Abstract
We propose a new concept of weak solution to the equations of compressible magnetohydrodynamics driven by ihomogeneous boundary data. The system of the underlying field equations is solvable globally in time in the out of equilibrium regime characteristic for turbulence. The weak solutions comply with the weak–strong uniqueness principle; they coincide with the classical solution of the problem as long as the latter exists. The choice of constitutive relations is motivated by applications in stellar magnetoconvection.
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1 Introduction
The large scale dynamics arising in stellar magnetoconvection is driven in an essential way by the boundary conditions, see Gough [12]. The flux separation and pattern formation arise in the turbulent regime when the fluid flow is far from equilibrium. As the underlying field equations are non-linear, the existence of global in time smooth solutions in such a regime is not known. Indeed the vast majority of global existence results in the class of classical solutions is restricted to the initial state close to a stable equilibrium, see e.g. Matsumura and Nishida [17], Valli [24], among others. Moreover, the recent results of Buckmaster et al. [4], and Merle et al. [18, 19] indicate that classical solutions may develop singularities in a finite time.
In view of the above arguments, the concept of weak solution represents a suitable alternative to restore global-in-time existence even for problems with large data and solutions remaining out of equilibrium in the long run. We consider a mathematical model of fully compressible three dimensional fluid convection driven by an externally imposed magnetic field. The fluid occupies a bounded domain \(\Omega \subset R^3\) with regular boundary. The time evolution of the density \(\varrho = \varrho (t,x)\), the (absolute) temperature \(\vartheta = \vartheta (t,x)\), the velocity field \(\textbf{u}= \textbf{u}(t,x)\), and the magnetic field \(\textbf{B}= \textbf{B}(t,x)\) is governed by the compressible MHD system.
Equation of continuity
Momentum equation
Induction equation
Internal energy balance
The pressure \(p = p(\varrho , \vartheta )\) and the internal energy \(e=e(\varrho , \vartheta )\) are interrelated through Gibbs’ equation
where \(s = s(\varrho , \vartheta )\) is the entropy. Consequently, the internal energy balance (1.4) may be reformulated in the form of entropy balance equation
see e.g. Gough [12], Weiss and Proctor [26], Tao et al. [22].
The right-hand side of equation (1.4) represents the entropy production rate, which, in accordance with the Second law of thermodynamics, must be non-negative. Accordingly, we consider Newtonian fluid, with the viscous stress tensor
where the viscosity coefficients \(\mu > 0\) and \(\eta \ge 0\) are continuously differentiable functions of the temperature. Similarly, the heat flux obeys Fourier’s law,
where the heat conductivity coefficient \(\kappa > 0\) is a continuously differentiable function of the temperature.
The existence of global-in-time weak solutions for the compressible MHD system (1.1)–(1.8) was shown in [7] under certain physically relevant restrictions imposed on constitutive equations and transport coefficients. The boundary conditions considered in [7] are conservative,
characteristic for closed systems. If the driving force \(\textbf{g}\) in the momentum equation is conservative, meaning \(\textbf{g} = \nabla _xG\), \(G = G(x)\), the total energy of the system is conserved and the dynamics obeys the rather “boring” scenario formulated by the celebrated statement of Clausius:
The energy of the world is constant; its entropy tends to a maximum.
A rigorous mathematical proof of this statement for the Navier–Stokes–Fourier system was given in [11].
As pointed out by Gough [12], a rich fluid behaviour is conditioned by a proper choice of boundary conditions. Motivated by models in astrophysics, we suppose the boundary \(\partial \Omega \) is impermeable and the tangential component of the normal viscous stress vanishes on it,
However, the theory presented in this paper can accommodate more general boundary conditions for the velocity field discussed in Sect. 6.
Similarly to the well known Rayleigh–Bénard problem, see e.g. Davidson [6], we impose the inhomogeneous Dirichlet boundary conditions for the temperature,
To incorporate the effect of an exterior magnetic field, we suppose that \(\textbf{B}\) is (not necessarily small) perturbation of a background magnetic field \(\textbf{B}_B\), \(\textrm{div}_x\textbf{B}_B= 0\),
In the context of astrophysics, we may think of \(\textbf{B}_B\) as being the magnetic field imposed by a massive star. As for \(\textbf{b}\), we suppose either
or
The boundary condition (1.12) is of Dirichlet type, compatible with the requirement of solenoidality of \(\textbf{B}\). The condition (1.13) is of flux type, similar to the first condition (1.10) and must be accompanied by another flux condition related to the electric field, namely
The flux in (1.14) may be inhomogeneous as well. The borderline case \(\textbf{B}_B = 0\) in (1.12) and (1.13) corresponds to a perfectly isolating and perfectly conducting boundary, respectively (see e.g. Alekseev [1]). There are other types of boundary conditions discussed in the literature, see e.g. Ladyzhenskaya and Solonnikov [14], Lassner [15], Ströhmer [21], among others.
Due to possibly only technical but so far unsurmountable difficulties, a mathematically tractable weak formulation of compresible MHD system cannot be based on merely rewriting the system (1.1)–(1.4) in the sense of distributions. The available a priori bounds are not strong enough to render certain terms, notably \(p(\varrho , \vartheta ) \textrm{div}_x\textbf{u}\) in the internal energy balance (1.4), integrable. A remedy proposed in [7] is replacing (1.4) by the entropy equation (1.6). Unfortunately, the entropy production rate is a priori bounded only in the non-reflexive space \(L^1\) of integrable functions. As a result, any approximation scheme provides merely an inequality
satisfied in the sense of distributions.
Replacing entropy equation by inequality definitely enlarges the class of possible solutions and gives rise to an underdetermined problem. To save well posedness, at least formally, the integrated total energy balance/inequality is appended to the system (1.1)–(1.3), (1.15) in [7]. The total energy balance reads
If the system is conservative, the total energy flux vanishes on the boundary and (1.16) integrated over \(\Omega \) yields
It turns out that the equations (1.1)–(1.3), the inequality (1.15), together with the integral identity (1.17) represent a suitable weak formulation for the conservative system. Indeed the existence of global-in-time weak solutions for any finite energy (initial) data was proved in [7]. The weak solutions comply with a natural compatibility principle. Any sufficiently smooth weak solution is a classical solution of the problem. In addition, the weak solutions satisfy the weak–strong uniqueness principle: A weak solution coincides with the strong solution on the life span of the latter, see [8].
The situation becomes more delicate for open (non-conservative) systems, with inhomogeneous Dirichlet boundary conditions. The heat flux \(\textbf{q} \cdot \textbf{n}\) as well as the induction flux (the normal component of the electric field) \([ \textbf{B}\times \textbf{u}\times \textbf{B}+ \zeta \textbf{curl}_x\textbf{B}\times \textbf{B}] \cdot \textbf{n}\) do not necessarily vanish on \(\partial \Omega \) and give rise to additional uncontrollable sources of energy in (1.17). In [5] (see also the monograph [10]), a new approach has been developed to handle the inhomogeneous Dirichlet boundary conditions for the temperature. The total energy balance (1.17) is replaced by a similar inequality for the ballistic energy
where \({\tilde{\vartheta }}\) is an arbitrary extension of the boundary temperature \(\vartheta _B\) inside \(\Omega \). Pursuing the same strategy, we introduce a “magnetic” variant of the ballistic energy
After a straightforward manipulation, we deduce equality
as soon as \({\tilde{\vartheta }}|_{\partial \Omega } = \vartheta _B\), \({\tilde{\vartheta }}> 0\), \(\textbf{B}\times \textbf{n}|_{\partial \Omega } = \textbf{B}_B\times \textbf{n}|_{\partial \Omega }\). The same is true if \(\textbf{B}\cdot \textbf{n}|_{\partial \Omega } = \textbf{B}_B\cdot \textbf{n}|_{\partial \Omega }\) and the vanishing tangential flux condition (1.14) holds. Note carefully that (1.19) does not contain the boundary flux integrals; whence it is suitable for a weak formulation of the problem.
The mathematical theory we propose is based on imposing the weak form of the equations (1.1)–(1.3), together with the entropy inequality (1.15), and the ballistic energy balance (1.19) as a weak formulation of the compressible MHD system. The basic hypotheses concerning the state equation and the transport coefficients as well as the exact definition of weak solution are given in Sect. 2. In Sect. 3 we introduce the basic tool to investigate stability properties in the class of weak solutions—the relative energy inequality. In Sect. 4, we show that the weak solutions enjoy the weak–strong uniqueness property. In Sect. 5, the existence of global-in-time weak solutions is established by means of a multilevel approximation scheme. The paper is concluded in Sect. 6 by a brief discussion on possible extensions of the theory to a larger class of boundary conditions.
2 Weak Formulation
We start by a list of structural hypotheses imposed on the equation of state and the transport coefficients.
2.1 Equation of State
Our choice of the equation of state is motivated by [10, Chapter 4]. In particular, we consider the radiation contribution to the pressure/internal energy relevant to problems in astrophysics, cf. Battaner [3]. In accordance with Gibbs’ relation (1.5) we suppose that
where the function \(P \in C^1[0,\infty )\) satisfies
This implies, in particular, that \(Z \mapsto P(Z)/ Z^{\frac{5}{3}}\) is decreasing, and we suppose
Note that \(p_M\), \(e_M\) satisfy the relation characteristic for monoatomic gas,
\(p_R\) is the radiation pressure with the associated internal energy \(e_R\), and (2.3) corresponds to the presence of electron pressure under the degenerate gas regime. It follows from (2.2) that p and e satisfy the hypothesis of thermodynamic stability:
In accordance with (1.5), the entropy takes the form
where
2.2 Transport Coefficients
We suppose the viscosity coefficients in the Newtonian stress \(\mathbb {S}(\vartheta , \nabla _x\textbf{u})\) are continuously differentiable functions of the temperature satisfying
Note that \(\alpha = \frac{1}{2}\) corresponds to Sutherland’s law relevant in astrophysics, see Yang et al. [27]. We also allow \(\underline{\eta } = 0\).
Similarly, the heat conductivity coefficient in the Fourier heat flux \(\textbf{q}(\vartheta , \nabla _x\vartheta )\) is a continuously differentiable function of the temperature satisfying
Here, the case \(\beta = 3\) reflects the effect of radiation.
Finally, we suppose the magnetic diffusivity coefficient \(\zeta = \zeta (\vartheta )\) is a continuously differentiable function of the temperature,
2.3 Boundary Data
We suppose the background magnetic field \(\textbf{B}_B= \textbf{B}_B(t,x)\) is sufficiently smooth at least of class \(C^{1,2}([0,T] \times \overline{\Omega }; R^3)\),
Similarly, we suppose the boundary temperature \(\vartheta _B\) can be extended inside \(\Omega \), \(\vartheta _B\in C^{1,2}([0,T] \times \overline{\Omega } )\),
2.4 Weak Solutions
Before discussing the concept of weak solution, it is useful to introduce the function spaces
Both \(H_{0,\tau }\) and \(H_{0,n}\) are endowed with the Hilbert norm
see e.g. Alexander and Auchmuty [2].
Definition 2.1
(Weak solution to the compressible MHD system) A quantity \((\varrho , \vartheta , \textbf{u}, \textbf{B})\) is termed weak solution of the compressible MHD system (1.1)–(1.4), with the boundary conditions (1.10), (1.11), (1.12) (respectively (1.13)), and the initial data
if the following holds:
-
Equation of continuity. \(\varrho \in C_\textrm{weak}([0,T]; L^{\frac{5}{3}} (\Omega ))\), \(\varrho \ge 0\) in \((0,T) \times \Omega \), and the integral identity
$$\begin{aligned} \int _0^\tau \int _{\Omega } \Big ( \varrho \partial _t \varphi + \varrho \textbf{u}\cdot \nabla _x\varphi \Big ) \ \,\textrm{d} {x} \ dt = - \left[ \int _{\Omega } \varrho \varphi \ \,\textrm{d} {x} \right] _{t=0}^{t = \tau },\ \varrho (0, \cdot ) = \varrho _0, \end{aligned}$$(2.13)holds for any \(\varphi \in C^1([0,T] \times \overline{\Omega })\). In addition, the renormalized version of (2.13)(2.13)
$$\begin{aligned} \int _0^\tau \int _{\Omega } \left( b(\varrho ) \partial _t \varphi + b(\varrho ) \textbf{u}\cdot \nabla _x\varphi + \Big ( b(\varrho ) - b'(\varrho ) \varrho \Big ) \textrm{div}_x\textbf{u}\right) \ \,\textrm{d} {x} \,\textrm{d} t = \left[ \int _{\Omega } b(\varrho ) \varphi \ \,\textrm{d} {x} \right] _{t = 0}^{t= \tau } \end{aligned}$$(2.14)holds for any \(\varphi \in C^1([0,T] \times \overline{\Omega })\) and any \(b \in C^1(R)\), \(b' \in C_c(R)\).
-
Momentum equation. \(\varrho \textbf{u}\in C_\textrm{weak}([0,T]; L^{\frac{5}{4}}(\Omega ; R^3)\), \(\textbf{u}\in L^2(0,T; W^{1,q}(\Omega ; R^3))\), \(q = \frac{8}{5 - \alpha }\), \(\textbf{u}\cdot \textbf{n}|_{\partial \Omega } = 0\), and the integral identity
$$\begin{aligned}&\int _0^\tau \int _{\Omega } \Big ( \varrho \textbf{u}\cdot \partial _t \varvec{\varphi }+ \varrho \textbf{u}\otimes \textbf{u}: \nabla _x\varvec{\varphi }+ p(\varrho , \vartheta ) \textrm{div}_x\varvec{\varphi }\Big ) \ \,\textrm{d} {x} \nonumber \\&\quad = \int _0^\tau \int _{\Omega } \mathbb {S}(\vartheta , \nabla _x\textbf{u}) : \nabla _x\varvec{\varphi } \ \,\textrm{d} {x} \,\textrm{d} t - \int _0^\tau \int _{\Omega } \left( \textbf{B}\otimes \textbf{B}- \frac{1}{2} |\textbf{B}|^2 \mathbb {I} \right) : \nabla _x\varvec{\varphi } \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad - \int _0^\tau \int _{\Omega } \varrho \textbf{g} \cdot \varvec{\varphi } \ \,\textrm{d} {x} \,\textrm{d} t + \left[ \int _{\Omega } \varrho \textbf{u}\cdot \varvec{\varphi } \ \,\textrm{d} {x} \right] _{t = 0}^{t = \tau },\ (\varrho \textbf{u})(0, \cdot ) = \varrho _0 \textbf{u}_0 \end{aligned}$$(2.15)holds for any \(\varvec{\varphi }\in C^1([0,T] \times \overline{\Omega }; R^3)\), \(\varvec{\varphi }\cdot \textbf{n}|_{\partial \Omega } = 0\).
-
Induction equation. \(\textbf{B}\in C_\textrm{weak}([0,T]; L^2(\Omega ; R^3))\),
$$\begin{aligned} \textrm{div}_x\textbf{B}(\tau , \cdot ) = 0 \ \text{ for } \text{ any }\ \tau \in [0,T]. \end{aligned}$$(2.16)In addition
$$\begin{aligned} (\textbf{B}- \textbf{B}_B) \in L^2(0, T; H_{0, \tau }(\Omega ; R^3)) \end{aligned}$$(2.17)if the boundary condition (1.12) is imposed or
$$\begin{aligned} (\textbf{B}- \textbf{B}_B) \in L^2(0, T; H_{0, n}(\Omega ; R^3)) \end{aligned}$$(2.18)for the boundary conditions (1.13), (1.14). The integral identity
$$\begin{aligned}&\int _0^\tau \int _{\Omega } \Big ( \textbf{B}\cdot \partial _t \varvec{\varphi }- (\textbf{B}\times \textbf{u}) \cdot \textbf{curl}_x\varvec{\varphi }- \zeta (\vartheta ) \textbf{curl}_x\ \textbf{B}\cdot \textbf{curl}_x\ \varvec{\varphi }\Big ) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad = \left[ \int _{\Omega } \textbf{B}\cdot \varvec{\varphi } \ \,\textrm{d} {x} \right] _{t=0}^{t = \tau },\ \textbf{B}(0, \cdot ) = \textbf{B}_0, \end{aligned}$$(2.19)holds for any \(\varvec{\varphi }\in C^1([0,T] \times \overline{\Omega }; R^3)\), \(\textrm{div}_x\varvec{\varphi }= 0\),
$$\begin{aligned} \varvec{\varphi }\times \textbf{n}|_{\partial \Omega } = 0 \end{aligned}$$(2.20)in the case of boundary conditions (1.12),
$$\begin{aligned} \varvec{\varphi }\cdot \textbf{n}|_{\partial \Omega } = 0 \end{aligned}$$(2.21) -
Entropy inequality. \(\vartheta \in L^\infty (0,T; L^4(\Omega ))\), \(\vartheta > 0\) a.a. in \((0,T) \times \Omega \), \(\vartheta - \vartheta _B\in L^2(0,T; W^{1,2}_0 (\Omega ))\), \(\log (\vartheta ) \in L^2(0,T; W^{1,2}(\Omega ))\), and the integral inequality
$$\begin{aligned}&\int _{\tau _1}^{\tau _2} \int _{\Omega } \left( \varrho s (\varrho , \vartheta ) \partial _t \varphi + \varrho s (\varrho , \vartheta ) \textbf{u}\cdot \nabla _x\varphi + \frac{\textbf{q} (\vartheta , \nabla _x\vartheta ) }{\vartheta } \cdot \nabla _x\varphi \right) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le - \int _{\tau _1}^{\tau _2} \int _{\Omega } \frac{1}{\vartheta } \left( \mathbb {S}(\vartheta , \nabla _x\textbf{u}) : \nabla _x\textbf{u}- \frac{\textbf{q} (\vartheta , \nabla _x\vartheta ) \cdot \nabla _x\vartheta }{\vartheta } + \zeta (\vartheta ) |\textbf{curl}_x\textbf{B}|^2 \right) \varphi \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\qquad + \left[ \int _{\Omega } \varrho s(\varrho , \vartheta ) \varphi \ \,\textrm{d} {x} \right] _{t = \tau _1-}^{t = \tau _2+},\ 0 \le \tau _1 \le \tau _2 \le T,\ \varrho s(\varrho , \vartheta )(0-, \cdot ) = \varrho _0 s(\varrho _0, \vartheta _0), \end{aligned}$$(2.22)holds for any \(\varphi \in C^1_c([0,T] \times \Omega )\), \(\varphi \ge 0\).
-
Ballistic energy inequality. The inequality
$$\begin{aligned}&\int _{\Omega } \left( \frac{1}{2} \varrho |\textbf{u}|^2 + \varrho e(\varrho , \vartheta ) + \frac{1}{2} |\textbf{B}|^2 - {\tilde{\vartheta }}\varrho s (\varrho , \vartheta ) - \textbf{B}_B\cdot \textbf{B}\right) (\tau , \cdot ) \ \,\textrm{d} {x}\nonumber \\&\qquad + \int _0^\tau \int _{\Omega } \frac{{\tilde{\vartheta }}}{\vartheta } \left( \mathbb {S}(\vartheta , \nabla _x\textbf{u}) : \nabla _x\textbf{u}- \frac{\textbf{q} (\vartheta , \nabla _x\vartheta ) \cdot \nabla _x\vartheta }{\vartheta } + \zeta (\vartheta ) | \textbf{curl}_x\textbf{B}|^2 \right) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \le - \int _0^\tau \int _{\Omega } \left( \varrho s (\varrho , \vartheta ) \partial _t {\tilde{\vartheta }}+ \varrho s (\varrho , \vartheta ) \textbf{u}\cdot \nabla _x{\tilde{\vartheta }}+ \frac{\textbf{q}(\vartheta , \nabla _x\vartheta )}{ \vartheta } \cdot \nabla _x{\tilde{\vartheta }}\right) \ \,\textrm{d} {x} \nonumber \\&\quad \quad - \int _0^\tau \int _{\Omega } \Big ( \textbf{B}\cdot \partial _t \textbf{B}_B- (\textbf{B}\times \textbf{u}) \cdot \textbf{curl}_x\textbf{B}_B- \zeta (\vartheta ) \textbf{curl}_x\ \textbf{B}\cdot \textbf{curl}_x\textbf{B}_B\Big ) \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \quad + \int _0^\tau \int _{\Omega } \varrho \textbf{g} \cdot \textbf{u} \ \,\textrm{d} {x} \,\textrm{d} t \nonumber \\&\quad \quad + \int _{\Omega } \left( \frac{1}{2} \varrho _0 |\textbf{u}_0|^2 + \varrho _0 e(\varrho _0, \vartheta _0) + \frac{1}{2} |\textbf{B}_0 |^2 - {\tilde{\vartheta }}(0, \cdot ) \varrho _0 s (\varrho _0, \vartheta _0) - \textbf{B}_B(0, \cdot ) \cdot \textbf{B}_0 \right) \ \,\textrm{d} {x} \end{aligned}$$(2.23)holds for any \({\tilde{\vartheta }}\in C^1([0,T] \times \overline{\Omega })\), \({\tilde{\vartheta }}> 0\), \({\tilde{\vartheta }}|_{\partial \Omega } = \vartheta _B\).
Remark 2.2
As the magnetic field \(\textbf{B}\) satisfies (2.16), validity of (2.19) can be extended to any (not necessarily solenoidal) test function \(\varvec{\varphi }\) satisfying (2.20), (2.21), respectively. Indeed, if \(\varvec{\varphi }\) satisfies the boundary condition (2.20), we consider the potential \(\Psi \) solving
and write
On the one hand, as \(\Psi \) vanishes on \(\partial \Omega \) and \(\textbf{B}\) is solenoidal, we have
whence the integral identity (2.19) holds for \(\nabla _x\Psi \). On the other hand, the function \((\varvec{\varphi }- \nabla _x\Psi )\) is solenoidal, and, as \(\nabla _x\Psi \times \textbf{n}|_{\partial \Omega } = 0\), a legal test function in (2.19).
If \(\varvec{\varphi }\) satisfies (2.21), we simply consider
- the standard potential in the Helmholtz decomposition.
Remark 2.3
Unlike \(\varrho \), \(\varrho \textbf{u}\), and \(\textbf{B}\), the total entropy \(\varrho s(\varrho , \vartheta )\) is not weakly continuous in time. However, it can be deduced from the entropy inequality that the one sided limits
exist for any \(\phi \in C(\overline{\Omega })\), and
whenever \(\phi \ge 0\), see [10, Chapter 3, Section 3.3.3] for details.
Remark 2.4
As we shall see below, the “magnetic” ballistic energy \(E_{BM}\) introduced in (1.18) is in fact a strictly convex l.s.c function of the variables \((\varrho , S = \varrho s(\varrho , \vartheta ), \textbf{m}= \varrho \textbf{u}, \textbf{B})\). In particular, its integral is a weakly lower semi-continuous function; whence (2.23) holds for any time \(\tau \) with the convention for entropy discussed Remark 2.3. Alternatively, we may impose a stronger integrated version of (2.23) in the form
for any \(\psi \in C^1_c[0,T)\), \(\psi \ge 0\).
Remark 2.5
Note carefully that the ballistic energy inequality (2.23) remains valid if we replace \(\textbf{B}_B\) by any other extension \(\widetilde{\textbf{B}}\) such that
and
respectively. Indeed the difference \(\textbf{B}_B- \widetilde{\textbf{B}}\) becomes an eligible test function for for the weak form of the induction equation (2.19), endowed with the boundary conditions (2.20), (2.21), respectively.
The existence of global-in-time weak solutions in the sense of Definition 2.1 will be shown in Sect. 5 below.
3 Relative Energy
A suitable form of the relative energy for the compressible MHD system reads
In applications, the quantity \((\varrho , \vartheta , \textbf{u}, \textbf{B})\) stands for a weak solution of the compressible MHD system while \((r, \Theta , \textbf{U}, \textbf{H})\) are arbitrary sufficiently smooth functions satisfying the compatibility conditions
As a consequence of hypothesis of thermodynamic stability stated in (2.4), the energy
is a strictly convex l.s.c. function if expressed in the variables \((\varrho , S = \varrho s(\varrho , \vartheta ), \textbf{m}= \varrho \textbf{u}, \textbf{B})\), see [10, Chapter 3, Section 3.1] for details. Moreover, we have
where \(\left( e(\varrho , \vartheta ) - \vartheta s(\varrho , \vartheta ) + \frac{p(\varrho , \vartheta ) }{\varrho } \right) \) is Gibbs’ free energy. Consequently, the relative energy can be seen as Bregman divergence associate to the convex function E. In particular,
as long as \(r > 0\).
3.1 Relative Energy Inequality
In the remaining part of this section, we derive a relative energy inequality provided \((\varrho , \vartheta , \textbf{u}, \textbf{B})\) is a weak solution of the compressible MHD system specified in Definition 2.1 and \((r, \Theta , \textbf{U}, \textbf{H})\) arbitrary smooth functions satisfying (3.2). Our starting point is the ballistic energy inequality (2.24). As noted in Remark 2.5, we may flip \(\textbf{B}_B\) for \(\textbf{H}\), and, of course, \({\tilde{\vartheta }}\) for \(\Theta \) obtaining
3.1.1 Momentum Perturbation
Our first goal is to replace \(|\textbf{u}|^2\) by \(|\textbf{u}- \textbf{U}|^2\). This can be achieved by considering \(\textbf{U}\) as a test function in the momentum balance equation (2.15). After a straightforward manipulation explained in detail in [10, Chapter 3, Section 3.2.1], we obtain
for any \((\Theta , \textbf{U}, \textbf{H})\) as in (3.2).
3.1.2 Density Perturbation
Next, exactly as in [10, Chapter 3, Section 3.4], we use the quantity
as a test function in the equation of continuity (2.13):
whenever \(r \in C^1([0,T] \times \overline{\Omega })\), \(r > 0\).
3.1.3 Magnetic Field Perturbation and the Final form of Relative Energy Inequality
Since
we may rewrite (3.5) in the final form
for any “test” functions \((r, \Theta , \textbf{U}, \textbf{H})\) specified in (3.2).
Remark 3.1
In view of future applications, in particular the weak–strong uniqueness principle discussed in the forthcoming section, the class of functions \((r, \Theta , \textbf{U}, \textbf{H})\) in (3.2) can be extended to the “maximal regularity” framework
by a density argument. The exponent q must be taken large enough for all terms in (3.7) to be well defined.
4 Weak–Strong Uniqueness Principle
The first important property of the weak solutions introduced in Definition 2.1 is the weak–strong uniqueness principle. We suppose that \((r, \Theta , \textbf{U}, \textbf{H})\) is a smooth solution of the problem, specifically,
together with the relevant initial and boundary conditions.
Theorem 4.1
(Weak–strong uniqueness) Let \(\Omega \subset R^3\) be a bounded Lipschitz domain. Suppose the thermodynamic functions p, e, s satisfy the hypotheses (2.1)–(2.6), and the transport coefficients \(\mu \), \(\eta \), \(\kappa \), \(\zeta \) comply with (2.7)–(2.9), with
Let \((\varrho , \vartheta , \textbf{u}, \textbf{B})\) be a weak solution of the compressible MHD system (1.1)–(1.4) in \((0,T) \times \Omega \) specified in Definition 2.1. Suppose that the same problem, with the same initial and boundary data, admits a classical solution \((r, \Theta , \textbf{U}, \textbf{H})\) in the class
Then
Remark 4.2
As we assume existence of a classical solution in Theorem 4.1, the initial data are also regular, in particular,
Remark 4.3
Remarkably, the result holds in the full range of exponents \(\alpha \), \(\beta \) specified in (4.2). In particular, we remove the gap between the weak–strong uniqueness principle stated in [5, 10] for \(\beta = 3\) and the existence result requiring \(\beta > 6\). The main novelty is formulated in Lemma 4.4 below.
The rest of this section is devoted to the proof of Theorem 4.1. Note that existence of local in time strong solutions to the compressible MHD system can be established by the nowadays well understood technique proposed by Valli [24], Valli and Zajaczkowski [25], or by a more recent approach via maximal regularity in the spirit of Kotschote [13]. As noted in Remark 3.1, the conclusion of Theorem 4.1 remains valid for the strong solutions in the maximal regularity class used in [13].
The idea of the proof of Theorem 4.1 is simple: We consider the strong solution \((r, \Theta , \textbf{U}, \textbf{H})\) as test functions in the relative energy inequality (3.7) and use Gronwall’s type argument. We proceed in several steps specified below.
4.1 Magnetic Field
As \(\textbf{U}\) solves the momentum equation, we get
Similarly, \(\textbf{H}\) being the exact solution of the magnetic field equation,
Regrouping the terms in (3.7), (4.4), (4.5) that do not contain magnetic dissipation, we get the expression
Now, we claim the identity
Indeed the integrals on the right-hand side can be treated as follows:
Summing up the previous discussion, we may rewrite inequality (3.7) in the form
In addition, we have
We claim
Indeed either \(\textbf{B}\), \(\textbf{H}\) satisfy the boundary condition (1.12) and then \((\textbf{B}- \textbf{H}) \times \textbf{n} = 0\) or \(\textbf{H}\) satisfies the flux condition (1.14). In both cases the surface integral in (4.9) vanishes.
Thus relation (4.7) takes the form
4.2 Pressure and Entropy
After a simple manipulation, we can rewrite (4.10) in the form
with a quadratic remainder
Using the identity
we compute, exactly as in [5, Section 3.3],
Consequently, inequality (4.11) can be written as
with a remainder
4.3 Diffusive Terms
First, as \(\vartheta \), \(\Theta \) coincide on \(\partial \Omega \), we can integrate by parts:
Putting all diffusive terms on the left-hand side of the relative energy inequality (4.13), we get
Similarly
Finally,
Thus (4.13) can be rewritten in the final form
with \(\mathcal {R}_2\) given in (4.14).
4.4 Refined Estimates
Similarly to [10, Chapter 4, Section 4.2.1], we introduce the “essential” and “residual” component of a measurable function. We set
The remaining part of the proof of weak–strong uniqueness principle follows essentially the arguments of [10, Chapter 4, Section 4], with the necessary modifications to accommodate the terms containing the magnetic field as well as the complete slip/Dirichlet boundary conditions. First, we recall the coercivity property of the relative energy:
where the constant depend on \(\inf r\), \(\sup r\), \(\inf \Theta \), \(\sup \Theta \).
4.4.1 Temperature Gradient
The following crucial estimate was proved in [10, Chapter 4, Section 4.2.2] for \(\beta = 3\). Here, we extend its validity to general \(\beta \ge 3\).
Remark 4.4
Under the hypotheses of Theorem 4.1, there is a constant \(c > 0\),
where \(\xi > 0\) depends on \(\Theta \), \(\inf \Theta \), and \(\Vert \vartheta \Vert _{L^4(\Omega )}\).
Remark 4.5
As we shall see below, the result holds whenever \(\kappa (\vartheta ) \approx 1 + \vartheta ^\beta \), \(\beta \ge 2\).
Proof
We consider first the essential part. Rewrite
Applying Hölder’s inequality we deduce from (4.21)
where \(\xi \) depends on \(\Theta \). As a matter of fact, the only hypotheses to be imposed on \(\kappa \) is that \(\kappa (\vartheta ) > 0\) whenever \(\vartheta > 0\).
As for the residual part, we first rewrite
Obviously,
Consequently, as \(\inf \Theta > 0\), it remains to handle a quantity
where C is a positive constant depending on \(\Theta , \inf \Theta \). Note that this step requires \(\vartheta ^2 {\mathop {\sim }\limits ^{<}}\kappa (\vartheta )\).
To proceed we consider \(\overline{\vartheta } > 0\) to be fixed below. We write
Obviously, there is \(\xi = \xi (\overline{\vartheta }, \Theta )\) such that
Thus it remains to handle the integral
As \(\kappa (\vartheta ) \approx (1 + \vartheta ^\beta )\), our task reduces to
First observe
where
Finally, it remains to handle the integral
On the one hand, using Sobolev–Poincaré inequality we get
On the other hand, by Hölder’s inequality,
Next, by Jensen’s inequality,
whence
Our final observation is that as \(\Vert \vartheta \Vert _{L^4(\Omega )}\) is bounded, for any \(\delta > 0\), there is \(\overline{\vartheta }(\delta )\) such that
Combining (4.25), (4.26) we may infer
Putting together (4.22), (4.24), and (4.27) we obtain (4.20). \(\square \)
4.4.2 Viscous Stress and Magnetic Diffusion
Exactly as in [10, Chapter 4, Section 4.2.3], we estimate the viscous stress dissipation:
where
Next, repeating the same arguments, we get
Finally, we recall the generalized Korn–Poincaré inequality [9, Chapter 11, Theorem 11.23]:
whenever
The previous estimates, together with (4.32), allow us to rewrite the inequality (4.18) in the form
where the remainder \(\mathcal {R}_2\) is specified in (4.14).
4.5 Completing the Proof of Weak–Strong Uniqueness
The rest of the proof of weak–strong uniqueness consists in absorbing all integrals appearing in the remainder \(\mathcal {R}_2\) by the left-hand side of (4.33) and applying Gronwall’s argument. We omit the details as the whole procedure is described in [10, Chapter 4].
We have proved Theorem 4.1.
Remark 4.6
The Lipschitz regularity of the domain \(\Omega \) required in Theorem 4.1 may not be sufficient for the existence of the strong solution to the MHD system.
5 Existence of Weak Solutions
The proof of existence of weak solutions can be carried out by means of the approximate scheme, a priori estimates, and compactness arguments specified in Chapter 5 and Chapter 12 of the monograph [10].
The magnetic field equations can be incorporated exactly as in [7] without any additional difficulties. Similarly to [10, Chapter 12], two additional assumptions must be imposed on the constitutive relations, specifically,
where \(\mathcal {S}\) is the function defining the entropy \(s_M\). Moreover, we require
where \(\beta \) is the exponent in (2.8). While (5.1) corresponds to the Third Law of Thermodynamics, hypothesis (5.2) is purely technical to ensure the necessary a priori bounds.
5.1 Approximation Scheme
The existence of weak solutions can be shown by solving the following multi-level approximation scheme. The equation of continuity (1.1) is replaced by its parabolic regularization:
supplemented by the Neumann boundary conditions
and the initial condition
where \(\varrho _{0, \delta } > 0\) is a smooth regularization of the initial density \(\varrho _0\). Here and hereafter, \(\varepsilon \) and \(\delta \) are small parameters to be sent to zero successively in the limit passage.
The momentum equation (1.2) is replaced by a Galerkin approximation
for any test function \(\varvec{\varphi }\in C^1([0,T]; X_N)\), where \(X_N\) is a finite dimensional space spanned by the eigenvectors of the Lamé operator with the complete slip boundary conditions (1.10). Here,
Similarly, the induction equation is replaced by a Galerkin approximation. We look for solutions in the form
where
for any \(\varvec{\varphi }\in C^1([0,T]; Y_N)\). The finite-dimensional space \(Y_N\) is chosen in accordance with the boundary conditions. The functions in \(Y_N\) are smooth, solenoidal,
and \(\varvec{\varphi }\times \textbf{n}|_{\partial \Omega } = 0\), \(\cup _{N > 0} Y_N\) is dense in \(H_{0, \tau }\) in the case of boundary condition (1.12), and \(\varvec{\varphi }\cdot \textbf{n}|_{\partial \Omega } = 0\), \(\cup _{N > 0} Y_N\) is dense in \(H_{0, n}\) in the case of boundary conditions (1.13), (1.14).
Finally, the internal energy balance is regularized, adapted to the viscous approximation of the equation of continuity, and solved exactly,
with the regularized initial/boundary conditions
Here, we have denoted
5.2 Existence of Weak Solutions
The existence of weak solutions to the Navier–Stokes–Fourier system (without magnetic field) follows by solving the approximate scheme (5.3)–(5.7), (5.9)–(5.11). The proof in the absence of magnetic field and for the Dirichlet boundary conditions for the velocity field is given with full details in [10, Part II, Chapter 5]. The crucial observation made in [7] is that the basic energy estimate consists in considering the scalar product of (5.6) with \(\textbf{u}\) and (5.8) with \(\textbf{b}\)—a step compatible with the Galerkin approximation. The proof can be reproduced without any essential difficulties in the present setting as soon as the complete slip boundary conditions (1.10) are accommodated.
The main problem associated to the complete slip boundary conditions (1.10) is the absence of the Korn–Poincaré inequality necessary to control the term
or, more specifically, its radiation component
in the ballistic energy inequality, where \({\tilde{\vartheta }}\) is the unique harmonic extension of the boundary temperature,
cf. [10, Chapter 12, Section 12.4.2].
To overcome this difficulty, and extra hypothesis is needed depending on the specific shape of the spatial domain \(\Omega \). Denote
where \(\underline{\mu } > 0\), \(\underline{\eta } \ge 0\) are the structural constants introduced in (2.7). If \(\underline{\mu } > 0\), \(\underline{\eta } = 0\), the set \(\mathcal {N}\) contains all conformal Killing vectors with vanishing normal trace, see Schirra [20, Proposition 2.5]. If, in addition, \(\underline{\eta } > 0\), then the set \(\mathcal {N}\) consists of all rigid motions \(\textbf{w} = \varvec{\omega }\times x + \textbf{a}\) with zero normal trace. In the latter case, \(\mathcal {N} \ne \{ 0 \}\) if and only if \(\Omega \) is rotationally symmetric. Consequently, a suitable hypothesis to handle the integral (5.12) reads:
Note that (5.13) is satisfied in the case when \(\Omega \) is an annulus bounded by two concentric spheres,
provided \(\underline{\mu }, \underline{\eta } > 0\), and \(\Theta _1\), \(\Theta _2\) are possibly time dependent positive constants. Indeed we recall Korn–Poincaré inequality, see e.g. Lewintan, Müller, Neff [16]:
for any \(\textbf{v} \in W^{1,q}(\Omega ;R^3)\), \(\textbf{v} \cdot \textbf{n}|_{\partial \Omega } = 0\), where \(\Pi _{\mathcal {N}}\) denotes a projection onto the space \(\mathcal {N}\). Accordingly, the integral (5.12) can be written as
and estimated via (5.14), exactly as in [10, Chapter 12, Section 12.4.2].
We may therefore state the following result concerning the existence of weak solutions to the compressible MHD system.
Theorem 5.1
(Existence of weak solution) Let \(\Omega \subset R^3\) be a bounded domain of class at least \(C^3\). Suppose the data \(\vartheta _B\), \(\textbf{B}_B\) are twice continuously differentiable,
Let the thermodynamic functions p, e, s satisfy the hypotheses (2.1)–(2.6) and the transport coefficients \(\mu \), \(\eta \), \(\kappa \), and \(\zeta \) comply with (2.7)–(2.9), where
In addition, suppose the harmonic extension of the boundary temperature satisfies (5.13).
Then for any initial data
and any \(\textbf{g} \in L^\infty ((0,T) \times \Omega ; R^3)\), the compressible MHD system admits a weak solution in \((0,T) \times \Omega \) in the sense specified in Definition 2.1.
Remark 5.2
As a matter of fact, positivity of the initial density and temperature is not really necessary in the weak framework. Expressing the initial data in terms of the initial density \(\varrho _0\), the momentum \(\textbf{m}_0\), the total entropy \(S_0\), and the magnetic field \(\textbf{B}_0\), we can show global in time existence under the condition that the initial energy
is finite.
Remark 5.3
The weak solutions resulting from the approximate scheme specified above satisfy the differential version of the ballistic energy inequality (2.24).
Remark 5.4
The hypothesis \(\lim _{Z \rightarrow \infty } \mathcal {S}(Z) = 0\) can be omitted and \(\beta \ge 3\) allowed as soon as the boundary temperature \(\vartheta _B\) is constant.
6 Concluding Remarks
The theory can be extended in a straightforward manner to other types of boundary conditions.
-
The complete slip boundary conditions (1.10) can be replaced by the Dirichlet (no slip) boundary conditions
$$\begin{aligned} \textbf{u}|_{\partial \Omega } = \textbf{u}_B\end{aligned}$$as long as \(\textbf{u}_B\cdot \textbf{n} = 0\). This enables to include problems of Taylor–Coutte type. The extension to general in/out flow boundary conditions for the velocity would require a more elaborated treatment in view of the presence of the magnetic field. For the in/out flow boundary conditions for the Navier–Stokes–Fourier system in the absence of magnetic field see [10].
-
The Dirichlet boundary conditions for the temperature can be replaced by a flux type condition on a part of the boundary:
$$\begin{aligned} \partial \Omega = \Gamma _1 \cup \Gamma _2,\ \vartheta |_{\Gamma _1} = \vartheta _B,\ \textbf{q} \cdot \textbf{n}|_{\Gamma _2} = \textbf{h}. \end{aligned}$$Note that this type of condition is particularly relevant in the modelling of stellar magnetoconvection, see Thompson and Christensen–Dalsgaard [23].
-
The induction flux (electric field) can be prescribed in (1.14):
$$\begin{aligned} \Big [ \textbf{B}\times \textbf{u}+ \zeta \textbf{curl}_x\textbf{B}\Big ] \times \textbf{n}|_{\partial \Omega } = \textbf{E}_B \times \textbf{n}. \end{aligned}$$ -
Mixed type boundary conditions on different components of \(\partial \Omega \) can be accommodated without essential difficulties.
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Funding was provided by Grantová Agentura Ceské Republiky (Grant No. 21-02411S), Korea National Science Foundation (Grant No. NRF2022R1F1A1073801). Narodowe Centrum Badań i Rozwoju (Grant No. 2021/43/B/ST1/02851).
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The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21-02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. This work was partially supported by the Thematic Research Programme, University of Warsaw, Excellence Initiative Research University. The work of P.G. and A. Ś-G. was partially supported by National Science Centre (Poland), Agreement No. 2021/43/B/ST1/02851. The work of Y.-S. Kwon was partially supported by the National Research Foundation of Korea (NRF2022R1F1A1073801).
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Feireisl, E., Gwiazda, P., Kwon, YS. et al. Mathematical Theory of Compressible Magnetohydrodynamics Driven by Non-conservative Boundary Conditions. J. Math. Fluid Mech. 25, 84 (2023). https://doi.org/10.1007/s00021-023-00827-2
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DOI: https://doi.org/10.1007/s00021-023-00827-2