Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion

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1 J. Math. Fluid Mech. 17 (15), c 15 Springer Basel /15/467-1 DOI 1.17/s Journal of Mathematical Fluid Mechanics Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion Dongho Chae, Renhui Wan, and Jiahong Wu CommunicatedbyY.Giga Abstract. The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, ( Δ) α.due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need α 1 in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any α> 1. The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations. Mathematics Subject Classification. 35Q35, 35B65, 35Q85, 76W5. Keywords. Hall-MHD equations, fractional magnetic diffusion, local well-posedness. 1. Introduction This paper focuses on the Hall-magnetohydrodynamics (Hall-MHD) equations with fractional magnetic diffusion, t u + u u + p = B B, t B + u B + (( B) B)+( Δ) α B = B u, (1.1) u =, B =, u(x, ) = u (x), B(x, ) = B (x), where x R d with d, u = u(x, t) andb = B(x, t) are vector fields representing the velocity and the magnetic field, respectively, p = p(x, t) denotes the pressure, α > is a parameter and the fractional Laplacian ( Δ) α is defined through the Fourier transform, ( Δ) α f(ξ) = ξ α f(ξ). For notational convenience, we also use Λ for ( Δ) 1. The Hall-MHD equations with the usual Lapalcian dissipation were derived in [1] from kinetic models. The Hall-MHD equations differ from the standard incompressible MHD equations in the Hall term (( B) B), which is important in the study of magnetic reconnection (see, e.g., [8,13]). The Hall-MHD equations have been mathematically investigated in several works ([1, 4 7]). Global weak solutions of (1.1) with both Δu and ΔB and local classical solutions of (1.1) with ΔB (with or without Δu) were obtained in [4]. In addition, a blowup criterion and the global existence of small classical solutions were also established in [4]. These results were later sharpened by [5]. We examine the issue of whether or not (1.1) is locally well-posedness when the fractional power α<1. Previously local solutions of (1.1) were obtained for α = 1([4, 5]). Standard energy estimates appear to indicate that α 1 is necessary in order to obtain local bounds for the solutions in Sobolev spaces. This

2 68 D. Chae et al. JMFM requirement comes from the estimates of the regularity-demanding Hall term (( B) B). To understand more precisely the issue at hand, we perform a short energy estimate on the essential part of the equation for B, The global L -bound t B + (( B) B)+( Δ) α B =. t B(t) L + Λ α B(τ) L dτ = B L (1.) follows from the simple fact (( B) B) B = (( B) B) ( B) =. To obtain the H 1 -bound, we invoke the equation for B L, 1 d d dt B L + Λα B L = i (( B) B) i B. We can indeed shift one-derivative, namely i (( B) B) i B = (( B) i B) i B. i=1 Hölder s inequality allows us to conclude that 1 d dt B L + Λα B L B L B L B L. Therefore, it appears that we need α 1 in order to bound the term B L on the right-hand side. More generally, the energy inequality involving the H σ -norm 1 d dt B H + σ Λα B H C B σ H σ B L B H σ. also appears to demand that α 1 in order to bound B H σ. This paper obtains the local existence and uniqueness of solutions to (1.1) with any α> 1.More precisely, we prove the following theorem. Theorem 1.1. Consider (1.1) with α> 1.Assume(u,B ) H σ (R d ) with σ>1+ d, and u = B =. Then there exist T = T ( (u,b ) H σ) > and a unique solution (u, B) of (1.1) on [,T ] such that In addition, for any σ <σ, (u, B) L ([,T ]; H σ (R d )). (u, B) C([,T ]; H σ (R d )) and (u(t),b(t)) H σ is continuous from the right on [,T ). The essential idea of proving Theorem 1.1 is to fully exploit the dissipation in the equation for B and estimate the Sobolev norm (u, B) H σ via Besov space techniques. We identify H σ with the Besov space B, σ and suitably shift the derivatives in the nonlinear term. The definition of Besov spaces and related facts used in this paper are provided in the appendix. The rest of this paper is divided into two sections followed by an appendix. Section states and proves the result for the local a priori bound. Section 3 presents the complete proof of Theorem 1.1. The appendix supplies the definitions of the Littlewood Paley decomposition and Besov spaces.

3 Vol. 17 (15) Hall-MHD Equations 69. Local A Priori Bound This section establishes a local a priori bound for smooth solutions of (1.1), which is the key component in the proof of Theorem 1.1. The result for the local a priori bound can be stated as follows. Proposition.1. Consider (1.1) with α> 1. Assume the initial data (u,b ) H σ (R d ) with σ>1+ d. Let (u, B) be the corresponding solution. Then, there exists T = T ( (u,b ) H σ) > such that, for t [,T ], (u(t),b(t)) H σ C(α, T, (u,b ) H σ) and T Λ α B(s) H σ ds C(α, T, (u,b ) H σ). Proof of Proposition.1. The proof identifies the Sobolev space H σ with the Besov space B, σ and resorts to Besov space techniques. Let l Z be an integer and let Δ l denote the homogeneous frequency localized operator. Applying Δ l to (1.1) yields t Δ l u +Δ l (u u)+ Δ l p =Δ l (B B), t Δ l B +Δ l (u B)+Δ l (( B) B)+( Δ) α Δ l B =Δ l (B u). Taking the inner product with (Δ l u, Δ l B) and integrating by parts, we have 1 d ( Δl u L dt + Δ lb ) L + C αl Δ l B L K 1 + K + K 3 + K 4 + K 5, (.1) where K 1 = [Δ l,u ]u Δ l u, K = [Δ l,u ]B Δ l B, K 3 = [Δ l,b ]B Δ l u, K 4 = [Δ l,b ]u Δ l B, K 5 = Δ l (( B) B) Δ l B. Note that we have used the standard commutator notation, [Δ l,u ]u =Δ l (u u) u (Δ l u) and applied the lower bound, for a constant C >, Δ l B ( Δ) α Δ l B C αl Δ l B L. Using the notion of paraproducts, we write K 1 = K 11 + K 1 + K 13, where K 11 = (Δ l (S k 1 u Δ k u) S k 1 u Δ l Δ k u) Δ l u, K 1 = k l k l K 13 = (Δ l (Δ k u S k 1 u) Δ k u Δ l S k 1 u) Δ l u, ( Δ l (Δ k u Δ k u) Δ k u Δ l Δk u) Δ l u

4 63 D. Chae et al. JMFM with Δ k =Δ k 1 +Δ k +Δ k+1.byhölder s inequality and a standard commutator estimate, K 11 C S l 1 u L Δ l u L C u L Δ l u L k l k l Δ k u L Δ k u L. Since the summation over k for fixed l above consists of only a finite number of terms and, as we shall later in the proof, the norm generated by each term is a multiple of that generated by the typical term, it suffices to keep the typical term with k = l and ignore the summation. This would help keep our presentation concise. We will invoke this practice throughout the rest of the paper. By Hölder s inequality, K 1 is bounded by K 1 C u L Δ l u L. By Hölder s inequality and Bernstein s inequality, K 13 C Δ l u L u L l k Δ k u L. Therefore, K 1 C Δ l u L u L Δ l u L + l k Δ k u L. Similarly, K, K 3 and K 4 are bounded by K C u L Δ l B L + C B L Δ lu L Δ l B L + C u L Δ l B L l k Δ k B L, K 3 C B L Δ l u L ( Δ l B L + l k Δ k B L K 4 C B L Δ l u L Δ l B L + C u L Δ l B L + C u L Δ l B L l k Δ k B L. ), Using the cancelation property, B Δ l ( B) Δ l ( B) =, we can rewrite K 5 as K 5 = (Δ l (B ( B)) B (Δ l B)) Δ l B = [Δ l,b ]( B) Δ l ( B) =K 51 + K 5 + K 53,

5 Vol. 17 (15) Hall-MHD Equations 631 where K 51 = (Δ l (S k 1 B ( Δ k B)) S k 1 B ( Δ l Δ k B)) Δ l B, K 5 = k l k l K 53 = (Δ l (Δ k B ( S k 1 B)) Δ k B ( Δ l S k 1 B)) Δ l B, ( Δ l (Δ k B ( Δ k B)) Δ k B ( Δ l Δk B)) Δ l B. By Hölder s inequality and a standard commutator estimate, K 51 C S k 1 B L Δ k B L Δ l B L k l C l B L Δ l B L. By Hölder s inequality, K 5 C l B L Δ l B L. By Hölder s inequality and Bernstein s inequality, K 53 C l B L Δ l B L Δ k B L. Therefore, K 5 C l B L Δ l B L Δ l B L + Δ k B L. Inserting the estimates above in (.1), we obtain d ( Δl u L dt + Δ lb ) L + C αl Δ l B L C ( u, B) L ( Δ l u L + Δ lb L ) + C ( u, B) L l k Δ k u L + C l B L Δ l B L + C l B L Δ l B L + l k Δ k B L Δ k B L. Multiplying the inequality above by σl and summing over l Z, invoking the global bound for the L -norm of (u, B) and the equivalence of the norms f H σ f L + f Ḣ σ,

6 63 D. Chae et al. JMFM we have u(t) H + σ B(t) H + C σ t Λ α B(τ) H σ dτ t u H + B σ H + C ( u, B) σ L ( u(τ) + B(τ) ) dτ Ḣ σ Ḣ σ + C t (σ+1)l B L Δ l B L dτ + C t (σ+1)l B L Δ k B L dτ. (.) To derive the inequality above, we have used Young s inequality for series convolution σl l k Δ k u L = (σ+1)(l k) σk Δ k u L C σl Δ l u L C u. Ḣ σ We further bound the last two terms in (.), L 1 C t (σ+1)l B L Δ l B L dτ, L C t (σ+1)l B L Δ k B L Set θ =1 1 α.forα> 1, θ (, 1). By Hölder s inequality, t ( L 1 = C B L σl Δ l B ) θ ( ) (1 θ) L (σ+α)l Δ l B L dτ C C t t ( ) θ ( ) (1 θ) B L σl Δ l B L (σ+α)l Δ l B L dτ B 1 θ L B Ḣ σ dτ + C 4 t B(τ) Ḣ σ+α dτ. By Young s inequality for series convolution and an interpolation inequality, t L = C B L (l k)(σ+ 1 ) (σ+ 1 )k Δ k B L C C t t B L B dτ C Ḣ σ+ 1 B 1 θ L B Ḣ σ dτ + C 4 t t B L B θ B(τ) Ḣ σ+α dτ. Inserting the estimates above in (.) and invoking the embedding inequalities B L C B H σ for σ>1+ d, dτ. dτ B (1 θ) Ḣ σ Ḣ σ+α dτ

7 Vol. 17 (15) Hall-MHD Equations 633 we have t u(t) H + σ B(t) H + C σ B(τ) dτ Ḣ σ+α t u H + B σ H + C ( u(t) σ H σ + B(t) ) γ H dτ, (.3) σ for a constant γ>1. This inequality implies a local bound for u(t) H + σ B(t) Hσ, namely for some T = T ( (u,b ) H σ) > such that, for t [,T ], u(t) H σ + B(t) H σ C(u,B,α,T ) and T This completes the proof of Proposition.1. B(τ) Hσ+α dτ <. (.4) 3. Local Existence and Uniqueness This section proves Theorem 1.1. ProofofTheorem1.1. The local existence and uniqueness can be obtained through an approximation procedure. Here we use the Friedrichs method, a smoothing approach through filtering the high frequencies. For each positive integer n, we define Ĵ n f(ξ) =χ Bn (ξ) f(ξ), where B n denotes the closed ball of radius n centered at and χ Bn denotes the characteristic functions on B n. Denote Hn σ {f H σ (R d ), supp f } B n. We seek a solution (u, B) Hn σ satisfying t u + J n P(J n Pu J n Pu) =J n P(J n PB J n PB), t B + J n P(J n Pu J n PB)+J n P( (( J n PB) J n PB)) +( Δ) α J n PB = J n P(J n PB J n Pu), u(x, ) = (J n u )(x), B(x, ) = (J n B )(x), where P denotes the projection onto divergence-free vector fields. For each fixed n 1, it is not very hard, although tedious, to verify that the right-hand side of (3.1) satisfies the Lipschitz condition in Hn σ and, by Picard s theorem, (3.1) has a unique global (in time) solution. The uniqueness implies that J n Pu = u, J n PB = B and ensures the divergence-free conditions u = and B =. Then, (3.1) is simplified to { t u + J n P(u u) =J n P(B B), t B + J n P(u B)+J n P( (( B) B)) + ( Δ) α B = J n P(B u). We denote this solution by (u n,b n ). As in the proof of Proposition.1, we can show that (u n,b n ) satisfies t (u n,b n ) H σ (un,b n ) H + C σ (3.1) (u n (s),b n (s)) γ Hσ ds (3.)

8 634 D. Chae et al. JMFM for some γ>1. Due to (u n,b n ) H σ (u,b ) H σ, this inequality is uniform in n. This allows us to obtain a uniform local bound sup t [,T ] (u n (t),b n (t)) H σ M(α, T, (u,b ) H σ). (3.3) As in (.4), we also have the uniform local bound for the time integral T (Λ α B n )(s) H σ ds M(α, T, (u,b ) H σ). Furthermore, these uniform bounds allow us to show that (u n,b n ) (u m,b m ) L as n, m. (3.4) This is shown through standard energy estimates for (u n,b n ) (u m,b m ) L. The process involves many terms, but most of them can be handled in a standard fashion (see, e.g., [9, p. 17]). We provide the detailed energy estimate for the term that is special here, namely the Hall term (( B) B). In the process of the energy estimates, we need to bound the term ( (( B n ) B n ) (( B m ) B m )) (B n B m ) dx = ( (( (B n B m )) B n )) (B n B m ) dx + ( (( B m ) (B n B m )) (B n B m ) dx. (3.5) The first term on the right-hand side of (3.5) is zero, ( (( (B n B m )) B n )) (B n B m ) dx = (( (B n B m )) B n ) ( (B n B m )) dx =. For the second term on the right of (3.5), by the simple vector identity (( B m ) (B n B m )) =(B n B m ) ( B m ) ( B m ) (B n B m ), we have ( (( B m ) (B n B m )) (B n B m ) dx ( B m ) d L α B n B m d Bn B m L d α L wherewehaveused C Λ α B m H σ Bn B m L Λα (B n B m ) L, f L d α C Λ α f H σ, f L d d α C Λα f L. Putting together the estimates for all the terms, we obtain d dt Bn B m L C Λα B m H σ Bn B m L + C ( 1 n + 1 ). m Noticing that Λ α B m Hσ is time integrable, Gronwall s inequality yields the desired convergence (3.4). Let (u, B) be the limit. Due to the uniform bound (3.3), (u, B) H σ for t [,T ]. By the interpolation inequality, for any <σ <σ, f H σ C σ f 1 σ σ L f σ σ H σ,

9 Vol. 17 (15) Hall-MHD Equations 635 we further obtain the strong convergence (u n,b n ) (u, B) H σ as n and consequently, (u, B) C([,T ]; H σ ). This strong convergence makes it easy to check that (u, B) satisfies the Hall-MHD equation in (1.1). In addition, the time continuity in (u, B) C([,T ]; H σ ) allows to show the weak time continuity (u, B) C W ([,T ]; H σ ) or t (u(x, t),b(x, t)) φ(x) dx is continuous for any φ H σ. To show the right (in time) continuity of (u(t),b(t)) H σ, we make use of the energy inequality, for any t> t, t (u(t),b(t)) H (u( t),b( t)) σ H + C (u(s),b(s)) γ σ H ds, σ This inequality can be obtained in a similar fashion as (3.). Then, lim (u(t),b(t)) H σ (u( t),b( t)) H σ. t t+ By the weak continuity in time, (u( t),b( t)) H σ lim (u(t),b(t)) H σ. t t+ The desired right (in time) continuity of (u(t),b(t)) H σ then follows. This completes the proof of Theorem 1.1. t Appendix A. Besov Spaces This appendix provides the definitions of some of the functional spaces and related facts used in the previous sections. Materials presented in this appendix can be found in several books and many papers (see, e.g., [,3,1 1]). We start with several notations. S denotes the usual Schwartz class and S its dual, the space of tempered distributions. S denotes a subspace of S defined by { } S = φ S: φ(x) x γ dx =, γ =, 1,,... R d and S denotes its dual. S can be identified as S = S /S = S /P where P denotes the space of multinomials. To introduce the Littlewood Paley decomposition, we write for each j Z A j = { ξ R d : j 1 ξ < j+1}. The Littlewood Paley decomposition asserts the existence of a sequence of functions {Φ j } j Z Ssuch that supp Φ j A j, Φj (ξ) = Φ ( j ξ) or Φ j (x) = jd Φ ( j x), and j= Φ j (ξ) ={ 1, if ξ R d \{},, if ξ =. Therefore, for a general function ψ S,wehave Φ j (ξ) ψ(ξ) = ψ(ξ) for ξ R d \{}. j=

10 636 D. Chae et al. JMFM In addition, if ψ S, then That is, for ψ S, and hence j= Φ j (ξ) ψ(ξ) = ψ(ξ) for any ξ R d. j= j= Φ j ψ = ψ Φ j f = f, f S in the sense of weak- topology of S. For notational convenience, we define Δ j f =Φ j f, j Z. (A.1) Definition A.1. For s R and 1 p, q, the homogeneous Besov space Ḃs p,q consists of f S satisfying f Ḃs js Δ j f L p l q <. p,q We now choose Ψ S such that Then, for any ψ S, and hence Ψ(ξ) =1 Ψ ψ + Ψ f + Φ j (ξ), ξ R d. j= Φ j ψ = ψ j= Φ j f = f j= in S for any f S. To define the inhomogeneous Besov space, we set, if j, Δ j f = Ψ f, if j = 1, (A.) Φ j f, if j =, 1,,... Definition A.. The inhomogeneous Besov space Bp,q s with 1 p, q and s R consists of functions f S satisfying f B s p,q js Δ j f L p l q <. The Besov spaces Ḃs p,q and Bp,q s with s (, 1) and 1 p, q can be equivalently defined by the norms (Rd ( f(x + t) f(x) L p) q ) 1/q f Ḃs = dt, p,q t d+sq (Rd ( f(x + t) f(x) L p) q ) 1/q f B s p,q = f L p + t d+sq dt. When q =, the expressions are interpreted in the normal way. Many frequently used function spaces are special cases of Besov spaces. The following proposition lists some useful equivalence and embedding relations.

11 Vol. 17 (15) Hall-MHD Equations 637 Proposition A.3. For any s R, For any s R and 1 <q<, In particular, Ḃ q,min{q,} Lq Ḃ q,max{q,}. Ḣ s Ḃs,, H s B s,. Ḃ s q,min{q,} Ẇ s q Ḃs q,max{q,}. For notational convenience, we write Δ j for Δ j. There will be no confusion if we keep in mind that Δ j s associated with the homogeneous Besov spaces is defined in (A.1) while those associated with the inhomogeneous Besov spaces are defined in (A.). Besides the Fourier localization operators Δ j,the partial sum S j is also a useful notation. For an integer j, S j j 1 k= 1 where Δ k is given by (A.). For any f S, the Fourier transform of S j f is supported on the ball of radius j. Bernstein s inequalities are useful tools in dealing with Fourier localized functions and these inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives. Proposition A.4. Let α. Let1 p q. (1) If f satisfies supp f {ξ R d : ξ K j }, for some integer j and a constant K>, then ( Δ) α f L q (R d ) C 1 αj+jd( 1 p 1 q ) f L p (R ). d () If f satisfies supp f {ξ R d : K 1 j ξ K j } for some integer j and constants <K 1 K, then C 1 αj f L q (R d ) ( Δ) α f L q (R d ) C αj+jd( 1 p 1 q ) f L p (R ), d where C 1 and C are constants depending on α, p and q only. Δ k, Acknowledgements. Chae was partially supported by NRF Grant Nos and Wu was partially supported by NSF Grant DMS19153 and the AT&T Foundation at Oklahoma State University. Wan was partially supported by NSFC (No ). References [1] Acheritogaray, M., Degond, P., Frouvelle, A., Liu, J.-G.: Kinetic formulation and global existence for the Hall-Magnetohydrodynamics system. Kinet. Relat. Model. 4, (11) [] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg (11) [3] Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer-Verlag, Berlin-Heidelberg-New York (1976) [4] Chae, D., Degond, P., Liu, J.-G.: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, (14) [5] Chae, D., Lee, J.: On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. J. Differ. Equ. 56, (14) [6] Chae, D., Schonbek, M.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 55, (13)

12 638 D. Chae et al. JMFM [7] Chae D., Weng S.: Singularity formation for the incompressible Hall-MHD equations without resistivity. Ann. Inst. H. Poincaré Anal. Non Linéaire. doi:1.116/j.anihpc (15) [8] Homann, H., Grauer, R.: Bifurcation analysis of magnetic reconnection in Hall-MHD systems. Physica D 8, 59 7 (5) [9] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (1) [1] Miao, C., Wu, J., Zhang, Z.: Littlewood Paley Theory and Its Applications in Partial Differential Equations of Fluid Dynamics. Science Press, Beijing (1) (in Chinese) [11] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin, New York (1996) [1] Triebel, H.: Theory of Function Spaces II. Birkhauser Verlag, Basel (199) [13] Wardle, M.: Star formation and Hall effect. Astrophys. Space Sci. 9, (4) Dongho Chae Department of Mathematics College of Natural Science Chung-Ang University Seoul , Republic of Korea dchae@cau.ac.kr Renhui Wan School of Mathematics Zhejiang University Hanzhou 317, China rhwanmath@163.com; rhwanmath@zju.edu.cn; 135@zju.edu.cn Jiahong Wu Department of Mathematics Oklahoma State University Stillwater, OK 7478, USA jiahong@math.okstate.edu and Department of Mathematics Chung-Ang University Seoul , Republic of Korea (accepted: June 18, 15; published online: September 9, 15)