{b{\ell }}\Vert _{L^\infty ((0,{\mathcal T})\times \Gamma _{\ell })}\}$ and $64^8C_*^6M^{-2}C_{**}\le \frac{1}{2}$. Then, it follows from (39) and (40) that

$$\begin{aligned} 64^{k+8}C_*^6M^{-2}J_{k+1}=S_{k+1}\le \left( \frac{1}{2}\right) ^{\left( \frac{7}{6}\right) ^k} \end{aligned}$$

and, thus, $\lim _{k\rightarrow \infty }J_k=0.$ This leads to the desired bound $T\le M$ on ${\mathcal M}\times (0,{\mathcal T})$. $\square$

Proposition 7

If Assumption 1 holds, we have the estimates

$$\begin{aligned} \sup _{0\le t\le \mathcal T}\Vert \nabla (q_v,q_c,q_r,T)\Vert ^2(t) +\int _0^\mathcal T\Vert \nabla ^2(q_v,q_c,q_r,T)\Vert ^2(t)\textrm{d}t\le K_3({\mathcal T}), \end{aligned}$$

for a continuous bounded function $K_3({\mathcal T})$ determined by the quantities $q_v^*, q_c^*, q_r^*$, $K_2({\mathcal T})$, $\Vert T_0\Vert _{L^\infty {({\mathcal M})}}$, $\Vert (T_0, q_{v0}, q_{r0}, q_{c0})\Vert _{H^1({\mathcal M})}$, and $\Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^r(0,{\mathcal T}; L^q({\mathcal M}))}.$

Proof

We only give the details about the proof for the estimate of T, those for the moisture components are similar (actually simpler).

We first estimate the vertical derivative $\partial _pT$. Multiplying the thermodynamic equation by $-\partial _p^2T$ and integrating the resultant over $\mathcal M$ yields

$$\begin{aligned}{} & {} \int _\mathcal M(-\partial _tT+\mathcal D^TT)\partial _p^2T\hbox {d}\mathcal M \nonumber \\{} & {} \quad =\int _\mathcal M\left[ \overline{\textbf{v}}_h\cdot \nabla _hT+{{\overline{\omega }}}\partial _pT -\widetilde{\kappa }\frac{T}{p}{{\overline{\omega }}}+\frac{c_lq_r}{{\widetilde{C}}} V \partial _p T+{\widetilde{L}}(S_{cd}-S_{ev})\right] \partial _p^2T\hbox {d}\mathcal M.\nonumber \\ \end{aligned}$$

(41)

Following the derivations in Hittmeir et al. (2020) (see (87) and (88) there) we have

$$\begin{aligned}{} & {} -\int _\mathcal M\partial _tT\partial _p^2T\hbox {d}\mathcal M \ge \frac{\hbox {d}}{\hbox {d}t}\left( \frac{\Vert \partial _pT\Vert ^2}{2}+\alpha _{0T} \left. \int _{\mathcal M'}\left( \frac{T^2}{2}-TT_{b0}\right) \hbox {d}\mathcal M'\right| _{p_0}\right) \nonumber \\{} & {} \quad -C(\Vert \partial _pT\Vert +C_3), \end{aligned}$$

(42)

$$\begin{aligned}{} & {} \int _\mathcal M\mathcal D^{T}T\partial _p^2T\hbox {d}\mathcal M \ge \frac{3}{4}\left( \mu _{T}\Vert \nabla _h\partial _pT\Vert ^2+\nu _{T} \Vert \partial _p^2T\Vert _w^2\right) \nonumber \\{} & {} \quad -C(\Vert \partial _pT\Vert ^2+C_4). \end{aligned}$$

(43)

By the Hölder, Sobolev, and Young inequalities, one deduces

$$\begin{aligned}{} & {} \int _\mathcal M(\overline{\textbf{v}}_h\cdot \nabla _hT+{{\overline{\omega }}}\partial _pT) \partial _p^2T\hbox {d}\mathcal M\nonumber \\{} & {} \quad \le C\Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}\Vert \nabla T\Vert _{L^{\frac{2q}{q-2}}({\mathcal M})}\Vert \partial _p^2T\Vert \nonumber \\{} & {} \quad \le C\Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}\Vert \nabla T\Vert ^{1-\frac{3}{q}}\Vert \nabla T\Vert _{H^1({\mathcal M})}^{1+\frac{3}{q}}\nonumber \\{} & {} \quad \le \eta \Vert \nabla ^2T\Vert ^2+C_\eta \left( \Vert (\overline{\textbf{v}}_h, {{\overline{\omega }}})\Vert _{L^q({\mathcal M})}^{\frac{2q}{q-3}}+C_5\right) \Vert \nabla T\Vert ^2, \end{aligned}$$

(44)

for any positive number $\eta$. Moreover, by (25) and Proposition 6, we obtain by the Young inequality that

$$\begin{aligned}{} & {} -\int _\mathcal M\widetilde{\kappa }\frac{T}{p}{{\overline{\omega }}}\partial _p^2T\hbox {d}\mathcal M+\int _\mathcal M \frac{c_l q_r}{{\widetilde{C}}} V \partial _pT \partial _p^2 T \hbox {d}\mathcal M \le \eta \Vert \partial _p^2T\Vert _w^2+C_\eta (\Vert {{\overline{\omega }}}\Vert ^2\nonumber \\{} & {} \quad +C_6\Vert \partial _pT\Vert ^2), \end{aligned}$$

(45)

for any positive number $\eta$. Due to the nonnegativity and uniform boundedness of $T, q_v, q_c, q_r$, it follows from the Young inequality that

$$\begin{aligned} \int _\mathcal M\widetilde{L}(S_{cd}-S_{ev})\partial _p^2T\hbox {d}\mathcal M \le C\int _\mathcal M|\partial _p^2T|\hbox {d}\mathcal M \le \eta \Vert \partial _p^2T\Vert ^2+C_\eta , \end{aligned}$$

(46)

for any positive number $\eta$. Substituting (42)–(46) into (41), one obtains

$$\begin{aligned}{} & {} \frac{\hbox {d}}{\hbox {d}t}\left( \frac{\Vert \partial _pT\Vert ^2}{2}+\alpha _{0T} \left. \int _{\mathcal M'}\left( \frac{T^2}{2}-TT_{b0}\right) \hbox {d}\mathcal M'\right| _{p_0}\right) + \frac{3}{4}(\mu _{T} \Vert \nabla _h\partial _pT\Vert ^2+\nu _{T} \Vert \partial _p^2T\Vert _w^2) \nonumber \\{} & {} \quad \le 3\eta \Vert \nabla ^2T\Vert ^2+C_\eta \left( \Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}^{\frac{2q}{q-3}} +\Vert {{\overline{\omega }}}\Vert ^2+C_7\right) (\Vert \nabla T\Vert ^2+C_8), \end{aligned}$$

(47)

for any positive number $\eta$.

Next, we estimate the horizontal gradient $\nabla _hT$. Multiplying equation (20) by $-\Delta _hT$ and integrating the resultant over $\mathcal M$ yields

$$\begin{aligned}{} & {} \int _\mathcal M(-\partial _tT+\mathcal D^TT)\Delta _hT\hbox {d}\mathcal M \nonumber \\{} & {} \quad =\int _\mathcal M\left[ \overline{\textbf{v}}_h\cdot \nabla _hT+{{\overline{\omega }}}\partial _pT -\widetilde{\kappa }\frac{T}{p}{{\overline{\omega }}}+ \frac{c_l q_r}{{\widetilde{C}}} V \partial _pT+{\widetilde{L}}(S_{cd}-S_{ev})\right] \Delta _hT\hbox {d}\mathcal M.\nonumber \\ \end{aligned}$$

(48)

Following the derivations in Hittmeir et al. (2020) (see (94) and (95) there) we have

$$\begin{aligned} -\int _\mathcal M\partial _tT\Delta _hT\hbox {d}\mathcal M\ge & {} \frac{\hbox {d}}{\hbox {d}t}\left( \frac{\Vert \nabla _hT\Vert ^2}{2}+\alpha _{\ell T} \int _{\Gamma _\ell }\left( \frac{T^2}{2}-TT_{b\ell }\right) \hbox {d}\Gamma _\ell \right) \nonumber \\{} & {} -C(C_9+\Vert \nabla _hT\Vert ), \end{aligned}$$

(49)

$$\begin{aligned} \int _\mathcal M\mathcal D^TT\Delta _hT\hbox {d}\mathcal M\ge & {} \mu _T\Vert \Delta _hT\Vert ^2+\nu _T\Vert \nabla _h\partial _pT\Vert _w^2 -C. \end{aligned}$$

(50)

Similar to (44)–(46), we have

$$\begin{aligned}{} & {} \int _\mathcal M(\overline{\textbf{v}}_h\cdot \nabla _hT+{{\overline{\omega }}}\partial _pT) \Delta _hT\hbox {d}\mathcal M\le \eta \Vert \nabla ^2T\Vert ^2\nonumber \\{} & {} \quad +C_\eta \left( \Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}^{\frac{2q}{q-3}}+C_{10}\right) \Vert \nabla T\Vert ^2, \end{aligned}$$

(51)

$$\begin{aligned}{} & {} \quad -\int _\mathcal M\widetilde{\kappa }\frac{T}{p}{{\overline{\omega }}}\Delta _hT\hbox {d}\mathcal M+ \int _\mathcal M \frac{c_l q_r}{{\widetilde{C}}} V \partial _pT\Delta _h T \hbox {d}{\mathcal M}\le \eta \Vert \Delta _hT\Vert ^2+C_\eta (\Vert {{\overline{\omega }}}\Vert ^2\nonumber \\{} & {} \quad +C_{11}\Vert \partial _pT\Vert _w^2), \end{aligned}$$

(52)

$$\begin{aligned}{} & {} \int _\mathcal M {\widetilde{L}}(S_{cd}-S_{ev})\Delta _hT\hbox {d}\mathcal M \le \eta \Vert \Delta _hT\Vert ^2+C_\eta , \end{aligned}$$

(53)

for any positive number $\eta$. Substituting (49)–(53) into (48) gives

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left( \frac{\Vert \nabla _hT\Vert ^2}{2}+\alpha _{\ell T} \int _{\Gamma _\ell }\left( \frac{T^2}{2}-TT_{b\ell }\right) \hbox {d}\Gamma _\ell \right) +\frac{3}{4}(\mu _T\Vert \Delta _hT\Vert ^2+\nu _T\Vert \nabla _h\partial _pT\Vert _w^2)\nonumber \\ \le 3\eta \Vert \nabla ^2T\Vert ^2+C_\eta \left( \Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}^{\frac{2q}{q-3}} +\Vert {{\overline{\omega }}}\Vert ^2+C_{12}\right) (\Vert \nabla T\Vert ^2+C_{13}),\nonumber \\ \end{aligned}$$

(54)

for any positive number $\eta$.

Summing (47) with (54) yields

$$\begin{aligned}{} & {} \frac{3}{4}\Big (\mu _{T} \Vert \nabla _h\partial _pT\Vert ^2+\nu _{T} \Vert \partial _p^2T\Vert _w^2+\mu _T\Vert \Delta _hT\Vert ^2+\nu _T\Vert \nabla _h\partial _pT\Vert _w^2\Big ) \nonumber \\{} & {} \qquad +\frac{\hbox {d}}{\hbox {d}t}\left( \frac{\Vert \nabla T\Vert ^2}{2}+\alpha _{0T} \left. \int _{\mathcal M'}\left( \frac{T^2}{2}-TT_{b0}\right) \hbox {d}\mathcal M'\right| _{p_0} +\alpha _{\ell T} \int _{\Gamma _\ell }\left( \frac{T^2}{2}-TT_{b\ell }\right) \hbox {d}\Gamma _\ell \right) \nonumber \\{} & {} \quad \le 6\eta \Vert \nabla ^2T\Vert ^2+C_\eta \left( \Vert (\overline{\textbf{v}}_h,{{\overline{\omega }}})\Vert _{L^q({\mathcal M})}^{\frac{2q}{q-3}} +\Vert {{\overline{\omega }}}\Vert ^2+C_{14}\right) (\Vert \nabla T\Vert ^2+C_{15}), \end{aligned}$$

(55)

for any positive number $\eta$. Applying the elliptic estimate (see Proposition A.2 in Hittmeir et al. (2017)) to the elliptic equation $\mathcal D^TT=f$ subject to the boundary condition (17)–(19) leads to

$$\begin{aligned} \Vert \nabla ^2T\Vert\le & {} C(\Vert \mathcal D^TT\Vert +C_{16}\Vert \nabla T\Vert +C_{17})\\\le & {} C(\Vert \Delta _hT\Vert +\Vert \partial _p^2T\Vert _w^2+C_{18}\Vert \nabla T\Vert +C_{19}). \end{aligned}$$

Thanks to this, and noticing that

$$\begin{aligned} \int _{\mathcal M'}\left( \frac{T^2}{2}-TT_{b0}\right) \hbox {d}\mathcal M' =\frac{1}{2} \int _{\mathcal M'}\left[ (T-T_{b0})^2-T_{b0}^2\right] \hbox {d}\mathcal M' \ge - \int _{\mathcal M'}T_{b0}^2 \hbox {d}\mathcal M' \ge -C, \end{aligned}$$

and

$$\begin{aligned} \int _{\Gamma _\ell }\left( \frac{T^2}{2}-TT_{b\ell } \right) \hbox {d} \Gamma _\ell= & {} \frac{1}{2}\int _{\Gamma _\ell }\left[ (T-T_{b\ell })^2-T_{b\ell }^2 \right] \hbox {d} \Gamma _\ell \ge -\frac{1}{2}\int _{\Gamma _\ell }T_{b\ell }^2 \hbox {d}\Gamma _\ell \ge -C, \end{aligned}$$

the conclusion follows by applying the Grönwall inequality to (55). $\square$

We are now ready to give the proof of our main result, Theorem 1.

Proof of Theorem 1

By Proposition 1, there is a unique local solution $( T, q_v, q_c, q_r)$ to system (20)–(23), subject to (17)–(19). Due to Proposition 2 and Proposition 4, $q_c, q_c, q_r,$ and T are all nonnegative and, thus, $(T, q_v, q_c, q_r)$ is a local solution to the original system, subject to the corresponding initial and boundary conditions. By applying Proposition 1 inductively, one can extend the local solution to the maximal time of existence ${\mathcal T}_{\text {max}}$ characterized by (24). We need to prove $\mathcal T_{\text {max}}=\infty$. Assume, by contradiction, that $\mathcal T_{\text {max}}<\infty$. Then, by Propositions 2–7, we have the estimate

$$\begin{aligned} \sup _{0\le t\le \mathcal T} \Vert (T,q_v, q_c, q_r)\Vert _{H^1(\mathcal M)}\le C_0, \end{aligned}$$

for any $\mathcal T\in (0,\mathcal T_{\text {max}})$, and $C_0$ is a positive constant independent of $\mathcal T\in (0,\mathcal T_{\text {max}})$. This contradicts (24) and, thus, ${\mathcal T}_{\text {max}}=\infty$, proving the conclusion. $\square$