On a Navier–Stokes–Ohm problem from plasma physics in multi connected domains

We consider a model from electro-magneto-hydrodynamics describing a plasma in bounded multi connected domains. A nontrivial solution exists for magnetic fields as the equilibrium of this model. Nonlinear stability of the nontrivial solution is proved based on time weighted maximal Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-regularity.


Introduction
Let Ω be a bounded multi connected domain in R 3 and its boundary Σ = ∂Ω be class C 2 . We consider the following electro-magneto-hydrodynamics model which describes a plasma, namely completely ionized gas. 1 Here v 1 denotes the velocity of electrons, v 2 that of ions, E and H the electric and the magnetic fields, respectively. The numbers j , n j , μ j > 0, j = 1, 2 denote densities, number densities, viscosities. The numbers e, z, 0 , μ 0 , α, σ > 0 are physical constants which denote elementary charge, charge number, dielectricity and permeability of vacuum, as well as friction and conductivity. In this model we adopt Ohm's law, namely intense of electricity is proportional to the electric field with the electric conductivity constant σ . Therefore we call the problem (1) a Navier-Stokes-Ohm problem. For more background, we refer to Van Kampen-Felderhof [18] and Miyamoto [11]. The energy functional E is given by Energy dissipation reads by the boundary condition E × ν = 0. We identify the equilibria of the system and show that the energy is even a strict Lyapunov functional. We assume that ∂ t E(t) = 0 for t ∈ (t 1 , t 2 ). Then by the energy identity it holds that ∇v 1 = ∇v 2 = E = 0, (t, x) ∈ (t 1 , t 2 ) × Ω, hence by the Poincaré inequality, v 1 = v 2 = E = 0 on (t 1 , t 2 ) × Ω. This implies further that the pressures π j are constant, and rot H = 0, div H = 0 in Ω, H · ν = 0 on Σ.
If Ω is a bounded simply connected domain, then by Proposition 1 below there exists a potential ϕ satisfying Therefore, ϕ is constant in Ω, and H = ∇ϕ ≡ 0. These arguments show that the only equilibrium is the trivial solution v 1 = v 2 = E = H = 0, and that the energy E is even a strict Lyapunov functional. On the other hand, if Ω is a bounded multi connected domain, the set of H satisfying the conditions (2) has a nontrivial solution whose dimension is coincide with genus of Ω, in other words the second Betti number of Ω, which was proved by Foias-Temam [3] for L 2 (Ω). Kozono-Yanagisawa [8], [9] and Amrouche-Seoula [1] extended the result for L r (Ω). We set the function space which was used in Kozono-Yanagisawa [8], [9].
Prüss-Shimizu [13] proved nonlinear stability of the trivial solution which is the equilibrium of (1) in a bounded simply connected domain. In this paper, for the case of a bounded multi connected domain, we prove nonlinear stability of the non-trivial solution which is the equilibrium of (1). Especially we construct a concrete example of the non-trivial solution in X har (Ω). In order to prove nonlinear stability of the non-trivial equilibrium, we first set the problem (1) abstract formulation by using the linear operator A. Next we see that the kernel of A is the set of non-trivial equilibrium. Then we prove local well-posedness of the problem in time weighted L p space by maximal regularity of the linear operator as the same way in [13]. Based on the linear stability, non-trivial solution of the problem is exponentially stable.
The Navier-Stokes-Ohm problem (1) consists of a Navier-Stokes system and a Maxwell system which are coupled in a semi-linear way. We regard it as a system of evolution equations. Maximal regularity with time weights enables us to obtain well-posedness for initial values in the scale critical space: v ∈ H 1/2 2 (Ω) 6 and (E 0 , H 0 ) ∈ L 2 (Ω) 6 , which was found by Fujita-Kato [4] for the Navier-Stokes equations (cf. Corollary 1, below). There are many other such nonlinear, weakly coupled hybrid systems, in other words nonlinear parabolichyperbolic systems, in the literature. We think that it would be worthwhile to study such systems also from the abstract point of view in the framework of evolution equations. This paper is organized as follows. In Sect. 2, we see characters of X har (Ω) and construct a concrete example of the non-trivial solution in X har (Ω). We formulate the Navier-Stokes-Ohm problem (1) as an abstract evolution equation in Sect. 3 in the same way as in [13], and state results of a linear problem in Sect. 4. In Sect. 5, we prove local well-posedness of the abstract form of (1). Sect. 6 is devoted to show nonlinear stability of the non-trivial solution.

Properties of solutions in X har (Ä)
In this subsection, we see properties of solutions in X har (Ω) defined by (3) based on Temam [17, Appendix 1].
Case 1: Ω is simply connected. In order to make the differences between a simply connected domain and a multiply connected domain, we state well-known results.

Proposition 1
Let Ω be simply connected. For h ∈ X har (Ω), there exists a scalar function ϕ such that h = ∇ϕ and satisfy Theorem 1 Proof For every scalar functions φ and ψ, by the Green formula we have Plugging ϕ in Proposition 1 in both φ and ψ, we obtain We know that h = 0 because h ∈ X har (Ω). This shows that X har (Ω) = {0}.

A concrete vector field of X har (Ä)
In this subsection, we construct a concrete vector field of X har (Ω) in the case when Ω is a solid torus. Let 0 < r < R 1 . In the xz-plane, we make a circle with radius r with center (R, 0). Turning it around z-axis, we obtain the solid torus Ω (cf. Fig. 1 1 ). For the solid torus, the number of smooth cut is equal to 1. By using parameters 0 ≤ u, v < 2π, Ω is represented by Inversely r , u, v are expressed by x, y, z Toroid ∂Ω is given by The unit outward normal ν to ∂Ω is given by We define a potential ϕ as A vector field h = ∇ϕ is expressed by Since the solid torus Ω is biconnected domain, we check that ϕ in (8) satisfies the conditions in Lemma 2 as i = 1. We denote Γ := Γ 1 . First we check the condition (4). Laplace operator in the sold torus is the following.
The potential ϕ = ϕ(u) in (8) is the function of u, it holds that (6) holds as follows Finally we check the condition (7). Since the normal direction to Γ is angle u, it holds that which shows that ϕ satisfies (7).

Abstract formulation
Abstract formulation of (1) is essentially the same as the simply connected domain case in [13]. We first recall some well-known results for the Stokes operator as well as for the Maxwell operator. For this we will need the projections of Helmholtz and Weyl. For given v ∈ L q (Ω) 3 , 1 < q < ∞, we consider the weak Neumann problem It is well-known that there is a solution ϕ ∈ H 1 q (Ω) which is unique up to a constant. Then we define the Helmholtz projection in L q (Ω) 3

by means of
This projection is bounded, and it is orthogonal in case q = 2. In a similar way we define the Weyl projection. For given v ∈ L q (Ω) 3 , 1 < q < ∞, solve the weak Dirichlet problem .
Then we define the Weyl projection in L q (Ω) 3 by means of This projection is also bounded and it is orthogonal in case q = 2.
Consider the Stokes problem To define the Stokes operator, we set X S 0 := P H L 2 (Ω) 3 , and A S := −P H Δ. It is well-known that −A S generates a compact analytic C 0 -semigroup in X S 0 , which is exponentially stable for bounded domains, and moreover that A S is positive definite in case q = 2.
Consider the Maxwell equations

on Σ}, and define the Maxwell operator by means of
Here (E, H ) T denotes the transposed of (E, H ).
Then we consider the Maxwell equations with conductivity In X M 0 we define the Maxwell operator with conductivity by means of This operator is a bounded accretive perturbation of A M and therefore is also m-accretive in the Hilbert space X M 0 , hence it is the negative generator of a C 0 -semigroup of contractions in X M 0 with compact resolvent. However, A MC is not strongly accretive. Exponential stability can be proved by means of the Gearhart-Prüss theorem (e.g. Prüss [12]). The problem (1) is written the following form.
So the structure of the problem is a system of the Stokes equations coupled with the Maxwell system with conduction. The coupling consists of a linear and bounded one, and an unbounded quadratic coupling which fortunately only acts on the velocities.
To formulate this problem abstractly, we define the space X 0 = X S 0 × X S 0 × X M 0 , whose elements are Then the regularity space is X 1 = X S 1 × X S 1 × X M 1 , and the principal linear part A 0 is given by Furthermore, the bounded linear perturbation B 0 + B is given by Then the problem becomeṡ which is a semilinear evolution equation in the base space X 0 . In a more detailed form, the abstract problem (11) readṡ

The linear operator
The base space The fully linearized problem is given bẏ where the operator A in X 0 is given by Here we have set A v is strongly maccretive, hence the semigroup e −A v t is also analytic, exponentially stable and has maximal L p -regularity in the base space X v 0 = X S 0 × X S 0 . Further, the perturbations B v , B w are also bounded and Bu|u is purely imaginary. Therefore A is also m-accretive, but not strongly accretive.

Lemma 4 (Thm. 3.1 in [13])
The operator A is m-accretive in X 0 , hence −A is the generator of the C 0 -semigroup e −At of contractions in X 0 .
We also know that the only eigenvalue of A on iR is possibly 0, all other eigenvalues have strictly positive real parts. By the theorem of Arendt-Batty-Lubich-Phong (cf. [2]) we have the following stability result.

where P A denotes the projection on to the kernel N (A) of A along the range R(A). The semigroup e −At is strongly stable in R(A).
We consider the kernel N (A) of A. If u = (v, w) T ∈ N (A), then v and w satisfy Taking the inner product in X 0 with u yields This shows that ∇v 1 = ∇v 2 = 0 and w 1 = 0. By the Poincaré inequality, it holds that (13) we obtain rot w 2 = 0. Now u ∈ X 0 , which implies that w 2 also satisfies div w 2 = 0 and w 2 · ν = 0. Therefore it holds that

Lemma 6 The kernel of operator A is
We see the formulation of e −At more precisely. Considering the resolvent problem λu + As λ + A w is invertible for {Re λ ≥ 0} \ {0}, this yields Inserting this into the first equation and setting which reads as By the operator-valued Paley-Wiener lemma (see e.g. [12]), there is R ∈ L 1,loc (R + ; B(X 0 )) such that Summarizing the above, we have By using the relation between C 0 -semigroup and the resolvent via the Laplace transform we obtain the proposition.
Proposition 4 C 0 -semigroup e −At on X 0 has the following expression: Finally in this section, we state the exponential stability result for e −At . Theorem 2 (Thm 3.5 in [13]) e −At is exponentially stable on R(A). There exist constants ω 1 > 0 and M 1 ≥ 1 such that which is equivalent to

Local Well-posedness
In order to obtain local well-posedness, time-weighted maximal L p -regularity for A v which gives parabolic regularization plays an essential role. For 1 < p < ∞ and 1/ p < μ ≤ 1 and some 0 < a ≤ ∞, we define v ∈ L p,μ (0, a; H 1 p,μ is defined in the similar way. We introduce the base space We introduce the solution space Time trace space of E 1,μ (a) is given by The case μ = 1 is the natural state space for the problem. Also we set We state the time weighted maximal L p -regularity result of the Stokes system (10) by Prüss-Simonett [14,Sect. 7] in the context of our problem setting.
The following result is local well-posedness of (11).
If blow up occurs, i.e. if t + (u 0 ) < ∞, then u([0, t + )) is not relatively compact in X γ,μ . The solutions generate a local semiflow in the state spaces X γ,μ .
Proof First we see that if the solution u = (v, w) T belongs to and the same estimate holds for ∂ t v.
The proof on this theorem is essentially the same as in [13,Thm. 4.1]. The solution of (11) is expressed by the integral equation for some a > 0 determined later which is decomposed as By Theorem 2, it holds that the first term of right hand side of (16) satisfies e −At u 0 R(A) ≤ M 1 u 0 X 0 for t ≥ 0. So our task is to estimate the second term of right hand side of (16). We set Now we estimate the nonlinear term which is the second term of the right hand side of (16). We define the space For the convection term v j · ∇v j , by the Hölder inequality, the boundedness of P H , and the Sobolev embedding theorem we have which yields Let 0 E 1,μ (a) denote the space of functions in E 1,μ (a) with times trace 0 at t = 0, and consider a ball B r in this space with center origin. We set ϕ(a) := u * E 1,μ (a) and observe that ϕ(a) → 0 as a → 0 uniformly for initial values u 0 belonging to a compact subset of X γ,μ . By the assumption 1/4 + 1/ p ≤ μ, using the embedding relation and the Lipschitz estimate For the nonlinear term v j × w 2 , the Hölder inequality yields If we take β as 3 4 < β < 1, then the embedding relation D([A v ] β ) → L ∞ holds. Choosing κ as μ < μ + 1 − β = κ ≤ 1, we have the embedding We set ψ(a) := v * L p,μ (0,a;L ∞ (Ω)) . The above embedding implies the estimate Here ψ(a) → 0 as a → 0 locally uniformly in X γ,μ . Now we set the map where S vv and S wv are defined in Proposition 4. S vv has maximal regularity with maximal regularity constant M v . S wv holds the estimate , where the constants M v , M w do not depend on a. Choosing first r > 0 and then a > 0 small enough, we know that T : B r → B r is a self-map and strictly contraction uniformly for initial values u 0 ∈ K ⊂ X γ,μ compact. Therefore the contraction mapping principle yields that there is a unique solution u(t) of (11) on [0, a], where a is uniform on K. The characterization of the maximal time of existence is standard argument (cf. [7]).
If we take 2μ − 2/ p = 1/2, then it is corresponding result of Fujita-Kato [4], in which obtains well-posedness of the Navier-Stokes equations for initial values in the scale critical space.

Corollary 1 Let 4/3 ≤ p < ∞. Then for each initial value
there exists a = a(u 0 ) > 0 and a unique solution u ∈ E 1,μ (a) of the problem (11) satisfies 0, a). The solution depends continuously on the data.

Nonlinear stability of equilibria
By Lemma 6, we know that the equilibria of (1) is where w E = (0, w E2 ) T ∈ X w 0 , w E2 ∈ X har (Ω). The following nonlinear stability of the problem (1) is the main result in this paper.
The right-hand side is independent of a, this shows that it is able to extend t + (u 0 ) = ∞. Therefore we conclude that sup t≥t 0 e ωtũ (t) X γ ≤ 4MCε 2 and (20) under u 0 X γ,μ ≤ δ.