On the Motion of Chemically Reacting Fluids Through Porous Medium

Abstract

We consider a parabolic-hyperbolic system of nonlinear partial differential equations modeling the motion of a chemically reacting mixture through porous medium. The existence of classical as well as weak solutions is established under several physically relevant choices of the constitutive equations and relevant boundary conditions.

1 Introduction

A simple model of the motion of a mixture of n chemically reacting fluids takes the form (see e.g. Giovangigli [10, Chaps. 2,3]):

$$\begin{aligned} \partial _t (\rho ^i) + \mathrm{div}_x(\rho ^i \mathbf{v} ) + \mathrm{div}_x\mathscr {F}^i = m_i \omega ^i, \end{aligned}$$

(1)

where $\rho ^i$ is the mass density of the i-th species, $\mathbf{v}$ is the fluid bulk velocity of the mixture, $m_i$ the molar mass of the i-th species, $\mathscr {F}^i$ the diffusive fluxes, and $\omega ^i$ represent the molar production, typically given functions of $(\rho ^1, \dots , \rho ^n)$ and of the temperature. We also denote

$$ \rho = \sum _{i=1}^n \rho ^i, $$

the total density of the mixture and introduce the mass fractions

$$ Y^i = \frac{\rho ^i}{\rho }, \ i = 1,\dots , n. $$

Obviously,

$$\begin{aligned} Y^i \ge 0,\ \sum _{i=1}^n Y^i = 1. \end{aligned}$$

(2)

We may sum up (1) to deduce the mass conservation (equation of continuity):

$$\begin{aligned} \partial _t \rho + \mathrm{div}_x(\rho \mathbf{v} ) = - \mathrm{div}_x\sum _{i = 1}^n \mathscr {F}^i + \sum _{i = 1}^n m_i \omega ^i = 0, \end{aligned}$$

(3)

where the last equality should be viewed as a natural constraint to be imposed on $\mathscr {F}^i$, $\omega ^i$ enforced by the principle of mass conservation.

The diffusion fluxes are typically given through the empirical Fick’s law:

$$\begin{aligned} \mathscr {F}^i = - d_i \nabla _xY^i,\ d_i > 0, \ i = 1,\dots ,n \end{aligned}$$

(4)

If the motion takes place in the porous medium environment, we may close the system by imposing the standard hypothesis that the velocity $\mathbf{v}$ is given by the pressure gradient, more specifically

$$\begin{aligned} \mathbf{v} = - \nabla _xp + \rho \mathbf{g}, \end{aligned}$$

(5)

where $\mathbf{g}$ represents the gravitational force. For the one component compressible flow, the relation (5) has been rigorously identified as a homogenization limit of the compressible Navier–Stokes system, see Masmoudi [11]. The result has been extended to a more general class of pressure laws and also to the full Navier–Stokes–Fourier system in [8]. The Eq. (1) has been considered in many special situations, namely in connection with the Navier–Stokes system, see, e.g., [9] and references there. Here, the main point consists in considering mixed boundary conditions for a simplified model, and an application of the DiPerna, Lions theory to the general problem.

1.1 A Parabolic-Hyperbolic System

We consider the problem of solvability of the system described above for $(\rho ^i,\mathbf{v})$, or alternatively, for $(Y^i,p)$, in the set $\{(t,x)\in [0,\tau ]\times \varOmega \}$, where $\varOmega \subset R^3$ is a bounded smooth domain, under the following simplifying assumptions:

• The diffusion coefficients $d_i$ vanish for all $i=1,2,\dots$.

• The process is isothermal, the temperature $T > 0$ is constant.

• The effect of the gravitational force is neglected, $\mathbf{g} = 0$.

• The production rates $\omega ^i = \omega ^i(\rho ^1, \dots , \rho ^n)$ are given smooth functions of species densities. More specifically,

  $$\begin{aligned} \omega ^i = \mathscr {C}_i - \rho ^i \mathscr {D}_i , \end{aligned}$$

  (6)

  where $\mathscr {C}_i \ge 0$, $\mathscr {D}_i \ge 0$,

  $$\begin{aligned} \mathscr {C}_i = \sum _{j = 1}^m \left[ \nu ^b_{i,j} K^f_j (T) \Pi _{l=1}^n \left( \frac{\rho ^l}{m_l} \right) ^{\nu ^{f}_{l,j}} + \nu ^f_{i,j} K^b_j (T) \Pi _{l=1}^n \left( \frac{\rho ^l}{m_l} \right) ^{\nu ^{b}_{l,j}} \right] , \end{aligned}$$

  (7)
  $$\begin{aligned} \mathscr {D}_i = \frac{1}{m_i} \left[ \sum _{j = 1, \nu ^f_{i,j} \ge 1}^m \nu ^f_{i,j} K^f_j(T) \left( \frac{\rho ^i}{m_i} \right) ^{\nu ^f_{i,j} - 1} \Pi _{l=1, l \ne i}^n \left( \frac{\rho ^l}{m_l} \right) ^{\nu ^{f}_{l,j}} \right. \end{aligned}$$

  (8)
  $$ \left. + \sum _{j = 1, \nu ^b_{i,j} \ge 1}^m \nu ^b_{i,j} K^b_j(T) \left( \frac{\rho ^i}{m_i} \right) ^{\nu ^b_{i,j} - 1} \Pi _{l=1, l \ne i}^n \left( \frac{\rho ^l}{m_l} \right) ^{\nu ^{b}_{l,j}} \right] , $$

  where m is the number of chemical reactions, $K^f_j$, $K^b_j$ are positive functions of the temperature, and $\nu ^f_{i,j}$, $\nu ^b_{i,j}$ are non-negative integers (stoichiometric coefficients), see [10, Sect. 6.4.6].

• The pressure of the mixture is given by the perfect gas law,

  $$\begin{aligned} p = \sum _{i=1}^n \frac{1}{m_i} \rho ^i R T, \end{aligned}$$

  (9)

  where R is the perfect gas constant.

Remark 1

It is interesting to note that (9) with equal molar masses $m_i = m$ is the only choice of the pressure compatible with the Second Law of Thermodynamics as soon as Fick’s law is imposed, cf. [9].

Our goal in the present paper is to discuss the solvability and a proper choice of boundary conditions for system (1) under the simplifying conditions stated above. In Sect. 2, we study the case when the pressure p satisfies a parabolic equation of porous medium type independent of the species densities $\rho ^i$. The standard parabolic theory yields a regular pressure p that can be subsequently substituted in (1) to determine uniquely $\rho ^i$, $i = 1,\dots ,n$ by the method of characteristics. Relevant boundary conditions are easy to discuss in this context.

In Sect. 3, we address the general situation when all equations in (1) are strongly coupled. The resulting system is of mixed parabolic-hyperbolic type. We derive a priori bounds and show weak sequential stability of the family of solutions. To this end, a variant of DiPerna, Lions [7] theory for the transport equation is used.

2 The Case of “Independent” Pressure

We start with the simple situation of equal molar masses $m_i = m > 0$ for all $i=1,\dots , n$. In accordance with (9) and (3), (5), we may sum up the Eq. (1) to obtain

$$\begin{aligned} \partial _t p - \mathrm{div}_x(p \nabla _xp) = 0. \end{aligned}$$

(10)

Thus the pressure satisfies a parabolic type differential equation that may be solved separately and independently of the other quantities. Note that the same situation occurs in the absence of chemical reactions, meaning $\omega _i = 0$ for all $i=1,\dots ,n$. Most generally, we have (10) whenever

$$\begin{aligned} \sum _{j \in S_j} \omega _j = 0, \ m_j = m_{S_j} > 0 \ \text{ for } \text{ all }\ j \in S_j, \ S_i \cap S_j = \emptyset \ \text{ if }\ i \ne j,\ \cup _{j} S_j = \{1, \dots , n\}. \end{aligned}$$

(11)

2.1 Boundary Value Problem for the Pressure Equation

Equation (10) represents the standard porous medium equation studied frequently in the literature, see e.g. Di Benedetto [6]. Here, in addition, we avoid the “vacuum” problem by imposing positive initial and boundary conditions on p.

2.1.1 Mixed Neumann–Dirichlet Boundary Conditions

We suppose the boundary $\partial \varOmega$ can be decomposed as

$$\begin{aligned} \partial \varOmega = \varGamma _D\cup \varGamma _N,\ \varGamma _D,\ \varGamma _N \; \text{ smooth } \text{ and } \text{ compact } \text{ with } \; \varGamma _D\cap \varGamma _N = \emptyset \,. \end{aligned}$$

(12)

We impose the (non-homogeneous) Dirichlet boundary condition

$$\begin{aligned} { p = p_b \text{ on } \varGamma _D,\ \ p_b \text{ is } \text{ a } \text{ positive } \text{ constant, } } \end{aligned}$$

(13)

together with the (homogenous) Neumann boundary condition

$$\begin{aligned} \nabla _xp\cdot \mathbf{n} = 0 \ \text{ on } \; \varGamma _N \,. \end{aligned}$$

(14)

As usual, $\mathbf{n}$ denotes the outer unit normal vector to the boundary $\partial \varOmega$ of $\varOmega$.

Remark 2

This choice of boundary conditions corresponds to the presence of a “well” in the container $\varOmega$ on the boundary of which a (constant) pressure is maintained, with the rest of $\partial \varOmega$ being an impermeable wall.

In order to deal with a well-posed problem, we prescribe the initial pressure distribution

$$\begin{aligned} p(0,x) = p_0(x) \ \text{ in }\ \varOmega . \end{aligned}$$

(15)

(i) Consider the situation

$$ p_0(x) \ge p_b > 0 \ \text{ for } \text{ all } \ x \in \varOmega ,\ p_0 \in W^{2,\infty } (\varOmega ),\ p_0 \not \equiv p_b. $$

By virtue of the standard parabolic theory, problem (10), (12–15) admits a unique solution

$$ p(t,x)\ge p_b \; \text{ for } \text{ any } \; (t,x)\in (0,\tau )\times \varOmega \,. $$

Moreover, the solution is smooth in the open set $(0,\tau )\times \varOmega$ and, by virtue of the strong maximum principle (Hopf’s boundary point lemma),

$$\begin{aligned} \nabla _xp\cdot \mathbf{n} < 0 \; \text{ on } \; \varGamma _D \,. \end{aligned}$$

(16)

Now, equation (1) reduces to the transport problem

$$\begin{aligned} \partial _t (\rho ^i) - \mathrm{div}_x(\rho ^i \nabla _xp ) = m_i \omega ^i(\rho ^1, \dots , \rho ^n),\ i = 1,\dots n, \end{aligned}$$

(17)

with a given (regular) velocity field $\mathbf{v} = - \nabla _xp$. Keeping (14), (16) in mind, the method of characteristics yields that Eq. (17) admits a unique solution for any initial data

$$\begin{aligned} \rho ^i (0, \cdot ) = \rho ^i_0 \,, \quad \text{ in } \, \varOmega \end{aligned}$$

(18)

satisfying the obvious compatibility condition

$$\begin{aligned} \sum _{i = 1}^n \frac{1}{m_i} \rho ^i_0 RT = p_0 \quad \text{ in } \, \varOmega \,. \end{aligned}$$

(19)

(ii) Now, we examine the complementary situation

$$ 0 < p_0(x) \le p_b \quad \text{ for } \text{ all } \, x\in \overline{\varOmega } ,\ p_0\in W^{2,\infty } (\varOmega ) ,\ p_0\not \equiv p_b \,. $$

It is easy to check, by means of the same arguments as above, that

$$\begin{aligned} \nabla _xp \cdot \mathbf{n} > 0 \ \text{ on }\ \varGamma _D. \end{aligned}$$

(20)

Consequently, for the transport problem (17), (18) to be uniquely solvable, we have to prescribe the boundary conditions

$$ \rho ^i |_{\varGamma _D} = \rho ^i_b ,\ i = i, \dots , n, $$

with the compatibility condition

$$ \sum _{i = 1}^n \frac{1}{m_i} \rho ^i_b RT = p_b. $$

(iii) In general, the sign of the normal component of the velocity $- \nabla _xp\cdot \mathbf{n}$ on $\varGamma _D$ is determined by the pressure. In particular, the relevant boundary conditions for $\rho ^i$ must be prescribed a posteriori, after having solved problem (10), (12–15).

2.2 Other Boundary Conditions

More general boundary conditions can be handled in a similar fashion. One should always keep in mind that the boundary conditions for the species densities $\rho ^i_b$ must be determined after having identified the sign of $\nabla _xp \cdot \mathbf{n}$ together with p on $\partial \varOmega$.

3 General System

We focus on the general case in which the equations for the pressure and the species densities are coupled. It turns out that it is more convenient to consider p, together with the mass fractions $Y^i$, as independent variables. Taking into account (3) with $\mathbf{v}=-\nabla _x p$, the resulting system of equations reads:

$$\begin{aligned} \partial _t p - \mathrm{div}_x(p \nabla _xp) = RT \sum _{i=1}^n \omega ^i, \end{aligned}$$

(21)

$$\begin{aligned} \partial _t Y^i - \nabla _xp \cdot \nabla _xY^i = \frac{m_i}{\rho } \omega ^i ,\ i = 1,\dots , n. \end{aligned}$$

(22)

Recalling the pressure-density relation

$$\begin{aligned} \rho = p \left( \sum _{i=1}^n \frac{1}{m_i} Y^i RT \right) ^{-1},\ \rho ^i = Y^i \rho , \end{aligned}$$

(23)

and using the specific form of $\omega ^i$ stated in (6–8), we view the right-hand sides of the above equations as functions of p and $Y^1, \dots , Y^n$.

System (21–23) is nonlinear of parabolic-hyperbolic type. To avoid unnecessary technicalities, we impose the homogeneous Neumann boundary conditions for the pressure,

$$\begin{aligned} \nabla _xp\cdot \mathbf{n}|_{\partial \varOmega } = 0 \,. \end{aligned}$$

(24)

Accordingly, only the initial conditions for $Y^i$ are necessary to make the problem, at least formally, well-posed.

3.1 A Priori Estimates

We start by deriving suitable a priori estimates for (smooth) solutions of problem (21), (22), (24).

3.1.1 Uniform Bounds on the Pressure

Uniform bounds on the pressure are usually derived by application of some form of the maximum principle. A short inspection of the pressure Eq. (21) and the structure (6) of the functions $\omega ^i$ reveals that

$$ \sum _{i=1}^n \omega _i = \sum _{i=1}^n \mathscr {C}_i - \rho ^i \mathscr {D}_i {\mathop {\sim }\limits ^{<}}\sum _{ j = 1}^n \left( p^{ \sum _{l=1}^m \nu ^f_{l,j} } + p^{ \sum _{l=1}^m \nu ^b_{l,j} } \right) . $$

Consequently, in view of the standard maximum principle estimates, we get a uniform bound

$$\begin{aligned} 0 \le p (t,x) \le \overline{p} \ \text{ on } \text{ the } \text{ time } \text{ interval }\ (0,\tau ), \end{aligned}$$

(25)

where $\tau > 0$ depends, in general, on $\Vert p(0,\cdot ) \Vert _{L^\infty (\varOmega )}$. Moreover, the estimate is uniform, meaning extendable to any positive $\tau$ if at least one of the following situations occurs:

• $$ \sum _{i=1}^n \mathscr {C}_i - \rho ^i \mathscr {D}_i {\mathop {\sim }\limits ^{<}}(p + 1), $$

  for specific examples see [10, Sect. 3.2.3];

• $$ \Vert p(0,\cdot ) \Vert _{L^\infty (\varOmega )} \ \text{ is } \text{ sufficiently } \text{ small, } $$

  where “small” means in terms of $\tau$ and the structural constants appearing in (7), (8).

Accordingly, in the remaining part of this section, we assume the validity of the bound (25). Note that, in view of the structure of $\omega ^i$ stated in (6), relation (25) implies that

$$\begin{aligned} p(t, \cdot ) \ge \underline{p}> 0 \ \text{ for } \text{ any }\ t \in (0,\tau ) \ \text{ as } \text{ soon } \text{ as }\ \inf _{x \in \varOmega } p(0,x) > 0, \end{aligned}$$

(26)

where the lower bound $\underline{p}$ may depend on $\tau$.

3.1.2 Maximal Regularity Estimates

In view of (25), (26) we may use the maximal regularity estimates for (non-degenerate) parabolic equations, see Denk, Hieber, and Pruess [4] or Ashyralyev and Sobolevskii [5], to deduce the bounds

$$\begin{aligned} \partial _t p,\ \nabla _x^l p \,;\ l = 0,1,2 \,, \; \text{ bounded } \text{ in } \; L^q((0,\tau )\times \varOmega ) \; \text{ for } \text{ any } \text{ finite } \; 1< q < \infty \,. \end{aligned}$$

(27)

Unfortunately, the bounds (27) are still not sufficient for the transport Eq.  (22) to be well-posed. The available DiPerna, Lions theory [7] (see also Ambrosio [2], Crippa and De Lellis [3]) require that, at least,

$$\begin{aligned} \mathrm{div}_x\nabla _xp = \Delta _x p\in L^1(0,\tau ; L^\infty (\varOmega )) \,. \end{aligned}$$

(28)

In order to guarantee (28), higher order regularity estimates are needed that will be established in the next section.

3.1.3 Higher Order Regularity

Taking the time derivative of (21) with respect to t and denoting $P = \partial _t p$, we obtain

$$\begin{aligned} \partial _t P - \mathrm{div}_x(p \nabla _xP) = \mathrm{div}_x( \partial _t p \nabla _xp ) + RT \sum _{i=1}^n \partial _t \omega ^i. \end{aligned}$$

(29)

To evaluate $\partial _t \omega ^i$ we realize that, thanks to (6–8),

$$ \omega _i = \sum _{k = 1}^{k_i} \rho ^k G_{k,i} (\rho ^1, \dots , \rho ^n),\ i = 1,\dots , n, $$

where $G_{k,i}$ are continuously differentiable functions. Using (17) we compute

$$ \partial _t \left( \rho ^k G_{k,i} (\rho ^1, \dots , \rho ^n) \right) = \partial _t \rho ^k G_{k,i} (\rho ^1, \dots , \rho ^n) + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \partial _t \rho ^j $$

$$ = \mathrm{div}_x( \rho ^k \nabla _xp) G_{k,i} (\rho ^1, \dots , \rho ^n) + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \mathrm{div}_x(\rho ^j \nabla _xp) $$

$$ + m_i \omega _i G_{k,i} (\rho ^1, \dots , \rho ^n) + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} m_j \omega _j. $$

Furthermore,

$$ \mathrm{div}_x( \rho ^k \nabla _xp) G_{k,i} (\rho ^1, \dots , \rho ^n) + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \mathrm{div}_x(\rho ^j \nabla _xp) $$

$$ = \mathrm{div}_x\left[ \rho ^k \nabla _xp G_{k,i} (\rho ^1, \dots , \rho ^n) \right] - \rho _k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \nabla _xp \cdot \nabla _x\rho ^j $$

$$ + \rho _k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \nabla _xp \cdot \nabla _x\rho ^j + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \rho ^j \Delta _x p $$

$$ = \mathrm{div}_x\left[ \rho ^k \nabla _xp G_{k,i} (\rho ^1, \dots , \rho ^n) \right] + \rho ^k \sum _{j=1}^n \frac{ G_{k,i}(\rho ^1, \dots , \rho ^n) }{\partial \rho ^j} \rho ^j \Delta _x p. $$

Summing up the previous observations and going back to (29) we infer that

$$ \partial _t P - \mathrm{div}_x(p \nabla _xP) = \mathrm{div}_x( \mathbf{F} ) + G \,, $$

with

$$ \mathbf{F}, \ G \quad \text{ bounded } \text{ in } \; L^q((0,\tau )\times \varOmega ) \; \text{ for } \text{ any } \text{ finite } \; 1< q < \infty ,\ \mathbf{F}\cdot \mathbf{n}|_{\partial \varOmega } = 0 \,. $$

Thus, applying the (weak) maximal regularity theory for parabolic equations (see Amann [1]), we conclude that

$$\begin{aligned} \partial _t p = P \quad \text{ is } \text{ bounded } \text{ in } \; L^q(0,\tau ; W^{1,q}(\varOmega )) \; \text{ for } \text{ any } \; 1< q < \infty \,. \end{aligned}$$

(30)

Note that this step requires higher regularity of the initial data (at $t=0$), specifically,

$$ \partial _t p(0,\,\cdot \,) = P(0,\,\cdot \,) \in B^{1-(2/q);q,q}(\varOmega ) \,, $$

see Amann [1, Theorem 2.1]. This kind of initial regularity hypothesis is not unusual for a parabolic problem.

Finally, embedding $W^{1,q}(\varOmega )$ into $L^\infty (\varOmega )$ for $q>3$, together with boundedness of the right hand side of (21), (see (25)) yields the desired conclusion

$$\begin{aligned} \mathrm{div}_x\nabla _xp = \Delta _x p\in L^q (0,\tau ; L^\infty (\varOmega )) \quad \text{ for } \text{ any } \; 1< q < \infty \,. \end{aligned}$$

(31)

3.2 Weak Sequential Stability

Our goal is to establish the following result:

Theorem 1

Let $\{p_\varepsilon \}_{\varepsilon > 0}$, $\{ Y^i_\varepsilon \}_{\varepsilon > 0}$; $i=1,\dots ,n$, be a family of (smooth) solutions of problem (21), (22) such that:

$$\begin{aligned} \begin{array}{c} p_\varepsilon \rightarrow p ,\ \nabla _xp_\varepsilon \rightarrow \nabla _xp \text{ in } C([0,\tau ] \times \varOmega ),\\ \\ \Delta _x p_\varepsilon \rightarrow \Delta _x p \text{ weakly-(*) } \text{ in } L^q (0,\tau ; L^\infty (\varOmega )),\ 1< q < \infty , \end{array} \end{aligned}$$

(32)

$$\begin{aligned} Y^i_\varepsilon \rightarrow Y^i \ \text{ weakly-(*) } \text{ in }\ L^\infty ((0,\tau ) \times \varOmega ), \end{aligned}$$

(33)

$$\begin{aligned} Y^i_\varepsilon (0, \cdot ) \rightarrow Y^i_0 \ \text{ in }\ L^1(\varOmega ). \end{aligned}$$

(34)

Then

$$\begin{aligned} Y^i_\varepsilon \rightarrow Y^i \text{ a.e. } \text{ in } (0,\tau )\times \varOmega \,, \end{aligned}$$

(35)

where p and $Y^1, \dots , Y^n$ satisfy (22), specifically,

$$\begin{aligned} \partial _t Y^i - \mathrm{div}_x(Y^i \nabla _xp) + Y^i \Delta _x p = \frac{1}{p} \omega _i (p, Y^1, \dots , Y^n) \sum _{j=1}^n \frac{m_i}{m_j} Y^j RT,\ i = 1,\dots , n.\qquad \end{aligned}$$

(36)

The rest of the paper is devoted to the proof of Theorem 1. We use the approach proposed in the seminal paper by DiPerna and Lions [7].

3.2.1 Existence for the Limit Problem

We show that the limit problem (36) admits a weak solution $Y^1, \dots , Y^n$ such that

$$ Y^i\ge 0 \, \text{ for } \text{ any } \, i = 1,\dots , n,\ \sum _{i=1}^n Y^i = 1 \,, $$

provided the initial data satisfy

$$ Y^i_0\ge 0 \,,\quad \sum _{i=1}^n Y^i_0 = 1 \,. $$

Step 1

We approximate the pressure p by a family of smooth functions $\{ p_\delta \}_{\delta > 0}$,

$$ p_\delta \rightarrow p,\ \nabla _xp_\delta \rightarrow \nabla _xp \ \text{ uniformly } \text{ in }\ [0,\tau ] \times \overline{\varOmega }, $$

$$ \Delta _x p_\delta \rightarrow \Delta _x p \; \text{ a.e. } \text{ in } \, (0,\tau )\times \varOmega \,,\quad \Vert \Delta _x p_\delta \Vert _{ L^q(0,\tau ; L^\infty (\varOmega )) } {\mathop {\sim }\limits ^{<}}1 \quad \text{ for } \text{ any } \, 1< q < \infty \,. $$

as $\delta \rightarrow 0$. Using the standard method of characteristics, we find a unique solution $Y^1_\delta , \dots , Y^n_\delta$ emanating from the initial data $Y^1_0, \dots , Y^n_0$.

Thanks to hypothesis (6),

$$ Y^i_\delta \ge 0 \quad \text{ for } \text{ all } \, i = 1, \dots ,n \,, $$

and, by virtue of (3),

$$ \sum _{i=1}^n Y^i_\delta = 1 \,. $$

Consequently, passing to a suitable subsequence if necessary, we may assume that

$$ Y^i_\delta \rightarrow Y^i \ \text{ weakly-(*) } \text{ in }\ L^\infty ((0,\tau ) \times \varOmega ) \cap C_\mathrm{weak}([0,\tau ]; L^1(\varOmega )) \ \text{ as }\ \delta \rightarrow 0, $$

where

$$\begin{aligned} \partial _t Y^i - \mathrm{div}_x(Y^i \nabla _xp) + Y^i \Delta _x p = \frac{1}{p}\, \overline{ \omega _i (p, Y^1, \dots , Y^n) \sum _{j=1}^n \frac{m_i}{m_j}\, Y^j } RT,\ i = 1,\dots ,n \,. \end{aligned}$$

(37)

$$\begin{aligned} Y^i(0, \cdot ) = Y^i_0 \,. \end{aligned}$$

(38)

Here and hereafter, the upper bar denotes a weak limit of compositions of smooth functions applied to weakly convergent sequences.

Step 2

In order to complete the proof, we have to show strong convergence

$$\begin{aligned} Y^i_\delta \rightarrow Y^i \ \text{ a.a. } \text{ in }\ (0,\tau ) \times \varOmega \ \text{ as } \ \delta \rightarrow 0. \end{aligned}$$

(39)

To this end, we write down a renormalized formulation of the $\delta -$problem in the form:

$$ \partial _t |Y_\delta |^2 - \mathrm{div}_x( |Y_\delta |^2 \nabla _xp_\delta ) + |Y_\delta |^2 \Delta _x p_\delta = \frac{2 RT}{p_\delta } \sum _{i,j=1}^n \frac{m_i}{m_j} \omega _i (p_\delta , Y^1_\delta , \dots , Y^n_\delta ) Y^i_\delta Y^j_\delta . $$

Letting $\delta \rightarrow 0$ we obtain

$$\begin{aligned} \partial _t \overline{|Y|^2} - \mathrm{div}_x( \overline{|Y|^2} \nabla _xp) + \overline{|Y|^2} \Delta _x p = \frac{2 RT}{p} \sum _{i,j=1}^n \frac{m_i}{m_j} \overline{\omega _i (p, Y^1, \dots , Y^n) Y^i Y^j}. \end{aligned}$$

(40)

Now, applying the regularization procedure of DiPerna and Lions [7] to (37) we deduce that

$$\begin{aligned} \partial _t {|Y|^2} - \mathrm{div}_x( {|Y|^2} \nabla _xp) + {|Y|^2} \Delta _x p = \frac{2 RT}{p} \sum _{i,j=1}^n \frac{m_i}{m_j} \overline{\omega _i (p, Y^1, \dots , Y^n) Y^j} Y^i. \end{aligned}$$

(41)

Step 3

Finally, we integrate the difference of (40), (41) over $\varOmega$:

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int _{\varOmega } \left( \overline{|Y|^2} - |Y|^2 \right) \ \mathrm{d} {x} = - \int _{\varOmega } \Delta _x p \left( \overline{|Y|^2} - |Y|^2 \right) \ \mathrm{d} {x} $$

$$ + \int _{\varOmega } \frac{2 RT}{p} \sum _{i,j=1}^n \frac{m_i}{m_j} \left[ \overline{\omega _i (p, Y^1, \dots , Y^n) Y^i Y^j} - \overline{\omega _i (p, Y^1, \dots , Y^n) Y^j} Y^i \right] \ \mathrm{d} {x}, $$

where

$$ \int _{\varOmega } \left[ \overline{\omega _i (p, Y^1, \dots , Y^n) Y^i Y^j} - \overline{\omega _i (p, Y^1, \dots , Y^n) Y^j} Y^i \right] \ \mathrm{d} {x} $$

$$ = \lim _{\delta \rightarrow 0} \int _{\varOmega } \left[ \omega _i (p_\delta , Y^1_\delta , \dots , Y^n_\delta ) Y^i_\delta - \omega _i (p_\delta , Y^1, \dots , Y^n) Y^i \right] (Y^j_\delta - Y^j ) \ \mathrm{d} {x} $$

$$ {\mathop {\sim }\limits ^{<}}\lim _{\delta \rightarrow 0} \int _{\varOmega } | Y_\delta - Y |^2 \ \mathrm{d} {x} = \int _{\varOmega } \left( \overline{|Y|^2} - |Y|^2 \right) \ \mathrm{d} {x}. $$

Thus, applying Gronwall’s lemma and using the fact that the initial values converge strongly, we conclude

$$ \overline{|Y|^2} = {|Y|^2} $$

yielding (39).

3.2.2 Compactness

Our ultimate goal is to show (35), (36). As $Y_\varepsilon$ are smooth, we may rewrite (36) as

$$\begin{aligned} \partial _t Y^i_\varepsilon - \nabla _xY^i_\varepsilon \cdot \nabla _xp_\varepsilon = \frac{RT}{p_\varepsilon } \sum _{j=1}^n \frac{m_i}{m_j} \omega _i (p_\varepsilon , Y^1_\varepsilon , \dots , Y^n_\varepsilon ) Y^j_\varepsilon ,\ i = 1,\dots , n. \end{aligned}$$

(42)

At this stage, we employ once more the regularization procedure of DiPerna, Lions [7] to Eq. (36):

$$\begin{aligned} \partial _t Y^i_r - \nabla _xY^i_r \nabla _xp = \frac{RT}{p} \sum _{j=1}^n \frac{m_i}{m_j} \omega _i (p, Y^1_r, \dots , Y^n_r) Y^j_r + e_r,\ i = 1,\dots , n, \end{aligned}$$

(43)

where

$$ e_r \rightarrow 0 \ \text{ in }\ L^1((0,\tau ) \times \varOmega ) \ \text{ as }\ r \rightarrow 0. $$

Similarly to the above, we subtract (42), (43), multiply the resulting expression by $Y^i_\varepsilon - Y^i_r$, and integrate over $\varOmega$ obtaining

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int _{\varOmega } |Y_\varepsilon - Y_r|^2 \ \mathrm{d} {x} + \int _{\varOmega } \Delta _x p_\varepsilon |Y_\varepsilon - Y_r|^2 \ \mathrm{d} {x} = \int _{\varOmega } \sum _{i = 1}^n (\nabla _xp_\varepsilon - \nabla _xp) \cdot \nabla _xY^i_r (Y^i_\varepsilon - Y^i_r ) \ \mathrm{d} {x} $$

$$ = \int _{\varOmega } \frac{RT}{p_\varepsilon } \sum _{i,j=1}^n \frac{m_i}{m_j} \left[ \omega _i (p_\varepsilon , Y^1_\varepsilon , \dots , Y^n_\varepsilon ) Y^j_\varepsilon -\omega _i (p_\varepsilon , Y^1_r, \dots , Y^n_r) Y^j_r \right] \ \mathrm{d} {x} + e_\varepsilon (r) + e_r, $$

where

$$ e_\varepsilon (r) \rightarrow 0 \ \text{ in }\ L^1((0,\tau ) \times \varOmega )\ \text{ as }\ \varepsilon \rightarrow 0 \ \text{ for } \text{ any } \text{ fixed }\ r . $$

Finally, letting first $\varepsilon \rightarrow 0$, then $r\rightarrow 0$, and realizing that

$$ Y^i_r \rightarrow Y^i \ \text{ in }\ C([0,\tau ]; L^2 (\varOmega )), $$

we get the desired conclusion (35), (36).