on global regular solutions to magnetohydrodynamics in axi symmetric domains

Z Angew Math Phys (2016) 67:142 c 2016 The Author(s) This article is published with open access at Springerlink.com 0044-2275/16/060001-22 published online October 31, 2016 DOI 10.1007/s00033-016-0734-z Zeitschrift fă ur angewandte Mathematik und Physik ZAMP On global regular solutions to magnetohydrodynamics in axi-symmetric domains Bernard Nowakowski and Wojciech M Zajaczkowski ˛ Abstract We consider mhd equations in three-dimensional axially symmetric domains under the Navier boundary conditions for both velocity and magnetic fields We prove the existence of global, regular axi-symmetric solutions and examine their stability in the class of general solutions to the mhd system As a consequence, we show the existence of global, regular solutions to the mhd system which are close in suitable norms to axi-symmetric solutions Mathematics Subject Classification 35Q35, 76D03, 76W05 Keywords Magnetohydrodynamics, Stability of axially symmetric solutions, Global existence of regular solutions Introduction We examine viscous, incompressible magnetohydrodynamics (mhd) flows in axially symmetric domains in R3 The governing equations read v,t + (v · ∇) v − νΔv + ∇q − (H · ∇) H = f in ΩkT := Ω × (kT, (k + 1)T ) div v = in ΩkT , H,t + (v · ∇) H − (H · ∇) v − μΔH = in ΩkT , div H = in ΩkT , (1.1) where k is a natural number including and: • • • • • • • v = (v1 (x, t), v2 (x, t), v3 (x, t)) ∈ R3 is the velocity of the fluid, q = p(x, t) + |H| ∈ R is the total pressure, p = p(x, t) ∈ R is the pressure, H = (H1 (x, t), H2 (x, t), H3 (x, t)) ∈ R3 is the magnetic field, f = (f1 (x, t), f2 (x, t), f3 (x, t)) ∈ R3 is the external force field, x = (x1 , x2 , x3 ) is the Cartesian system of coordinates, ν > and μ > are constant viscosity and resistivity coefficients, respectively Let S := ∂Ω denote the boundary of Ω Then, we supplement (1.1) with the following boundary conditions v · n = 0, (1.2) n × rot v = and H · n = 0, n × rot H = (1.3) on S kT = S × (kT, (k + 1)T ), k ∈ N ∪ {0}, where n is the unit outward vector normal to S As the initial conditions, we take 142 B Nowakowski and W M Zaja˛ czkowski Page of 22 v|t=kT = v(kT ) in Ω, H|t=kT = H(kT ) in Ω ZAMP (1.4) Throughout this paper, we assume that Ω is an axially symmetric, bounded domain located in a positive distance from the axis of symmetry Its geometry can be easily expressed in cylindrical coorΦ dinates (r, ϕ, z) which are introduced in the standard way through a mapping Φ, x = (x1 , x2 , x3 ) = (r cos ϕ, r sin ϕ, z) Then, the basis vectors read er = ∂r Φ = (cos ϕ, sin ϕ, 0), eϕ = ∂ϕ Φ = (− sin ϕ, cos ϕ, 0), ez = ∂z Φ = (0, 0, 1) Let x3 be the line of intersection of two planes: P (ϕ) and x2 = 0, where ϕ is the dihedral angle between P (ϕ) and x2 = Let ψ(r, z) = be a closed curve in the plane P (ϕ) such that < a ≤ r ≤ b (e.g one could take ψ(r, z) = (r − a0 )2 + z − r02 = 0, where a0 > r0 , so r ∈ [a0 − r0 , a0 + r0 ] and |z| ≤ r0 ) Then, we define Ω as a solid of revolution around the x3 -axis Ω := {(r, ϕ, z) : ψ(r, z) < 0, ϕ ∈ [0, 2π]} (1.5) Let ϕ0 ∈ [0, 2π] be fixed We introduce Ωϕ0 = P (ϕ0 ) ∩ Ω, Then n|Sϕ0 Sϕ0 = ∂Ωϕ0 ⎡ ⎛ ⎞ ⎛ ⎞⎤ x ⎣ ψ,r ⎝ ⎠ ∇ψ x2 + ψ,z ⎝0⎠⎦ = = |∇ψ| |∇ψ| r (1.6) (1.7) + ψ Let us note that the right-hand side in the above expression does not depend where |∇ψ| = ψ,r ,z on ϕ0 , which allows us to utilize the Cartesian coordinate system Our main goal (cf Theorem 3) is to prove the existence of global, regular solutions to problem (1.1)– (1.4) without any assumptions on smallness of the initial and the external data, however, fulfilling certain geometrical constraints which we will describe in the subsequent paragraphs Our proof is based on stability reasoning: we construct a special solution and examine its stability in the class of solutions to (1.1) As a by-product, we obtain a solution (1.1) This method has been recently utilized in e.g [1,2] The first step of our work is a construction of a special, axially symmetric solution (vs , Hs ) By axially symmetric, we mean ∂ϕ vsr = ∂ϕ vsϕ = ∂ϕ vsz = ∂ϕ Hsr = ∂ϕ Hsϕ = ∂ϕ Hsz = 0, (1.8) thus vs = vs (r, z, t), Hs = Hs (r, z, t) Since Ω is located in a positive distance from its axis of symmetry the construction of this special solution is much easier because it can be regarded as two-dimensional However, it is not exactly two-dimensional because vs and Hs have components along eϕ More precisely vs = vsr (r, z, t)er + vsϕ (r, z, t)eϕ + vsz (r, z, t)ez , Hs = Hsr (r, z, t)er + Hsϕ (r, z, t)eϕ + Hsz (r, z, t)ez , fs = fsr (r, z, t)er + fsϕ (r, z, t)eϕ + fsz (r, z, t)ez , qs = qs (r, z, t), where ur = u · er , uϕ = u · eϕ , uz = u · ez , (1.9) ZAMP Global regular solutions to magnetohydrodynamics Page of 22 142 for any u ∈ R3 In Sect 3, we show that a solution (vs , Hs ) to the following problem vst + (vs · ∇)vs − νΔvs + ∇qs − (Hs · ∇)Hs = fs in ΩkT := Ω × (kT, (k + 1)T ), div vs = in ΩT , Hst + (vs · ∇)Hs − (Hs · ∇)vs − μΔHs = in ΩkT , div Hs = in ΩkT , n × rot vs = 0, n · vs = on S n · Hs = n × rot Hs = 0, vs |t=kT = vs (kT ), on S Hs |t=kT = Hs (kT ) (1.10) kT kT := S × (kT, (k + 1)T ), , in Ω is global and regular, that is we prove the following theorem: Theorem Let div vs (0) = div Hs (0) = Assume that rot vs (0), rot Hs (0) ∈ L2 (Ω), fs ∈ L2 ΩkT , k ∈ {0, 1, 2, } and (k+1)T sup k fs (t) L2 (Ω) dt < +∞ kT Then, there exists a weak solution to (1.10) and a constant A3 (see Lemma 3.1) such that (k+1)T sup kT is so large that T ≥ 2c(Ω) A6 ν¯ and exp − and rot u(0), rot K(0) L2 (Ω) ν¯T ≤ , ≤ γ, where γ is sufficiently small number, then there exists a unique solution (u, K) to (1.11) such that u, K ∈ V21 (ΩkT ) [see (2.2)], k = 0, 1, 2, and there are two constants B4 and B5 (see Lemma 4.2) such that u, K V21 (ΩkT ) ≤ γ B42 + B52 In the last step, we prove the existence of global, strong solutions to (1.1) + (1.2) + (1.3) + (1.4) The main result reads: Theorem Let the assumptions of Theorems and hold Then, there exists a global, strong solution to (1.1) + (1.2) + (1.3) + (1.4) such that v = vs + u, H = Hs + K, q = qs + σ, v, H ∈ V21 ΩkT , ∇q ∈ L2 ΩkT , ZAMP Global regular solutions to magnetohydrodynamics Page of 22 142 where k = 0, 1, 2, For a brief description of past results concerning the regularity and existence of weak solutions, we refer the reader to the introduction in [1] Auxiliary facts From now on, we write N0 = N ∪ {0} and ΩkT = Ω × (kT, (k + 1)T ) We will also frequently use −Δ = rot rot −∇ div, which suggests the following “integration by parts” formula rot rot u · vdx = Ω rot u · rot vdx + Ω n × rot u · vdS S All constants are generic (i.e they may vary from line to line) and are denoted by c Additionally, if a constant depends on the domain (e.g in embedding inequalities), we write c(Ω) Below, we introduce functions spaces and recall some technical lemmas 2.1 Function spaces By Lp (Ω), p ∈ [1, ∞], we denote the Lebesgue space of integrable functions By H s (Ω), s ∈ N and Wp2,1 (ΩkT ), p > 1, we denote the Sobolev spaces equipped with the following norms ⎛ ⎞ 12 • u H s (Ω) =⎝ ⎠ , |Dα x u| dx |α|≤s Ω ⎛ • u Wp2,1 (ΩkT ) ⎜ =⎝ kT Dα x ⎞ p1 (k+1)T ⎟ p p p p |u,xx | + |u,x | + |u| + |u,t | dxdt⎠ Ω ∂xα11 ∂xα22 ∂xα33 , where = |α| = α1 + α2 + α3 and αi ∈ N0 , i = 1, 2, It is convenient to write 2 u, v X = u X + v X , (2.1) where X is a Banach space By V2k (Ω × (T1 , T2 )), we denote a space of all functions u such that ⎛ T ⎞ 12 u V2k (Ω×(T1 ,T2 )) = ⎝ess supt∈(T1 ,T2 ) u(t) H k (Ω) ∇u(t) + H k (Ω) dt⎠ , (2.2) T1 where k ∈ N0 , T1 < T2 Let us now fix ϕ0 ∈ [0, 2π] and define ∇ = (∂r , ∂z ) Since the distance between Ωϕ0 (cf (1.6)) and the axis of symmetry of Ω is always positive, we may write ⎞ 1s ⎛ u Ws1 (Ω0 ) ∼ =⎝ Ω0 s s |∇ u| + |u| drdz ⎠ (2.3) 142 B Nowakowski and W M Zaja˛ czkowski Page of 22 ZAMP We also note that for ψ ∈ {vs , Hs }, we have ψ X(Ω) ≤ c(Ωϕ0 ) ψ X(Ωϕ0 ) ≤ c(Ω) ψ X(Ω) , (2.4) where X is either a Lebesgue or a Sobolev space This inequality follows immediately from the definition of vs , Hs and the geometry of Ω More importantly, it justifies that whenever we use an embedding inequality for ψ we may take n = For function spaces defined above the following embedding will turn very useful Namely, if u ∈ V21 (ΩkT ), then u ∈ L10 (ΩkT ) (see [22, Lemma 3.7]) and u ≤ c(Ω) u L10 (ΩkT ) V21 (Ωt ) (2.5) We will also use the interpolation inequality ∇u L (ΩkT ) ≤ c1 u W22,1 (ΩkT ) + c2 −1 u L2 (ΩkT ) (2.6) 2.2 Auxiliary results Below, we gather crucial tools for establishing a-priori estimates for the solutions to problems (1.1), (1.10) and (1.11) Lemma 2.1 (see Theorem 1.1 in [19]) Let k ∈ N0 Assume that Ω is a bounded domain such that ∂Ω ∈ C k+1 In addition, let F ∈ H k (Ω), div F = Suppose that u is a solution to the following overdetermined problem rot u = F, div u = 0, u×n=0 or u · n = Then u H k+1 (Ω) ≤ c(Ω) F H k (Ω) , where k ∈ N0 Lemma 2.2 Let k ∈ N0 , Ω ∈ C k Suppose that F ∈ H k (Ω), div F = If u solves rot rot u = F on Ω, div u = u·n=0 on Ω, on S rot u × n = on S, then u H k+2 (Ω) ≤ c(k, Ω) F H k (Ω) For the proof of the above Lemma, we refer the reader to Lemma 2.1 and problem (2.7) in [6] Lemma 2.3 (cf Lemma 3.13 in [20]) Let us consider the Stokes problem v,t − νΔv + ∇p = F in ΩT , div v = in ΩT , v·n=0 on S T , rot v × n = on S T , v|t=0 = v(0) on Ω ZAMP Global regular solutions to magnetohydrodynamics Page of 22 142 If F ∈ Ls (ΩT ), where < s < ∞, then there exists a unique solution such that v ∈ Ws2,1 (ΩT ) and v Ws2,1 (ΩT ) + ∇p ≤ c(ν, Ω) Ls (ΩT ) F Ls (ΩT ) + v(0) 2− s Ws (Ω) Lemma 2.4 (cf Lemma 3.14 in [20]) Consider the following initial-boundary value problem in ΩT , H,t − μΔH = G H·n=0 on S T , rot H × n = on S T , H|t=0 = H(0) on Ω Assume that G ∈ Lp (ΩT ), where < p < ∞ Then, there exists a unique solution H such that H ∈ Wp2,1 (ΩT ) and H Wp2,1 (ΩT ) ≤ c(μ, Ω) G + H(0) Lp (ΩT ) 2− p Wp (Ω) Lemma 2.5 (Agmon inequalities; cf (1.2.44) in [21]) Let Ω ⊂ Rn , ∂Ω ∈ C n If ϕ ∈ H (Ω), then ϕ ϕ L∞ (Ω) L∞ (Ω) ≤ c(n, Ω) ϕ ≤ c(n, Ω) ϕ L2 (Ω) ϕ ϕ H (Ω) H (Ω) H (Ω) n = 2, n = 3 The existence and properties of solutions to (1.10) Using energy methods, we prove a priori estimates for a solution (vs , Hs ) to (1.10) Therefore, the existence of the solution will follow from the Faedo–Galerkin method We start with the basic global energy estimate Lemma 3.1 Let (vs , Hs ) be a solution to (1.10), div vs (0) = 0, div Hs (0) = Suppose that ν¯ = min{ν, μ}, k ∈ N0 , (k+1)T A21 c(Ω) sup ≡ ν¯ k∈N0 A22 ≡ 1− Then vs (kT ), Hs (kT ) fs (t) kT A21 + e−¯ν c(Ω)T L2 (Ω) L2 (Ω) dt < ∞, vs (0), Hs (0) (3.1) −¯ ν c(Ω)T L2 (Ω) e < ∞ ≤ A22 , t vs (t), Hs (t) L2 (Ω) + ν¯c(Ω) vs (τ ), Hs (τ ) H (Ω) dτ ≤ A21 + A22 ≡ A23 , (3.2) kT where t ∈ (kT, (k + 1)T ] Proof Multiplying (1.10)1,3 by vs and Hs , respectively, integrating over Ω, using (1.10)2,4 and the boundary conditions (1.10)5,6 , we obtain d vs , Hs dt L2 (Ω) + ν rot vs L2 (Ω) + μ rot Hs L2 (Ω) fs · vs dx = Ω 142 B Nowakowski and W M Zaja˛ czkowski Page of 22 ZAMP Utilizing Lemma 2.1 and the Hă older and Young inequalities, we get d 2 vs , Hs L2 (Ω) + ν¯c(Ω) vs , Hs L2 (Ω) ≤ fs L2 (Ω) , dt ν¯c(Ω) which implies d 2 fs L2 (Ω) eν¯c(Ω)t vs , Hs L2 (Ω) eν¯c(Ω)t ≤ dt ν¯c(Ω) Integrating the above inequality with respect to time from t = kT to t ∈ (kT, (k + 1)T ] yields L2 (Ω) vs (t), Hs (t) ≤ ν¯c(Ω) (3.3) t L2 (Ω) fs (τ ) dτ + e−¯ν c(Ω)(t−kT ) vs (kT ), Hs (kT ) L2 (Ω) kT Setting t = (k + 1)T gives (k+1)T vs ((k + 1)T ), Hs ((k + 1)T ) L2 (Ω) ≤ ν¯c(Ω) fs (t) L2 (Ω) dt + e−¯ν c(Ω)T vs (kT ), Hs (kT ) L2 (Ω) kT Iterating the above procedure, we get A21 + e−¯ν c(Ω)T vs (0), Hs (0) L2 (Ω) , − e−¯ν c(Ω)T which proves (3.2)1 To conclude (3.2)2 , we integrate (3.3) with respect to time and use the above inequality This ends the proof vs (kT ), Hs (kT ) L2 (Ω) ≤ In the below lemma, we establish higher-order estimates for weak solutions to (1.10) Lemma 3.2 Let the assumptions of Lemma 3.1 hold Assume that T > is so large that A24 ≡ A21 exp A25 ≡ A43 ν¯2 A24 − exp − ν¯2T A26 ≡ A25 2A43 ν ¯3 ≤ T Let , + rot vs (0), rot Hs (0) L2 (Ω) ν¯T < +∞, L2 (Ω) dτ ≤ A26 , exp − A43 + A21 + A25 ν¯2 Then rot vs (kT ), rot Hs (kT ) L2 (Ω) ≤ A25 , t rot vs (t), rot Hs (t) L2 (Ω) + ν¯ rot2 vs (τ ), rot2 Hs (τ ) (3.4) kT where t ∈ (kT, (k + 1)T ], k ∈ N0 Proof We begin with multiplying (1.10)1,3 by rot2 vs and rot2 Hs , respectively, integrating the result over Ω and using the boundary conditions (1.10)5,6 d rot vs , rot Hs dt L2 (Ω) + ν¯ rot2 vs , rot2 Hs L2 (Ω) ((vs · ∇) vs − (Hs · ∇) Hs ) · rot2 vs dx ≤− Ω ((vs · ∇) Hs − (Hs · ∇) vs ) · rot2 Hs dx + − Ω fs · rot2 vs dx Ω ZAMP Global regular solutions to magnetohydrodynamics Page of 22 142 Utilizing the Hă older and Young inequalities and Lemma 2.5, we get ((vs · ∇) vs ) · rot2 vs dx ≤ rot2 vs − ∇vs L2 (Ω) vs L2 (Ω) L∞ (Ω) Ω ≤ c(Ω) vs ≤ vs ∇vs H (Ω) H (Ω) +c L2 (Ω) ,Ω vs L2 (Ω) H (Ω) vs vs L2 (Ω) Analogously ((Hs · ∇) Hs ) · rot2 vs dx ≤ rot2 vs ∇Hs L2 (Ω) Hs L2 (Ω) L∞ (Ω) Ω ≤ c(Ω) rot2 vs ≤ ((vs · ∇) Hs ) · rot2 Hs dx ≤ rot2 Hs − L2 (Ω) rot2 vs L (Ω) L2 (Ω) + Hs Hs ∇Hs ∇Hs H (Ω) H (Ω) L2 (Ω) vs L2 (Ω) +c , Hs L2 (Ω) ,Ω Hs vs L2 (Ω) ,Ω Hs H (Ω) Hs H (Ω) vs L2 (Ω) , L∞ (Ω) Ω ≤ c(Ω) rot2 Hs ≤ rot2 Hs L2 (Ω) L2 (Ω) + vs ∇Hs H (Ω) vs H (Ω) +c ∇vs L2 (Ω) L2 (Ω) , 2 L2 (Ω) , and ((Hs · ∇) vs ) · rot2 Hs dx ≤ rot2 Hs L2 (Ω) Hs L∞ (Ω) Ω ≤ c(Ω) Hs ≤ Using Lemma 2.2 and taking 1, , 6 Hs H (Ω) H (Ω) ∇vs +c L2 (Ω) ,Ω Hs vs L2 (Ω) H (Ω) Hs L2 (Ω) sufficiently small, we obtain d 2 rot vs , rot Hs L2 (Ω) + ν¯ rot2 vs , rot2 Hs L (Ω) dt c (Ω) 2 vs , Hs H (Ω) vs , Hs L2 (Ω) + fs L2 (Ω) ≤ ν¯ ν¯ (3.5) In light of Lemma 2.1, we have d rot vs , rot Hs dt c(Ω) vs , Hs ≤ ν¯ L2 (Ω) + ν¯ rot vs , rot Hs H (Ω) vs , Hs L2 (Ω) L2 (Ω) rot vs , rot Hs L2 (Ω) + c(Ω) fs ν¯ L2 (Ω) 142 B Nowakowski and W M Zaja˛ czkowski Page 10 of 22 ZAMP From the above inequality it follows that ⎛ ⎛ ⎞⎞ t d ⎝ c(Ω) 2 rot vs , rot Hs L2 (Ω) exp ⎝ν¯t − vs (τ ), Hs (τ ) H (Ω) vs (τ ), Hs (τ ) L2 (Ω) dτ ⎠⎠ dt ν¯ kT ⎛ ⎞ t c(Ω) c(Ω) 2 ≤ fs L2 (Ω) exp ⎝ν¯t − vs (τ ), Hs (τ ) H (Ω) vs (τ ), Hs (τ ) L2 (Ω) dτ ⎠ ν¯ ν¯ kT Integrating with respect to time from t = kT to t ∈ (kT ; (k + 1)T ) yields rot vs (t), rot Hs (t) L2 (Ω) c(Ω) ≤ ν¯ + rot vs (kT ), rot Hs (kT ) ⎛ t fs (τ ) kT L2 (Ω) L2 (Ω) c(Ω) dτ exp ⎝ ν¯ ⎛ c(Ω) exp ⎝−¯ ν (t − kT ) + ν¯ ⎞ t vs (τ ), Hs (τ ) H (Ω) vs (τ ), Hs (τ ) L2 (Ω) kT dτ ⎠ ⎞ t vs (τ ), Hs (τ ) H (Ω) vs (τ ), Hs (τ ) L2 (Ω) dτ ⎠ kT From Lemma 3.1 it follows that c(Ω) sup ¯ k∈N0 ν t vs (τ ), Hs (τ ) H (Ω) vs (τ ), Hs (τ ) L2 (Ω) A43 , ν¯2 dτ ≤ kT thus rot vs (t), rot Hs (t) L2 (Ω) c(Ω) ≤ ν¯ + rot vs (kT ), rot Hs (kT ) Setting t = (k + 1)T and using that T ≥ rot vs ((k + 1)T ) , rot Hs ((k + 1)T ) 2A43 ν ¯3 t fs (τ ) L2 (Ω) A43 ν¯2 dτ exp kT L2 (Ω) exp −¯ ν (t − kT ) + A43 ν¯2 we have L2 (Ω) ≤ A24 + rot vs (kT ), rot Hs (kT ) L2 (Ω) exp − ν¯T Iterating the above procedure yields rot vs (kT ), rot Hs (kT ) L2 (Ω) ≤ A24 − exp − ν¯2T + rot vs (0), rot Hs (0) L2 (Ω) exp − ν¯T , which proves (3.4)1 To prove (3.4)2 , we integrate (3.5) with respect to time from t = kT to t ∈ (kT, (k + 1)T ] and use Lemma 2.1 Then t rot vs (t), rot Hs (t) L2 (Ω) + ν¯ rot2 vs (τ ), rot2 Hs (τ ) L2 (Ω) dτ kT c(Ω) ≤ sup rot vs (t), rot Hs (t) ν¯ kT ≤t≤(k+1)T c(Ω) + ν¯ t L2 (Ω) vs (τ ), Hs (τ ) H (Ω) kT t fs (τ ) kT This completes the proof L2 (Ω) vs (τ ), Hs (τ ) dτ + rot vs (kT ), rot Hs (kT ) L2 (Ω) ≤ A25 A43 + A21 + A25 ν¯2 L2 (Ω) dτ ZAMP Global regular solutions to magnetohydrodynamics Page 11 of 22 142 Remark 3.3 Under the assumptions of Lemma 3.2, we have vs W22,1 (ΩkT ) + Hs W22,1 (ΩkT ) + ∇p L2 (ΩkT ) ≤ c(Ω) A26 + A3 A6 + A1 + A5 ≡ A7 (3.6) Indeed, in light of Lemmas 2.3 and 2.4, we have vs W22,1 (ΩkT ) + Hs + (vs · ∇)Hs W22,1 (ΩkT ) L2 (ΩkT ) + ∇p L2 (ΩkT ) + (Hs · ∇)vs ≤ (vs · ∇)vs L2 (ΩkT ) L2 (ΩkT ) + fs L2 (ΩkT ) + (vs · ∇)Hs L2 (ΩkT ) + (Hs · ∇)Hs + vs (kT ) H (Ω) L2 (ΩkT ) + Hs (kT ) H (Ω) Using the Hă older inequality, we get (vs à )vs vs L2 (ΩkT ) L10 (ΩkT ) + (Hs · ∇)Hs + Hs L10 (ΩkT ) L2 (ΩkT ) + (Hs · ∇)vs L2 (ΩkT ) ∇vs L (ΩkT ) + ∇Hs ≤ A26 Combining that with (2.5) yields From Lemma 3.2 it follows that vs V21 (ΩkT ) + Hs V21 (ΩkT ) vs L10 (ΩkT ) + Hs L10 (ΩkT ) L (ΩkT ) ≤ c(Ω)A6 By (2.6), we have ∇vs L (ΩkT ) + ∇Hs L (ΩkT ) ≤ c1 + c2 vs − 12 W22,1 (ΩkT ) vs + Hs L2 (ΩkT ) W22,1 (ΩkT ) + Hs L2 (ΩkT ) Eventually (vs · ∇)vs L2 (ΩkT ) + (Hs · ∇)Hs L2 (ΩkT ) + (vs · ∇)Hs L2 (ΩkT ) + (Hs · ∇)vs L2 (ΩkT ) ≤ c(Ω)A26 +A3 A6 The above estimate with the estimates from Lemmas 3.1, 3.2 along with Lemma 2.1 ends this remark Proof of Theorem The proof is straightforward and follows from the energy estimates (see Lemmas 3.1, 3.2) and the Galerkin method As the basis functions, we can take the eigenfunctions of the Laplacian equipped with the Navier boundary conditions (cf Sections 2.3 and 3.2 in [6] and Section in [23]) Stability of solutions to (1.10) In this section, we examine the stability of solutions to (1.10) in the class of solutions to (1.1) + (1.2) + (1.3) + (1.4) The key point is the analysis of solutions to (1.11) Lemma 4.1 Let the assumptions of Lemma 3.2 hold In addition suppose that T > is so large that ν ¯T ≤ 12 , where A6 was introduced in (3.4) T ≥ 2c(Ω) ν ¯ A6 and exp − Let g ∈ L2 (kT, (k + 1)T ; L 65 (Ω)) satisfy (k+1)T sup k∈N0 g(t) kT L (Ω) dt ≤ B12 Suppose that B12 so small that c(Ω) B1 exp ν¯ c(Ω) A6 ν¯ ≤ γ 142 B Nowakowski and W M Zaja˛ czkowski Page 12 of 22 If u(0), K(0) L2 (Ω) • ≤ γ, then u(kT ), K(kT ) • ZAMP L2 (Ω) ≤ γ, u(t), K(t) sup kT ≤t≤(k+1)T L2 (Ω) c(Ω) A6 ν¯ ≤ γ exp ≡ γB22 (4.1) (k+1)T • rot u(τ ), rot K(τ ) L2 (Ω) dτ ≤ γ kT c(Ω) γ A6 A26 + + γ ≡ γB32 ν¯ c(Ω) exp ν¯ for k ∈ N0 Proof We multiply (1.11)1,3 by u and K, respectively, integrate the result over Ω and use the boundary conditions (1.11)5,6 d u, K dt L2 (Ω) + ν rot u + μ rot K L2 (Ω) =− (u · ∇)u · udx − Ω (K · ∇)K · udx + + L2 (Ω) Ω (K · ∇)Hs · udx + Ω (u · ∇)vs · udx − Ω (Hs · ∇)K · u dx + Ω g · udx − Ω (vs · ∇)u · udx Ω (u · ∇)K · Kdx Ω 13 (u · ∇)Hs · Kdx − − Ω (vs · ∇)K · Kdx + Ω (K · ∇)u · K + Ω (K · ∇)vs · Kdx + Ω (Hs · ∇)u · Kdx =: Jk k=1 Ω We easily note that J1 , J3 , J8 , J10 vanish We also have J4 = −J11 and J6 = −J13 By the Hă older and Young inequalities d u, K dt Choosing + + 2, d u, K dt L2 (Ω) + ν¯ rot u, rot K u u L6 (Ω) + K K 5, L2 (Ω) L6 (Ω) 7, + and 12 L2 (Ω) ≤ 2 u L2 (Ω) ∇vs L3 (Ω) 2 u L6 (Ω) + g L (Ω) 2 K L6 (Ω) + K L2 (Ω) ∇vs 12 L6 (Ω) u L2 (Ω) ∇Hs L3 (Ω) + L2 (Ω) ∇Hs L3 (Ω) + 12 + sufficiently small and using Lemma 2.1 to estimate L6 -norms, we get + ν¯ rot u, rot K L2 (Ω) ≤ c(Ω) u, K ν¯ L2 (Ω) ∇vs , ∇Hs L3 (Ω) + c(Ω) g ν¯ Applying Lemma 2.1 for the second term on the left-hand side, we obtain ⎛ d ⎝ u, K dt L3 (Ω) ⎛ ⎞⎞ t c(Ω) 2 ⎝ ¯c(Ω)t − ∇vs (τ ), ∇Hs (τ ) L3 (Ω) dτ ⎠⎠ L2 (Ω) exp ν ν¯ kT ⎛ ⎞ t c(Ω) c(Ω) 2 ≤ g L (Ω) exp ⎝ν¯c(Ω)t − ∇vs (τ ), ∇Hs (τ ) L3 (Ω) dτ ⎠ ν¯ ν¯ kT L (Ω) (4.2) ZAMP Global regular solutions to magnetohydrodynamics Page 13 of 22 142 Integration with respect to time from t = kT to t ∈ (kT, (k + 1)T ) yields L2 (Ω) u(t), K(t) c(Ω) ≤ ν¯ ⎛ · exp ⎝−¯ ν c(Ω)t + ⎛ t L (Ω) g(τ ) kT t c(Ω) ν¯ L3 (Ω) ∇vs (τ ), ∇Hs (τ ) kT ⎞ τ c(Ω) exp ⎝ν¯c(Ω)τ − ∇vs (t ) , ∇Hs (t ) ν¯ kT ⎞ L3 (Ω) dt ⎠ dτ dτ ⎠ ⎛ c(Ω) + u(kT ), K(kT ) L2 (Ω) exp ⎝−¯ ν c(Ω)(t − kT ) + ν¯ ⎞ t ∇vs (τ ), ∇Hs (τ ) L3 (Ω) dτ ⎠ (4.3) kT By the Hăolder inequality, Sobolev embedding and Lemmas 2.2 and 3.2, we have t ∇vs (τ ), ∇Hs (τ ) sup k∈N0 kT We take t = (k + 1)T and use that T ≥ u((k + 1)T ), K((k + 1)T ) L2 (Ω) ≤ L3 (Ω) dτ ≤ c(Ω)A26 , t ∈ (kT, (k + 1)T ) 2c(Ω) ν ¯ A6 c(Ω) B1 exp ν¯ c(Ω) A6 ν¯ + exp − ν¯T u(kT ), K(kT ) L2 (Ω) Iterating the above inequality, we get L2 (Ω) u((k + 1)T ), K((k + 1)T ) ≤ c(Ω) ν ¯ B1 c(Ω) ν ¯ A6 exp − ν¯2T − exp +exp − ν¯kT u(0), K(0) L2 (Ω) ≤ γ γ + = γ, 2 which proves (4.1)1 Next, we integrate (4.2) with respect to time from t = kT to t = (k + 1)T (k+1)T (k+1)T L2 (Ω) rot u(τ ), rot K(τ ) kT c(Ω) ≤ ν¯ L2 (Ω) u(τ ), K(τ ) ∇vs (τ ), ∇Hs (τ ) L3 (Ω) dτ kT (k+1)T + c(Ω) ν¯ g(τ ) L (Ω) dτ + u(kT ), K(kT ) kT L2 (Ω) (4.4) From (4.3) and (4.1) it follows that sup kT ≤t≤(k+1)T u(t), K(t) L2 (Ω) ≤ c(Ω) B1 exp ν¯ c(Ω) A6 ν¯ + γ exp c(Ω) A6 ν¯ which proves (4.1)2 Using the above inequality in (4.4) ends the proof ≤ 2γ exp c(Ω) A6 , ν¯ 142 B Nowakowski and W M Zaja˛ czkowski Page 14 of 22 ZAMP Lemma 4.2 Let the assumptions of Lemma 4.1 hold Suppose that g ∈ L2 (ΩkT ) If rot u(0), rot K(0) ≤ γ and γ is sufficiently small, then • • rot u(kT ), rot K(kT ) sup L2 (Ω) L2 (Ω) ≤ γ, rot u(t), rot K(t) kT ≤t≤(k+1)T L2 (Ω) ≤ c(Ω) B1 exp A26 + γ exp A26 ≡ γB42 , ν¯ (k+1)T • L2 (Ω) rot u(t), rot K(t) kT dt ≤ c(Ω) c(Ω) c(Ω) γB42 exp A26 + B1 + γ ≡ γB52 γ B4 γB3 + ν¯3 ν¯ ν¯ (4.5) for k ∈ N0 Proof Multiplying (1.11)1,3 by rot2 u and rot2 K, respectively, integrating the result over Ω and using the boundary conditions (1.11)5,6 , we get d rot u, rot K dt L2 (Ω) L2 (Ω) + ν¯ rot2 u, rot2 K ≤ (−(u · ∇)u − (u · ∇)vs − (vs · ∇)u Ω + (K · ∇)K + (K · ∇)Hs + (Hs · ∇)K) · rot2 udx + g · rot2 u dx − Ω ((u · ∇)K − (u · ∇)Hs Ω − (vs · ∇)K + (K · ∇)u + (K · ∇)vs + (Hs · ∇)u) · rot2 K dx ≡ I1 + I2 + I3 (4.6) By Lemma 2.5, the Hă older and Young inequalities, we have I11 ≤ u L∞ (Ω) ∇u L2 (Ω) rot2 u L2 (Ω) ≤ I12 ≤ rot2 u u L2 (Ω) L6 (Ω) ∇vs L3 (Ω) 11 I13 ≤ rot u I14 ≤ K L2 (Ω) L∞ (Ω) ∇u ∇K L2 (Ω) L2 (Ω) vs rot2 u L∞ (Ω) L2 (Ω) rot2 u L2 (Ω) 12 rot2 u ≤ c(Ω) rot u rot2 u L2 (Ω) +c ≤ 14 rot2 u L2 (Ω) +c ≤ L2 (Ω) rot u 14 +c 14 ,Ω 12 ∇u 13 ,Ω L6 (Ω) +c ≤ 11 u L2 (Ω) L2 (Ω) +c L2 (Ω) L2 (Ω) rot u L2 (Ω) ≤ c(Ω) rot2 u ≤ ≤ c(Ω) rot2 u vs ,Ω L2 (Ω) 13 14 rot u vs K , and I15 ≤ rot2 u L2 (Ω) K L6 (Ω) ∇Hs L3 (Ω) , where we proceed as in I12 , thus I15 ≤ 15 rot2 u L2 (Ω) +c 15 ,Ω rot K L2 (Ω) Hs L2 (Ω) vs H (Ω) vs L2 (Ω) H (Ω) L2 (Ω) L2 (Ω) ,Ω , vs L2 (Ω) rot u L2 (Ω) L2 (Ω) H (Ω) ,Ω + rot2 K rot K rot u K H (Ω) H (Ω) vs H (Ω) H (Ω) , ZAMP Global regular solutions to magnetohydrodynamics Page 15 of 22 142 Next I16 ≤ rot2 u L2 (Ω) ∇K Hs L2 (Ω) L∞ (Ω) , which we estimate analogously to I13 , therefore I16 ≤ rot2 u 16 L2 (Ω) +c 16 ,Ω For I2 , we have I2 ≤ 2 L2 (Ω) rot2 u + L2 (Ω) rot K g 2 L2 (Ω) H (Ω) Hs Finally, for terms in I3 , we have I31 ≤ u L∞ (Ω) ∇K L2 (Ω) L2 (Ω) rot2 K rot2 K L2 (Ω) ≤ 31 ≤ 31 rot2 K L2 (Ω) + rot2 u L2 (Ω) +c ≤ 31 rot2 K L2 (Ω) + rot2 u L2 (Ω) +c +c 31 ,Ω rot K u ,Ω 31 H (Ω) rot K L2 (Ω) L2 (Ω) rot u L2 (Ω) rot K ,Ω 31 u H (Ω) + rot u L2 (Ω) For I32 , we repeat the estimate we derived for I12 I32 ≤ 32 rot2 K L2 (Ω) +c 32 ,Ω rot u L2 (Ω) Hs H (Ω) vs H (Ω) The term I33 can be estimated in the same way as I13 I33 ≤ 33 ∇u L2 (Ω) rot2 K L2 (Ω) +c 33 ,Ω rot K L2 (Ω) For I34 , we have I34 ≤ rot2 K L2 (Ω) K L∞ (Ω) ≤ c(Ω) rot2 K ≤ 34 rot2 K ≤ 34 rot2 K L2 (Ω) L2 (Ω) L2 (Ω) rot u +c +c 34 34 K L2 (Ω) ,Ω H (Ω) rot u L2 (Ω) rot u ,Ω K L2 (Ω) H (Ω) rot K L2 (Ω) + rot K L2 (Ω) The last two terms in I3 are very similar to I12 and I16 , respectively, thus I35 ≤ 35 rot2 K L2 (Ω) +c ,Ω rot K ,Ω rot u 35 L2 (Ω) vs H (Ω) Hs H (Ω) and I36 ≤ Finally, if all i 36 rot2 K L2 (Ω) +c 36 L2 (Ω) are sufficiently small, we obtain d rot u, rot K dt L2 (Ω) + ν¯ rot2 u, rot2 K L2 (Ω) ≤ c(Ω) rot u, rot K L2 (Ω) ν¯3 c(Ω) rot u, rot K L2 (Ω) vs , Hs + ν¯ c(Ω) g L2 (Ω) , + ν¯ H (Ω) (4.7) 142 B Nowakowski and W M Zaja˛ czkowski Page 16 of 22 ZAMP By Lemma 2.2 the above inequality implies d rot u, rot K dt L2 (Ω) + ν¯c(Ω) rot u, rot K which is equivalent to ⎛ d ⎝ rot u, rot K dt L2 (Ω) ⎛ L2 (Ω) c(Ω) ≤ rot u, rot K ν¯ exp ⎝t¯ ν c(Ω) − L2 (Ω) vs (τ ), Hs (τ ) (4.8) dτ ⎠⎠ ⎞ t ν c(Ω) − exp ⎝t¯ H (Ω) vs (τ ), Hs (τ ) kT dτ ⎠ ⎞ t ν c(Ω) − exp ⎝t¯ H (Ω) ⎞⎞ H (Ω) kT ⎛ L2 (Ω) c(Ω) rot u, rot K L2 (Ω) ν¯3 c(Ω) rot u, rot K L2 (Ω) vs , Hs + ν¯ c(Ω) g L2 (Ω) , + ν¯ t ⎛ c(Ω) + g ν¯ ≤ vs (τ ), Hs (τ ) H (Ω) dτ ⎠ kT After integrating with respect to time from kT to t ∈ (kT, ((k + 1)T ), we obtain L2 (Ω) rot u(t), rot K(t) ⎛ ≤ τ · exp ⎝τ ν¯c(Ω) − kT c(Ω) g(τ ) ν¯ kT ⎛ H (Ω) vs (t ) , Hs (t ) L2 (Ω) exp ⎝τ ν¯c(Ω) − kT vs (t), Hs (t) kT kT ⎞ t ν c(Ω) + dt ⎠ dτ exp ⎝−t¯ vs (t), Hs (t) H (Ω) dt⎠ ⎞ H (Ω) L2 (Ω) H (Ω) vs (t ) , Hs (t ) dt ⎠ dτ ⎞ dt⎠ ⎛ + rot u(kT ), rot K(kT ) L2 (Ω) rot u(τ ), rot K(τ ) τ t · exp ⎝−t¯ ν c(Ω) + L2 (Ω) kT ⎛ t + t c(Ω) sup rot u(τ ), rot K(τ ) ν¯3 kT ≤τ ≤t ⎞ ⎛ ⎞ t exp ⎝−(t − kT )¯ ν c(Ω) + vs (t), Hs (t) H (Ω) dt⎠ (4.9) kT Using the assumptions, Lemmas 3.2 and 4.1, we get rot u(t), rot K(t) L2 (Ω) ≤ c(Ω) sup rot u(t), rot K(t) ν¯3 kT ≤t≤(k+1)T + L2 (Ω) γB32 exp A26 c(Ω) B1 exp A26 + rot u(kT ), rot K(kT ) ν¯ L2 (Ω) exp A26 , which for sufficiently small γ implies sup kT ≤t≤(k+1)T rot u(t), rot K(t) L2 (Ω) ≤ c(Ω) B1 exp A26 + rot u(kT ), rot K(kT ) ν¯ L2 (Ω) exp A26 (4.10) ZAMP Global regular solutions to magnetohydrodynamics Page 17 of 22 142 Next, we take t = (k + 1)T in (4.9), use the above inequality and the assumptions L2 (Ω) rot u((k + 1)T ), rot K((k + 1)T ) c(Ω) γ B4 γB3 exp A26 + B12 exp A26 + rot u(kT ), rot K(kT ) ν¯3 ≤ L2 (Ω) exp − ν¯T Iterating the above inequality, we get rot u((k + 1)T ), rot K((k + 1)T ) c(Ω) ν ¯3 γ B4 γB3 ≤ L2 (Ω) exp A26 + B12 exp A26 + rot u(0), rot K(0) − ν¯2T − exp L2 (Ω) exp − ν¯kT ≤ γ γ + , 2 which along with (4.10) proves (4.5)1,2 Finally, we integrate (4.7) with respect to t ∈ [kT, (k + 1)T ] We obtain (k+1)T (k+1)T rot2 u(t), rot2 K(t) L2 (Ω) kT c(Ω) dt ≤ ν¯ L2 (Ω) rot u(t), rot K(t) (k+1)T c(Ω) + ν¯ dt kT (k+1)T L2 (Ω) rot u(t), rot K(t) c(Ω) dt + ν¯ H (Ω) vs (t), Hs (t) kT + rot u(kT ), rot K(kT ) L2 (Ω) g(t) L2 (Ω) dt kT From (4.1)3 , (4.5)2 and the assumptions, we get (k+1)T rot2 u(t), rot2 K(t) L2 (ΩkT ) c(Ω) c(Ω) c(Ω) γB42 exp A26 + B1 + γ γ B4 γB3 + ν¯3 ν¯ ν¯ dt ≤ kT This ends the proof Remark 4.3 If we assume that (u, K) is a solution to (1.11) and the assumptions of Lemma 4.2 hold, then we can easily show + K W 2,1 (ΩkT ) + ∇σ L2 (ΩkT ) √ √ √ √ √ ≤ γc(Ω) (B4 + B5 ) ( γc(Ω) (B4 + B5 ) B3 + A7 ) + c2 γB3 A27 + B1 + γ ≡ γB6 u W22,1 (ΩkT ) (4.11) Indeed, using Lemmas 2.3 and 2.4, we have u W22,1 (ΩkT ) + K W22,1 (ΩkT ) + ∇σ L2 (ΩkT ) ≤ (u · ∇)u L2 (ΩkT ) + (vs · ∇)u L2 (ΩkT ) + (K · ∇)K L2 (ΩkT ) + (K · ∇)Hs + (u · ∇)K L2 (ΩkT ) + (u · ∇)Hs L2 (ΩkT ) + (vs · ∇)K + (K · ∇)vs L2 (ΩkT ) + (Hs · ∇)u L2 (ΩkT ) + u(kT ) + (u · ∇)vs L2 (ΩkT ) L2 (ΩkT ) H (Ω) L2 (ΩkT ) + (Hs · ∇)K + (K · ∇)u + K(kT ) H (Ω) L2 (ΩkT ) + g L2 (ΩkT ) L2 (ΩkT ) (4.12) By the Hăolder inequality, we get u à 2,1 W2 (kT ) + K ∇u L (ΩkT ) 2,1 W2 (ΩkT ) + ∇σ + ∇K · ∇u L (ΩkT ) L (ΩkT ) L2 (ΩkT ) + ∇vs + ∇K L (ΩkT ) ≤ u L (ΩkT ) L10 (ΩkT ) + ∇Hs + g L2 (ΩkT ) + u(kT ) + K L10 (ΩkT ) L (ΩkT ) + vs H (Ω) + K(kT ) L10 (ΩkT ) H (Ω) + Hs L10 (ΩkT ) (4.13) 142 B Nowakowski and W M Zaja˛ czkowski Page 18 of 22 ZAMP From (2.5), Lemma 4.2 and Remark 3.3, we infer that u ≤ L10 (ΩkT ) + K ∇u L10 (ΩkT ) √ γc(Ω) (B4 + B5 ) L (ΩkT ) + ∇K ∇u L (ΩkT ) + ∇vs L (ΩkT ) 2 + ∇K L (ΩkT ) L (ΩkT ) + ∇Hs L (ΩkT ) + A7 Similarly, by (2.6), Lemmas 4.1 and 2.1 we have ∇u ≤ c1 u (ΩkT ) + K √ u W 2,1 (ΩkT ) + K W 2,1 (ΩkT ) + c2 −1 γB3 2 L (ΩkT ) + ∇K L (ΩkT ) ≤ c1 2,1 W2 2,1 W2 (ΩkT ) + c2 −1 u L2 (ΩkT ) + K L2 (ΩkT ) Using the above inequalities in (4.13) yields u + ∇σ L2 (ΩkT ) √ √ √ γc(Ω) (B4 + B5 ) ( γc(Ω) (B4 + B5 ) B3 + A7 ) + c2 γB3 A27 + B1 + γ W22,1 (ΩkT ) ≤ √ + K W22,1 (ΩkT ) Proof of Theorem The proof of the existence of solutions to (1.11) is based on the Leray–Schauder fixed point theorem We follow the idea from [24, Sect 10] First, we rewrite (1.11) in the following form ut − νΔu + ∇σ = λ − (u · ∇) u − (u · ∇) vs − (vs · ∇)u in ΩkT , + K · ∇ K + K · ∇ Hs + (Hs · ∇)K + g ≡ w1 + g in ΩkT , div u = Kt − μΔK = λ − (u · ∇) K − (u · ∇) Hs − (vs · ∇)K in ΩkT , + (K · ∇)u + K · ∇ vs + (Hs · ∇)u ≡ w2 (4.14) in ΩkT , div K = n × rot u = 0, n·u=0 on S kT , n × rot K = 0, n·K=0 on S kT , u|t=kT = u(kT ), K|t=kT = K(kT ) in Ω This way, we introduce a mapping Φ : M × M × [0, 1] → M × M, Φ(u, K, λ) = (u, K), where we define M= z : ΩkT → R3 : z L 20 (ΩkT ) < ∞, ∇z L 20 (ΩkT ) 0, is compact and continuous Φ(u, K, ·) is uniformly continuous, there exists a bounded subset A × A ⊂ M × M such that any fixed point of Φ(·, ·, λ) for some λ ∈ [0, 1] belongs to A × A then Φ(·, ·, 1) will have at least one fixed point Ad (1) This property follows immediately from Lemmas 2.3 and 2.4 Ad (2) The embedding W22,1 (ΩkT ) → M is compact and by Lemmas 2.3 and 2.4, we have u, K M ≤ c(Ω) u, K W22,1 (ΩkT ) ≤ c(Ω) L2 (ΩkT ) w1 + w2 L2 (ΩkT ) + g L2 (ΩkT ) + u(kT ) H (Ω) + K(kT ) H (Ω) ZAMP Global regular solutions to magnetohydrodynamics Using the Hă older inequality (a à )b w1 · L2 (ΩkT ) vs + w2 L 20 (ΩkT ) L2 (ΩkT ) + Hs ≤ u L2 (ΩkT ) ≤ a L 20 (ΩkT ) + ∇vs L 20 (ΩkT ) L 20 (ΩkT ) , + K L 20 (ΩkT ) ∇b L 20 (ΩkT ) + ∇u we obtain L 20 (ΩkT ) + ∇K + ∇Hs 142 L 20 (ΩkT ) Page 19 of 22 L 20 (ΩkT ) ≤ c(Ω) u, K L 20 (ΩkT ) M , where the last inequality follows from Lemma 3.2 and the embedding theorem Thus, u, K M ≤ c(Ω) u, K M + g L2 (ΩkT ) + u(kT ) H (Ω) + K(kT ) H (Ω) This justifies the compactness of Φ To prove the continuity of Φ, we take two different sets of arguments of Φ, i.e Φ(u , K , λ) = (u1 , K1 ) and Φ(u2 , K , λ) = (u2 , K2 ) and consider the differences U = u1 − u2 , N = K1 − K2 , S = σ − σ Then, the triple (U, N, S) satisfies Ut − νΔU + ∇S = λ − U · ∇ u2 − u1 · ∇ U − U · ∇ vs − (vs · ∇)U in ΩkT , + N · ∇ K + K · ∇ N + N · ∇ Hs + (Hs · ∇)N in ΩkT , div U = Nt − μΔN = λ − U · ∇ K − u1 · ∇ N − U · ∇ Hs − (vs · ∇)N in ΩkT , + N · ∇ u + K · ∇)U + (N · ∇ vs + (Hs · ∇)U (4.15) in ΩkT , div N = n × rot U = 0, n·U=0 on S kT , n × rot N = 0, n·N=0 on S kT , U|t=kT = 0, N|t=kT = in Ω By Lemmas 2.3, 2.4 and the embedding W22,1 (ΩkT ) → M and Remarks 3.3 and 4.3, we have U, N M ≤ c(Ω) U L 20 (ΩkT ) + N L 20 (ΩkT ) · u1 L 20 (ΩkT ) + vs + ∇u2 √ L 20 (ΩkT ) + ∇U L 20 (ΩkT ) + ∇vs ≤ c(Ω) ( γB6 + A7 ) U, N M + ∇N + Hs L 20 (ΩkT ) L 20 (ΩkT ) L 20 (ΩkT ) + ∇Hs + K L 20 (ΩkT ) L 20 (ΩkT ) L 20 (ΩkT ) + ∇K L 20 (ΩkT ) , which justifies the continuity of Φ Ad (3) This property is evident Ad (4) We verified this condition in Remark 4.3 So far, we have the existence of at least one solution to (1.11) To prove its uniqueness let us assume that there exists another solution If we introduce the differences between these solutions (U, N, S) = (u1 , K1 , σ ) − (u2 , K2 , σ ), then the triple (U, N, S) will satisfy a system of equations which is analogous to (4.15) From energy estimates for that system, we have 142 B Nowakowski and W M Zaja˛ czkowski Page 20 of 22 d U, N dt L2 (Ω) + ν rot U Ω Ω Ω Ω Ω + ∇vs L3 (Ω) (N · ∇) u2 · N dx (U · ∇) Hs · Ndx + (N · ∇) vs · Ndx + Ω (Hs à ) U à Ndx By the Hăolder and Young inequalities, we obtain d 2 U, N L2 (Ω) + ν rot U L2 (Ω) + μ rot N dt L3 (Ω) (N · ∇) Hs · Udx Ω Ω Ω ∇u2 K1 · ∇ N · Udx + (U · ∇) K2 · Ndx − K · ∇ U · Ndx + · (U · ∇) u2 · Udx =− Ω + L2 (Ω) (N · ∇) K2 · Udx + (Hs · ∇) N · Udx − + + μ rot N Ω (U · ∇) vs · Udx + − L2 (Ω) ZAMP + ∇K2 L2 (Ω) L3 (Ω) ≤ c(Ω) U, N + K1 L∞ (Ω) L2 (Ω) + ∇Hs L3 (Ω) + Hs L∞ (Ω) Utilizing the Gronwall inequality, we get ⎛ t U(t), N(t) L2 (Ω) ≤ exp ⎝ ∇u2 L3 (Ω) kT + ∇Hs + ∇vs ⎞ L3 (Ω) + Hs L3 (Ω) + ∇K2 L3 (Ω) + K1 L∞ (Ω) ⎞ ⎠ dτ ⎠ L∞ (Ω) U(kT ), N(kT ) L2 (Ω) = 0, which implies U(t) = and N(t) = a.e This concludes the proof Proof of Theorem The proof follows immediately form Lemmas 3.2, 4.2 and Theorem Acknowledgements The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007–2013/ under REA Grant Agreement No 319012 and from the Funds for International Co-operation under Polish Ministry of Science and Higher Education Grant Agreement No 2853/7.PR/2013/2 The authors would like to express their gratitude to the referees for the valuable suggestions that helped to improve the paper Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made References [1] Nowakowski, B., Zaja˛ czkowski, W.M.: Stability of two-dimensional magnetohydrodynamic motions in the periodic case Math Methods Appl Sci 39(1), 44–61 (2016) doi:10.1002/mma.3459 ZAMP Global regular solutions to magnetohydrodynamics Page 21 of 22 142 [2] Zadrzy´ nska, E., Zaja˛ czkowski, W.M.: Stability of two-dimensional Navier–Stokes motions in the periodic case J Math Anal Appl 423(2), 956974 (2015) doi:10.1016/j.jmaa.2014.10.026 [3] Stră ohmer, G.: About an initial-boundary value problem from magnetohydrodynamics Math Z 209(3), 345362 (1992) doi:10.1007/BF02570840 [4] Stră ohmer, G.: About the equations of non-stationary magneto-hydrodynamics with non-conducting boundaries Nonlinear Anal Theory Methods Appl 39(5), 629–647 (2000) doi:10.1016/S0362-546X(98)00226-0 [5] Duvaut, G., Lions, J.-L.: In´equations en thermo´elasticit´e et magn´etohydrodynamique Arch Ration Mech Anal 46, 241–279 (1972) [6] Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations Commun Pure Appl Math 36(5), 635–664 (1983) doi:10.1002/cpa.3160360506 [7] Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion Adv Math 226(2), 1803–1822 (2011) doi:10.1016/j.aim.2010.08.017 [8] Cao, C., Regmi, D., Wu, J.: The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion J Differ Equations 254(7), 2661–2681 (2013) doi:10.1016/j.jde.2013.01.002 [9] Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global Cauchy problem of 2D generalized MHD equations Monatsh Math 175(1), 127–131 (2014) doi:10.1007/s00605-014-0652-0 [10] Huang, X., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system J Differ Equations 254(2), 511–527 (2013) doi:10.1016/j.jde.2012.08.029 [11] Jiu, Q., Zhao, J.: Global regularity of 2D generalized MHD equations with magnetic diffusion Z Angew Math Phys 66(3), 677–687 (2015) doi:10.1007/s00033-014-0415-8 [12] Kang, E., Lee, J.: Regularizing model for the 2D MHD equations with zero viscosity Abstr Appl Anal 2012, 2127861–212786-7 (2012) [13] Wei, Z., Zhu, W.: Global well-posedness of 2D generalized MHD equations with fractional diffusion J Inequal Appl 2013, 489 (2013) doi:10.1186/1029-242X-2013-489 [14] Zhou, Y., Fan, J.: Global Cauchy problem for a 2D Leray-α-MHD model with zero viscosity Nonlinear Anal 74(4), 1331–1335 (2011) doi:10.1016/j.na.2010.10.005 [15] Solonnikov, V.A.: Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations Trudy Mat Inst Steklov 70, 213–317 (1964) [16] Mucha, P.: Stability of nontrivial solutions of the Navier–Stokes system on the three-dimensional torus J Differ Equations 172(2), 359–375 (2001) doi:10.1006/jdeq.2000.3863 [17] Zaja˛ czkowski, W.: On global regular solutions to the Navier–Stokes equations in cylindrical domains Topol Methods Nonlinear Anal 37(1), 55–85 (2011) [18] Nowakowski, B.: Global existence of strong solutions to micropolar equations in cylindrical domains Math Methods Appl Sci 38(2), 311–329 (2015) doi:10.1002/mma.3070 [19] Solonnikov, V.: Overdetermined elliptic boundary-value problems J Math Sci 1(4), 477–512 (1973) [translation from: Zap Nauˇ cn Sem Leningrad Otdel Mat Inst Steklov (LOMI) 38 (1973), 153231 (Russian)] [20] Nowakowski, B., Stră ohmer, G., Zaja˛ czkowski, W.M.: Large time existence of special strong solutions to MHD equations in cylindrical domains J Math Fluid Mech (2015) (submitted) [21] Cholewa, J., Dlotko, T.: Global Attractors in Abstract Parabolic Problems Cambridge University Press, Cambridge (2000) [22] Zaja˛ czkowski, W.: Global special regular solutions to the Navier–Stokes equations in a cylindrical domain without the axis of symmetry Topol Methods Nonlinear Anal 24(1), 69–105 (2004) [23] Ukhovskii, M R., Iudovich, V I.: Axially symmetric flows of idealand viscous fluids filling the whole space J Appl Math Mech 32, 52–61 (1968) [translation from: Prikl Mat Meh 32 (1968), 59–69 (Russian)] [24] Nowakowski, B.: Large time existence of strong solutions to micropolar equations in cylindrical domains Nonlinear Anal Real World Appl 14(1), 635–660 (2013) doi:10.1016/j.nonrwa.2012.07.023 Bernard Nowakowski and Wojciech M Zaja˛ czkowski Institute of Mathematics Polish Academy of Sciences ´ Sniadeckich 00-656 Warsaw Poland e-mail: bernard@impan.pl Wojciech M Zaja˛ czkowski e-mail: wz@impan.pl 142 Page 22 of 22 B Nowakowski and W M Zaja˛ czkowski Wojciech M Zaja˛ czkowski Cybernetics Faculty, Institute of Mathematics and Cryptology Military University of Technology Kaliskiego 00-908 Warsaw Poland (Received: August 4, 2015; revised: October 10, 2016) ZAMP ... solutions to the Navier–Stokes equations in cylindrical domains Topol Methods Nonlinear Anal 37(1), 55–85 (2011) [18] Nowakowski, B.: Global existence of strong solutions to micropolar equations in. .. 3.13 in [20]) Let us consider the Stokes problem v,t − νΔv + ∇p = F in ΩT , div v = in ΩT , v·n=0 on S T , rot v × n = on S T , v|t=0 = v(0) on Ω ZAMP Global regular solutions to magnetohydrodynamics. .. of solutions to (1.10) in the class of solutions to (1.1) + (1.2) + (1.3) + (1.4) The key point is the analysis of solutions to (1.11) Lemma 4.1 Let the assumptions of Lemma 3.2 hold In addition