Mathematical Theory of Compressible Magnetohydrodynamics Driven by Non-conservative Boundary Conditions

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.


Introduction
The large scale dynamics arising in stellar magnetoconvection is driven in an essential way by the boundary conditions, see Gough [11].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 a.g.Matsumura and Nishida [13], Valli [18], among others.What is more, the recent results of Merle et al. [14], [15] 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 Ω ⊂ R 3 with regular boundary.The time evolution of the density ̺ = ̺(t, x), the (absolute) temperature ϑ = ϑ(t, x), the velocity field u = u(t, x), and the magnetic field B = B(t, x) is governed by the compressible MHD system of field equations (cf.[6]):
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 µ > 0 and η ≥ 0 are continuously differentiable functions of the temperature.Similarly, the heat flux obeys Fourier's law, where the heat conductivity coefficient κ > 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 [6] under certain physically relevant restrictions imposed on constitutive equations and transport coefficients.The boundary conditions considered in [6] are conservative, (1.9) characteristic for closed systems.If the driving force g in the momentum equation is conservative, meaning g = ∇ x G, 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 [10].
As pointed out by Gough [11], a rich fluid behaviour is conditioned by a proper choice of boundary conditions.Motivated by models in astrophysics, we suppose the boundary ∂Ω is impermeable and the tangential component of the normal viscous stress vanishes on it, (1.10) However, the theory presented in this paper can accommodate more general boundary conditions for the velocity field discussed in Section 6.
Similarly to the well known Rayleigh-Bénard problem, see e.g.Davidson [5], we impose the inhomogeneous Dirichlet boundary conditions for the temperature, To incorporate the effect of an exterior magnetic field, we suppose that B is (not necessarily small) perturbation of a background magnetic field B B , div x B B = 0, In the context of astrophysics, we may think of B B as being the magnetic field imposed by a massive star.As for b, we suppose either The boundary condition (1.12) is of Dirichlet type, compatible with the requirement of solenoidality of 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 B B = 0 in (1.12) and (1.13) corresponds to a perfectly isolating and perfectly conducting boundary, respectively (see e.g.Alekseev [1]).
For possibly technical but so far unsurmountable reasons, 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(̺, ϑ)div x u in the internal energy balance (1.4), integrable.A remedy proposed in [6] 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 [6].The total energy balance reads (1.16) If the system is conservative, the total energy flux vanishes on the boundary and (1.16) integrated over Ω yields d dt 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 [6].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 [7].
The situation becomes more delicate for open (non-conservative) systems, with inhomogeneous Dirichlet boundary conditions.The heat flux q • n as well as the induction flux (the normal component of the electric field) [B × u × B + ζcurl x B × B] • n do not necessarily vanish on ∂Ω and give rise to additional uncontrollable sources of energy in (1.17).In [4] (see also the monograph [9]), 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 1 2 where θ is an arbitrary extension of the boundary temperature ϑ B inside Ω. Pursuing the same strategy, we introduce a "magnetic" variant of the ballistic energy After a straightforward manipulation, we deduce equality 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 Section 2. In Section 3 we introduce the basic tool to investigate stability properties in the class of weak solutions -the relative energy inequality.In Section 4, we show that the weak solutions enjoy the weak-strong uniqueness property.In Section 5, the existence of global-in-time weak solutions is established by means of a multilevel approximation scheme.The paper is concluded in Section 6 by a brief discussion on possible extensions of the theory to a larger class of boundary conditions.

Weak formulation
We start by a list of structural hypotheses imposed on the equation of state and the transport coefficients.

Equation of state
Our choice of the equation of state is motivated by [9,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 ∈ C 1 [0, ∞) satisfies This implies, in particular, that Z → P (Z)/Z In accordance with (1.5), the entropy takes the form where (2.6)

Transport coefficients
We suppose the viscosity coefficients in the Newtonian stress S(ϑ, ∇ x u) are continuously differentiable functions of the temperature satisfying Note that α = 1 2 corresponds to Sutherland's law relevant in astrophysics, see Yang et al. [21].Similarly, the heat conductivity coefficient in the Fourier heat flux q(ϑ, ∇ x ϑ) is a continuously differentiable function of the temperature satisfying (2.8) Here, the case β = 3 reflects the effect of radiation.Finally, we suppose the magnetic diffusivity coefficient (2.9)

Boundary data
We suppose the background magnetic field Similarly, we suppose the boundary temperature ϑ B can be extended inside Ω,

Weak solutions
Before discussing the concept of weak solution, it is useful to introduce the function spaces Both H 0,τ and H 0,n are endowed with the Hilbert norm , see e.g.Alexander and Auchmuty [2].

and the integral identity
In addition if the boundary condition (1.12) is imposed or for the boundary conditions (1.13), (1.14).The integral identity in the case of boundary conditions (1.12), for the boundary conditions (1.13), (1.14).
, and the integral inequality • Ballistic energy inequality.The inequality Remark and write On the one hand, as Ψ vanishes on ∂Ω and B is solenoidal, we have whence the integral identity (2.19) holds for ∇ x Ψ.On the other hand, the function (ϕ − ∇ x Ψ) is solenoidal, and, as ∇ x Ψ × n| ∂Ω = 0, a legal test function in (2.19).
If ϕ satisfies (2.21), we simply consider -the standard potential in the Helmholtz decomposition.
Remark 2.3.Unlike ̺, ̺u, and B, the total entropy ̺s(̺, ϑ) is not weakly continuous in time.However, it can be deduced from the entropy inequality that the one sided limits 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 (̺, S = ̺s(̺, ϑ), m = ̺u, B).In particular, its integral is a weakly lower semi-continuous function; whence (2.23) holds for any time τ with the convention for entropy discussed Remark 2.3.Alternatively, we may impose a stronger integrated version of (2.23) in the form The existence of global-in-time weak solutions in the sense of Definition 2.1 will be shown in Section 5 below.

Relative energy
A suitable form of the relative energy for the compressible MHD system reads In applications, the quantity (̺, ϑ, u, B) stands for a weak solution of the compressible MHD system while (r, Θ, U, 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 (̺, S = ̺s(̺, ϑ), m = ̺u, B), see [9, Chapter 3, Section 3.1] for details.Moreover, we have where e(̺, ϑ) − ϑs(̺, ϑ) + p(̺,ϑ) ̺ is Gibbs' free energy.Consequently, the relative energy can be seen as Bregman divergence associate to the convex function E. In particular, E ̺, ϑ, u, B r, Θ, U, H ≥ 0, and E ̺, ϑ, u, B r, Θ, U, H = 0 ⇔ (̺, ϑ, u, B) = (r, Θ, U, H) as long as r > 0.

Relative energy inequality
In the remaining part of this section, we derive a relative energy inequality provided (̺, ϑ, u, B) is a weak solution of the compressible MHD system specified in Definition 2.1 and (r, Θ, U, 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 B B for H, and, of course, θ for Θ obtaining

Momentum perturbation
Our first goal is to replace |u| 2 by |u−U| 2 .This can be achieved by considering U as a test function in the momentum balance equation (2.15).After a straightforward manipulation explained in detail in [9, Chapter 3, Section 3.2.1],we obtain for any (Θ, U, H) as in (3.2).

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, Θ, U, H) specified in (3.2).

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, Θ, U, H) is a smooth solution of the problem, specifically, together with the relevant initial and boundary conditions.
Remark 4.2.As we assume existence of a classical solution in Theorem 4.1, the initial data are also regular, in particular, inf Remark 4.3.Remarkably, the result holds in the full range of exponents α, β specified in (4.2).
In particular, we remove the gap between the weak-strong uniqueness principle stated in [4], [9] for β = 3 and the existence result requiring β > 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 [18], Valli and Zajaczkowski [19], or by a more recent approach via maximal regularity in the spirit of Kotschote [12].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 [12].
The idea of the proof of Theorem 4.1 is simple: We consider the strong solution (r, Θ, U, H) as test functions in the relative energy inequality (3.7) and use Gronwall's type argument.We proceed in several steps specified below.

Magnetic field
As U solves the momentum equation, we get Similarly, 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 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 B, H satisfy the boundary condition (1.12) and then (B − H) × n = 0 or H satisfies the flux condition (1.14).In both cases the surface integral in (4.9) vanishes.Thus relation (4.7) takes the form

Pressure and entropy
After a simple manipulation, we can rewrite (4.10) in the form with a quadratic remainder Using the identity ∂s(r, Θ) Consequently, inequality (4.11) can be written as with a remainder

Diffusive terms
First, as ϑ, Θ coincide on ∂Ω, we can integrate by parts: Putting all diffusive terms on the left-hand side of the relative energy inequality (4.13), we get Finally, Thus (4.13) can be rewritten in the final form with R 2 given in (4.14).

Refined estimates
Similarly to [9, 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 [9, 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 Θ, sup Θ.

Temperature gradient
The following crucial estimate was proved in [9, Chapter 4, Section 4.2.2] for β = 3.Here, we extend its validity to general β ≥ 3.
Proof.We consider first the essential part.Rewrite Applying Hölder's inequality we deduce from (4.21) where ξ depends on Θ.As a matter of fact, the only hypotheses to be imposed on κ is that κ(ϑ) > 0 whenever ϑ > 0.
As for the residual part, we first rewrite Consequently, as inf Θ > 0, it remains to handle a quantity where C is a positive constant depending on Θ, inf Θ.Note that this step requires ϑ 2 < ∼ κ(ϑ).To proceed we consider ϑ > 0 to be fixed below.We write Thus it remains to handle the integral Finally, it remains to handle the integral ϑ≥ϑ On the one hand, using Sobolev-Poincaré inequality we get
Next, by Jensen's inequality, where q = 8 5 − α .(4.29) Next, repeating the same arguments, we get The previous estimates, together with (4.32), allow us to rewrite the inequality (4.18) in the form where the remainder R 2 is specified in (4.14).

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 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 [9, Chapter 4].
We have proved Theorem 4.1.
Remark 4.6.The Lipschitz regularity of the domain Ω required in Theorem 4.1 may not be sufficient for the existence of the strong solution to the MHD system.

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 [9].
The magnetic field equations can be incorporated exactly as in [6] without any additional difficulties.Similarly to [9,Chapter 12], two additional assumptions must be imposed on the constitutive relations, specifically, lim Z→∞ S(Z) = 0, ( where S is the function defining the entropy s M .Moreover, we require where β 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.

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 ̺(0, •) = ̺ 0,δ , ( where ̺ 0,δ > 0 is a smooth regularization of the initial density ̺ 0 .Here and hereafter, ε and δ are small parameters to be sent to zero successively in the limit passage.The momentum equation (1. as long as u B •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 [9].
• The Dirichlet boundary conditions for the temperature can be replaced by a flux type condition on a part of the boundary: Note that this type of condition is particularly relevant in the modelling of stellar magnetoconvection, see Thompson and Christensen-Dalsgaard [17].
• The induction flux (electric field) can be prescribed in (1.14): • Mixed type boundary conditions imposed on different components of ∂Ω can be accommodated without essential difficulties.
5. Note carefully that the ballistic energy inequality (2.23) remains valid if we replace B B by any other extension B such that div x B = 0, and B × n| ∂Ω = B B × n| ∂Ω , or B • n| ∂Ω = B B • n| ∂Ω , respectively.Indeed the difference B B − 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.