Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 54, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOW REGULARITY OF NON-L2(Rn) LOCAL SOLUTIONS TO GMHD-α SYSTEMS LORENZO RIVA, NATHAN PENNINGTON Abstract. The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell’s equations. Recently it has become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-α) system, which differs from the original MHD system by including an additional non- linear terms (indexed by α), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form −|ξ|γ/g(|ξ|). In [8], the problem was considered with initial data in the Sobolev space Hs,2(Rn) with n ≥ 3. Here we consider the problem with initial data in Hs,p(Rn) with n ≥ 3 and p > 2. Our goal is to minimizing the regularity required for obtaining uniqueness of a solution. 1. Introduction This article concerns the generalized Magneto-Hydrodynamic alpha (gMHD-α) system of equations, reported in its full generality as, ∂tv + (u · ∇)v + n∑ i=1 vi∇ui − ν1L1v + 1 2 ∇|B|2 = −∇p+ (B · ∇)B, (1.1) ∂tB + (u · ∇)B − (B · ∇)u− ν2L2B = 0, (1.2) v = (1− α2L3)u, (1.3) div u = divB = 0, (1.4) u(0, x) = u0(x), B(0, x) = B0(x), x ∈ Rn. (1.5) Since these equations govern the motion of fluids subject to a magnetic field, its terms have specific physical meaning: u the fluid velocity, B the magnetic field, and p the scalar-valued pressure of the fluid, ν1 > 0 is the fluid viscosity, ν2 > 0 the magnetic diffusion, and α > 0 a constant coming from varying the Hamiltonian that originally gave rise to the standard MHD equations (see [4]). Finally, the Li terms are Fourier multipliers with symbol −|ξ|γi/gi(|ξ|), where gi is a positive scalar function and γi > 0. The standard MHD system is the special case obtained when setting α = 0, g1 = g2 = 1, and γ1 = γ2 = 2, so that v = u and L1 = L2 = ∆. The existence of 2010 Mathematics Subject Classification. 35B65, 35A02, 76W05. Key words and phrases. Generalized MHD-α; local solution; low regularity. c©2020 Texas State University. Submitted July 31, 2019. Published May 28. 2020. 1 2 L. RIVA, N. PENNINGTON EJDE-2020/54 a global solution up to initial conditions is a classic result, at least in two dimen- sions. Unfortunately, the presence of nonlinear terms makes the MHD equations particularly complex to solve in arbitrary dimensions, and so a common strategy has been to study modified versions of them. One modification, termed Lagrangian Averaged MHD-α after the Lagrangian Averaged Navier-Stokes equation, is obtained from Equations (1.1)-(1.5) by setting Li = ∆ for i = 1, 2, 3 (γi = 2 and gi = 1). Linshiz and Titi proved the existence of a global solution for smooth initial data in three dimension [4]. Another version is obtained by setting α = 0 and g1 = g2 = 1 and leaving the γi’s unspecified. Zhao and Zhu used these generalized operators to guarantee a global solution to Equations (1.1)-(1.5) in the case of n = 3, provided that g1 = g2 = g3 = 1, γ1 = γ2 = n/2, and γ3 = 2 [14]. The first incorporation of a non-constant value for any gi appeared in [9], where Tao proved the existence of a unique global solution to the generalized Navier-Stokes equation (B0 = α = 0) when γ1 = n/2 + 1 and when g1 is a radial non-decreasing function bounded below satisfying∫ ∞ 1 ds sg1(s)4 =∞, (1.6) the prototypical example of which is essentially a logarithm. Wu obtained a similar result for the generalized MHD system in [12], specifically showing that there is a unique global solution provided that u0, B0 ∈ Hr,2(Rn) with r > n/2 + 1; γ1 ≥ n/2 + 1, γ2 > 0, and γ1 + γ2 ≥ 1; and g1, g2 are non-decreasing, bounded below by 1, and satisfy∫ ∞ 1 ds s(g1(s) + g2(s))2 =∞. (1.7) This work was ultimately extended to the gMHD-α system in [13], where Yamazaki obtained a unique global solution in three dimensions provided that γ1+γ2+γ3 ≥ 5, min{γ1, γ3} > γ2 > 0, γ3 + 2γ1 > 3, and the gi satisfy∫ ∞ 1 ds sg1(s)2g2(s)g3(s)2 =∞. (1.8) In [8], one of the authors considered a generalization of the equations in [14] with the incorporation of non-constant gi, i = 1, 2, 3, while still leaving L3 = ∆, and guaranteed a unique global solution. In this article we will extend those results for the case of γ3 6= 2 and non- constant g3. We will particularly focus on the case of low-regularity initial data in a non-L2(Rn) setting to then obtain, in the future, global Lp(Rn) solutions using an interpolation technique, the details of which can be found in [5] and [1]. The rest of this article is organized as follows. Section 2 is devoted to explaining the notation we will use and some supporting results necessary for the algorithm. Section 3 contains the main result (Theorem 3.1) of this paper and its proof. We end this section with two important spacial cases of Theorem 3.1. Theorem 1.1. Let g1, g2, g3 : [0,∞) → R be non-decreasing functions bounded below by 1, satisfying g (k) i (s) ≤ Cs−k (1.9) for i = 1, 2, 3 and 0 ≤ k ≤ n/2 + 1. Moreover, assume 0 ≤ γ3 ≤ 1 and p, q ≥ n with 2p > q. Then, for any divergence-free u0 ∈ Lp(Rn) and B0 ∈ Lq(Rn), there EJDE-2020/54 GMHD-α SYSTEMS 3 exists a unique local solution (u,B) to the generalized MHD-α system (1.1)-(1.5) provided that γ−1 > 6− γ3, γ−2 > 1 + n p . Condition (1.9) is a modification of the condition in the Mikhlin multiplier the- orem that is necessary for supporting estimates in Proposition (2.4) and (2.5). The functions that satisfy it are still essentially logarithms, the same type of functions that satisfy (1.6)-(1.8). Theorem 1.2. Let γ−3 − 1 ≤ n 2p ≤ γ−3 , n 2q − 1 + γ−3 ≤ n 2p , and let p, q ≥ n with q < 3p/2. Moreover, assume that g1, g2, g3 satisfy the inequality (1.9). Then for each divergence-free u0 ∈ Hn/2p,p(Rn) and B0 ∈ Hn/2q,q(Rn), there exists a unique local solution (u,B) to the generalized MHD system from Equations (1.1)-(1.5) provided that γ−1 > 6− γ−3 − n p , γ−2 > 1 + n 2p . Note that in the statement of Theorem 1.1 and 1.2, and in what follows, we use x− = x − ε for some positive ε, i.e. x− denotes a number arbitrarily close to, but strictly smaller than, x. 2. Notation and supporting facts We let Hr,p(Rn) be the usual Sobolev space, and we write ‖f‖r,p to mean ‖f‖Hr,p(Rn) and ‖f‖p for ‖f‖Lp(Rn). Because of the nature of the procedure we use, we require that the solutions live in an auxiliary continuous-in-time space CTa;r,p(Rn) defined by CTa;r,p(Rn) := { f ∈ C((0, T ), Hr,p(Rn)) : ‖f‖a;r,p <∞ } , where T > 0, a ≥ 0, C(X,Y ) is the space of continuous maps X → Y , and ‖f‖a;r,p := sup (0,T ) ta‖f(t)‖r,p. Finally, we denote by ĊTa;r,p(Rn) the subspace of CTa;r,p(Rn) consisting of functions f such that limt→0+ taf(t) = 0 and by BC(X,Y ) ⊂ C(X,Y ) the subspace of bounded continuous maps X → Y . The following are supporting propositions that we will use throughout this paper. The first proposition is a product estimate, its proof can be found in [10, Chapter 2]. Proposition 2.1. If r ≥ 0 and 1 < p ≤ ∞, then ‖fg‖r,p ≤ C ( ‖f‖p1‖g‖r,p2 + ‖f‖r,q1‖g‖q2 ) , where 1 p = 1 p1 + 1 p2 = 1 q1 + 1 q2 and p1, p2, q1, q2 ∈ [1,∞]. 4 L. RIVA, N. PENNINGTON EJDE-2020/54 The following is a useful Sobolev embedding which is a straightforward extension of a result from [11, Chapter 13]. Proposition 2.2. Let s ≥ r and (s− r)p < n. Then ‖f‖r,q ≤ C‖f‖s,p provided that 1 q − r n = 1 p − s n . Our next result follows from a simple calculus exercise. Proposition 2.3. If 0 < a, b ∈ R, then sup t∈[0,T ] ∫ t 0 (t− s)−as−b ds ≤ CT 1−a−b, provided that a+ b < 1. Our final two propositions consist of an estimate for the semigroup etLi analogous to similar results for the heat kernel et∆ and an estimate for the operator (1−Li)−1. The proofs of both propositions can be found in [7]. We recall that x− is a number arbitrarily close to, but strictly smaller than, x. Proposition 2.4. Let 1 < p1 ≤ p2 < ∞, r1 ≤ r2, g(x) be a non-decreasing function bounded below by 1, satisfying |g(k)(x)| ≤ C|x|−k for 1 ≤ k ≤ n/2 + 1. Then etLi : Hr1,p1(Rn)→ Hr2,p2(Rn) and ‖etLif‖r2,p2 ≤ t−(r2−r1+n/p1−n/p2)/γ−i ‖f‖r1,p1 . Note that this proposition necessitates the requirements on the gi’s. Proposition 2.5. Let 1 < p < ∞, r ∈ R, g(x) be a non-decreasing function bounded below by 1, satisfying |g(k)(x)| ≤ C|x|−k for all 1 ≤ k ≤ n/2 + 1. Then ‖(1− Li)−1f‖r,p ≤ C‖f‖r−γ−i ,p. 3. Main result and its proof In this section we state the most general form of the theorem and then proceed with its proof. Theorem 3.1. Let g1, g2, g3 : [0,∞) → R be non-decreasing functions bounded below by 1, satisfying g (k) i (s) ≤ Cs−k for i = 1, 2, 3 and 0 ≤ k ≤ n/2 + 1. Let r0, r1, r2 ≥ 0 and let p0, p1, p2 ≥ n with p0 ≤ p1 and p2 < 2p0. Moreover, assume that γ−3 − 1 ≤ r0 ≤ γ−3 ≤ r1, r2 − 1 + γ−3 ≤ r0, r2 ≤ r0 < n p1 , 2r1 ≥ max { 2, 1 + γ−3 − n p0 + 2n p1 } , r2 < min { n p2 , 2n p2 − n p0 } . EJDE-2020/54 GMHD-α SYSTEMS 5 Then, for any divergence-free u0 ∈ Hr0,p0(Rn) and B0 ∈ Hr2,p2(Rn), there exists a unique local solution (u,B) to the generalized MHD-α system (1.1)-(1.5) provided that γ−1 > 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 , γ−1 > 1− 2r2 + r1 − γ−3 − n p1 + 2n p2 , γ−2 > 1− r0 + n p0 . Note that Theorem 1.1 may be recovered by setting r0 = r2 = 0, r1 = 2, p := p0 = p1, and q := p2, while Theorem 1.2 may be recovered by setting r0 = n/(2p), r1 = 2, r2 = n/(2q), p := p0 = p1, and q := p2. Proof of Theorem 3.1. For the sake of clarity and to highlight some technical de- tails, the proof will be divided in subsections. We first write the generalized MHD-α system in a more helpful form. Without loss of generality, we set α = ν1 = ν2 = 1. We pass to divergence-free vector fields by applying the Hodge operator P to equa- tions (1.1) and (1.2) (more information about the Hodge operator can be found in [3, Chapter 11]), and then we apply (1 − L3)−1 to equation (1.1). By noting that P , (1− L3)−1, and ∂t all commute since they are Fourier multipliers, we obtain ∂tu+ P (1− L3)−1 ( (u · ∇)v + n∑ i=1 vi∇ui − (B · ∇)B ) − L1u = P ( −∇p− 1 2 ∇|B|2 ) = 0. An application of the divergence-free condition in (1.4) allows us to rewrite the terms of the form (x · ∇)y as div(x⊗ y). Note that x⊗ y is the matrix whose (i, j) entry is xiyj , so that the product estimate in Proposition 2.1 applies to x⊗ y. We then have the system ∂tu+ P (1− L3)−1 ( div(u⊗ v) + n∑ i=1 vi∇ui − div(B ⊗B) ) − L1u = 0, ∂tB + P ( div(u⊗B)− div(B ⊗ u) ) − L2B = 0, v = (1− L3)u, div u = divB = 0, u(x, 0) = u0(x), B(0, x) = B0(x), x ∈ Rn. An application of Duhamel’s principle shows that (u,B) is a solution to the sys- tem if and only if (u,B) is a fixed point of the map Φ(u,B) := (Φ1(u,B),Φ2(u,B)) defined by Φ1(u,B) := etL1u0 − ∫ t 0 e(t−s)L1 (W1(u, v) +W2(u, v)−W1(B,B)) ds, Φ2(u,B) := etL2B0 − ∫ t 0 e(t−s)L2 (W3(u,B)−W3(B, u)) ds, where W1(x, y) = P (1− L3)−1 div(x⊗ y), 6 L. RIVA, N. PENNINGTON EJDE-2020/54 W2(x, y) = P (1− L3)−1 ( n∑ i=1 yi∇xi ) , W3(x, y) = P div(x⊗ y). By the contraction mapping theorem, it suffices to show that Φ is a contraction on the space XT,M × YT,M , where XT,M := { f ∈ BC ( [0, T ), Hr0,p0(Rn) ) ∩ Ċa1,r1,p1(Rn) : sup (0,T ) ( ‖f(t)− etL1u0‖r0,p0 + ‖f(t)‖a1;r1,p1 ) < M } and YT,M := { f ∈ BC([0, T ), Hr2,p2(Rn)) : sup (0,T ) ‖f(t)− etL2B0‖r2,p2 < M } for some 0 < T < 1 and M > 0. Following the methods in [6, 2], we complete the proof by showing that I1 = sup (0,T ) ta1‖etL1u0‖r1,p1 < M/4, I2 = sup (0,T ) ∥∥∥∫ t 0 e(t−s)L1 (W1(u, v) +W2(u, v)−W1(B,B)) ds ∥∥∥ r0,p0 < M/4, I3 = sup (0,T ) ta1 ∥∥∥∫ t 0 e(t−s)L1 (W1(u, v) +W2(u, v)−W1(B,B)) ds ∥∥∥ r1,p1 < M/4, I4 = sup (0,T ) ∥∥∥∫ t 0 e(t−s)L2 (W3(u,B)−W3(B, u)) ds ∥∥∥ r2,p2 < M/4. We start with I1. If ϕ is in the Schwartz space, we have I1 = sup (0,T ) ta1‖etL1(u0 − ϕ+ ϕ)‖r1,p1 ≤ sup (0,T ) ta1‖etL1(u0 − ϕ)‖r1,p1 + sup (0,T ) ta1‖etL1ϕ‖r1,p1 ≤ sup (0,T ) ta1t−a1‖u0 − ϕ‖r0,p0 + sup (0,T ) ta1‖ϕ‖r1,p1 ≤ ‖u0 − ϕ‖r0,p0 + T a1‖ϕ‖r1,p1 , provided that (by Proposition 2.4) 0 ≤ a1 = r1 − r0 + n p0 − n p1 γ−1 < 1 and p0 ≤ p1. We can choose ϕ so that ‖u0 − ϕ‖r0,p0 is arbitrarily small, and then we can choose T small enough to reduce T a1‖ϕ‖r1,p1 so that the sum of the two is bounded by M/4. 3.1. I2 and I3. Minkowski’s inequality gives us I2 ≤ J1 + J2 + J3, I3 ≤ K1 +K2 +K3, EJDE-2020/54 GMHD-α SYSTEMS 7 where J1 := sup (0,T ) ∫ t 0 ‖e(t−s)L1W1(u, v)‖r0,p0ds, K1 := sup (0,T ) ta1 ∫ t 0 ‖e(t−s)L1W1(u, v)‖r1,p1ds, J2 := sup (0,T ) ∫ t 0 ‖e(t−s)L1W2(u, v)‖r0,p0ds, K2 := sup (0,T ) ta1 ∫ t 0 ‖e(t−s)L1W2(u, v)‖r1,p1ds, J3 := sup (0,T ) ∫ t 0 ‖e(t−s)L1W1(B,B)‖r0,p0ds, K3 := sup (0,T ) ta1 ∫ t 0 ‖e(t−s)L1W1(B,B)‖r1,p1ds. We will show that each term is bounded above by CM2T k for various values of k > 0, which, since T < 1, will imply I2, I3 < M/4 provided that CM2 < M/4. We begin our algorithm with J1 and K1, showing the details of the calculations and highlighting the choices of parameters. The argument for the other two pairs of integrals is very similar, and the details will be omitted. 3.2. J1 and K1. By Proposition 2.4, J1 is bounded by J1 ≤ sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 ‖W1(u, v)‖γ−3 −1,π1 ds provided that γ−3 − 1 ≤ r0 and where π1 is an intermediate parameter that will be specified later. Now we work towards bounding W1(u, v), and an application of Proposition 2.5 gives us ‖W1(u, v)‖γ−3 −1,π1 = ‖P (1− L3)−1 div(u⊗ v)‖γ−3 −1,π1 ≤ C‖u⊗ v‖π1 . We chose a regularity of γ−3 −1 when performing the semigroup estimate in order to end up with a product in zero regularity, so that we can apply Holder’s inequality: by Proposition 2.1, if 1 π1 = 1 p′ + 1 p1 , (3.1) we have ‖u⊗ v‖π1 ≤ C‖u‖p′‖v‖p1 ≤ C‖u‖p′‖u‖γ−3 ,p1 . Note that (3.1) specifies the required value of π1. We obtain ‖u‖p′ ≤ ‖u‖r0,p0 by Proposition 2.2 if r0 < n p0 and 1 p′ = 1 p0 − r0 n ; (3.2) we also get ‖u‖γ−3 ,p1 ≤ ‖u‖r1,p1 by requiring that r1 ≥ γ−3 . Combining the two bounds gives us ‖W1(u, v)‖γ−3 −1,π1 ≤ C‖u⊗ v‖π1 ≤ C‖u‖r0,p0‖u‖r1,p1 . (3.3) 8 L. RIVA, N. PENNINGTON EJDE-2020/54 Note that (3.1) and (3.2) give us 1 π1 = 1 p0 + 1 p1 − r0 n . With this new bound on W1(u, v), we come back to J1 and see that J1 ≤ sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 ‖W1(u, v)‖γ−3 −1,π1 ds ≤ C sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 ‖u‖r0,p0‖u‖r1,p1ds = C sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 s−a1‖u‖r0,p0sa1‖u‖r1,p1ds ≤ C‖u‖0;r0,p0‖u‖a1;r1,p1 sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 s−a1ds < CM2T 1−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 −a1 , where the last inequality holds by Proposition 2.3, if γ−1 > r0 − (γ−3 − 1) + n π1 − n p0 + γ1a1 = r0 − (γ−3 − 1) + n ( 1 p0 + 1 p1 − r0 n ) − n p0 + r1 − r0 + n p0 − n p1 = 1− r0 + r1 − γ−3 + n p0 . (3.4) We further note that the requirement in (3.4) also guarantees that the exponent on T is positive, as desired. We now turn our attention to K1. Proposition 2.4 guarantees that, if p1 ≥ π′1, then K1 ≤ sup (0,T ) ∫ t 0 (t− s)−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p1)/γ−1 ‖W1(u, v)‖r1−r0+γ−3 −1,π′1 ds. This time, we chose r1 − r0 + γ−3 − 1 in order to match the previous “jump” in regularity from r0 to γ−3 − 1. Propositions 2.1 and 2.5 give us ‖W1(u, v)‖r1−r0+γ−3 −1,π′1 = ‖P (1− L3)−1 div(u⊗ v)‖r1−r0+γ−3 −1,π′1 ≤ C‖u⊗ v‖r1−r0,π′1 ≤ C (‖u‖r1−r0,p′‖v‖p′′ + ‖v‖r1−r0,q′‖u‖q′′) , provided that 1 π′1 = 1 p′ + 1 p′′ = 1 q′ + 1 q′′ . Four applications of Proposition 2.2 lead us to the following bounds: ‖u‖r1−r0,p′ ≤ ‖u‖r1,p1 if r0 < n p1 and 1 p′ = 1 p1 − r0 n , (3.5) ‖v‖p′′ ≤ ‖v‖r0−γ−3 ,p0 if r0 < n p0 + γ−3 and 1 p′′ = 1 p0 − r0 − γ−3 n , (3.6) ‖v‖r1−r0,q′ ≤ ‖v‖r1−γ−3 ,p1 if r0 < n p0 + γ−3 and 1 q′ = 1 p1 − r0 − γ−3 n , (3.7) EJDE-2020/54 GMHD-α SYSTEMS 9 ‖u‖q′′ ≤ ‖u‖r0,p0 if r0 < n p0 and 1 q′′ = 1 p0 − r0 n . (3.8) Combining the parameters specified by (3.5)-(3.8), we obtain 1 π′1 = 1 p0 + 1 p1 − 2r0 − γ−3 n and ‖W1(u, v)‖r1−r0+γ−3 −1,π′1 ≤ C‖u⊗ v‖r1−r0,π′1 ≤ C‖u‖r0,p0‖u‖r1,p1 . Moreover, the integrability requirement from Proposition 2.4 necessitates 1 p1 ≤ 1 π′1 = 1 p0 + 1 p1 − 2r0 − γ−3 n , and so r0 ≤ 1 2 ( n p0 + γ−3 ) . (3.9) We can finally plug this bound into K1: K1 ≤ sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p1)/γ−1 × ‖W1(u, v)‖r1−r0+γ−3 −1,π′1 ds ≤ C sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p1)/γ−1 ‖u‖r0,p0‖u‖r1,p1ds = C sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p1)/γ−1 s−a1 × ‖u‖r0,p0sa1‖u‖r1,p1ds ≤ C‖u‖0;r0,p0‖u‖a1;r1,p1 sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p1)/γ−1 × s−a1ds < CM2T 1−(r1−(r1−r0+γ−3 −1)+n/π′1−n/p0)/γ−1 , where the last inequality follows by Proposition 2.3 if γ−1 > r1 − ( r1 − r0 + γ−3 − 1 ) + n π′1 − n p1 + γ−1 a1 = r0 − γ−3 + 1 + n ( 1 p0 + 1 p1 − 2r0 − γ−3 n ) − n p1 + r1 − r0 + n p0 − n p1 = 1− 2r0 + r1 + 2n p0 − n p1 . (3.10) Note that, once again, the requirement that Proposition 2.3 hold is sufficient to guarantee that the exponent on T be positive. To summarize, here is the list of inequalities needed to obtain the desired bounds on J1 and K1: r0 ≤ γ−3 (assumption), γ−3 − 1 ≤ r0 (semigroup estimate for J1), r1 − r0 + γ−3 − 1 ≤ r1 (semigroup estimate for K1) 10 L. RIVA, N. PENNINGTON EJDE-2020/54 r0 < n p0 (by (3.2)), r0 < n p1 (by (3.5)), r0 ≤ 1 2 ( n p0 + γ−3 ) (by (3.9)), r0 < n p0 + γ−3 (by (3.12)), r1 ≥ γ−3 (bound on ‖u‖γ−3 ,p1 in J1), γ−1 > 1− r0 + r1 − γ−3 + n p0 (by (3.4)), γ−1 > 1− 2r0 + r1 + 2n p0 − n p1 (by (3.10)). After some obvious simplifications and after noting that (3.10) implies (3.4) since( 1− 2r0 + r1 + 2n p0 − n p1 ) ︸ ︷︷ ︸ RHS of (3.10) − ( 1− r0 + r1 − γ−3 + n p0 ) ︸ ︷︷ ︸ RHS of (3.4) = γ−3 − r0 + n p0 − n p1 ≥ 0, the list reduces to γ−3 − 1 ≤ r0 ≤ γ−3 ≤ r1, r0 < n p1 , γ−1 > 1− 2r0 + r1 + 2n p0 − n p1 . 3.3. J2 and K2. We have J2 = sup (0,T ) ∫ t 0 ‖e(t−s)L1W2(u, v)‖r0,p0ds ≤ sup (0,T ) ∫ t 0 ‖W2(u, v)‖r0,p0ds. We now work towards bounding W2(u, v). We immediately see, thanks to Propo- sition 2.5, that ‖W2(u, v)‖r0,p0 = ∥∥P (1− L3)−1 n∑ i=1 vi∇ui ∥∥ r0,p0 ≤ C ∥∥ n∑ i=1 vi∇ui ∥∥ r0−γ−3 ,p0 ≤ C n∑ i=1 ‖vi∇ui‖p0 since r0 ≤ γ−3 . Now the product estimate is nothing more than Holder’s inequality, so if 1 p0 = 1 p′ + 1 p′′ EJDE-2020/54 GMHD-α SYSTEMS 11 we obtain ‖W2(u, v)‖r0,p0 ≤ C n∑ i=1 ‖vi‖p′‖∇ui‖p′′ ≤ C n∑ i=1 ‖v‖p′‖∇u‖p′′ ≤ C‖u‖γ−3 ,p′‖u‖1,p′′ . By Proposition 2.2 we have ‖u‖γ−3 ,p′ ≤ C‖u‖γ−3 +β,p1 ≤ C‖u‖r1,p1 and ‖u‖1,p′′ ≤ C‖u‖r1,p1 , where the first set of inequalities requires that 0 ≤ β < n p1 , 1 p′ = 1 p1 − β n , r1 ≥ γ−3 + β, (3.11) and the second inequality requires that 1 p′′ = 1 p1 − r1 − 1 n and r1 ≥ 1. (3.12) We finally obtain ‖W2(u, v)‖r0,p0 ≤ C‖u‖γ−3 ,p′‖u‖1,p′′ ≤ C‖u‖ 2 r1,p1 . We pause here to note that, without the presence of the space Ċa1;r1,p1(Rn) in the definition of XT,M , we would not be able to bound this W2 term. Returning to J2, we have J2 ≤ sup (0,T ) ∫ t 0 ‖W2(u, v)‖r0,π2ds ≤ C sup (0,T ) ∫ t 0 ‖u‖2r1,p1ds = C sup (0,T ) ∫ t 0 s−2a1sa1‖u‖r1,p1sa1‖u‖r1,p1ds ≤ C‖u‖2a1;r1,p1 sup (0,T ) ∫ t 0 s−2a1ds < CM2T 1−2a1 , provided 2a1 > 1, which is equivalent to γ−1 > 2r1 − 2r0 + 2n p0 − 2n p1 (3.13) and we recall that n p0 = n p′ + n p′′ = 2n p1 − β − r1 + 1. We choose β to be exactly β = 1− r1 − n p0 + 2n p1 , (3.14) and so the two requirements in (3.11) become r1 ≥ 1− n p0 + n p1 and 2r1 ≥ 1 + γ−3 − n p0 + 2n p1 . (3.15) 12 L. RIVA, N. PENNINGTON EJDE-2020/54 Turning to K2, noting that we go down to γ−3 instead of r0, we have K2 = sup (0,T ) ta1 ∫ t 0 ‖e(t−s)L1W2(u, v)‖r1,p1ds ≤ sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−γ−3 +n/p0−n/p1)/γ−1 ‖W2(u, v)‖γ−3 ,p0ds ≤ C sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−γ−3 +n/p0−n/p1)/γ−1 ‖u‖r1,p1‖u‖r1,p1ds = C sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−γ−3 +n/p0−n/p1)/γ−1 s−2a1sa1‖u‖r1,p1sa1‖u‖r1,p1ds ≤ C‖u‖2a1;r0,p0 sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−γ−3 +n/p0−n/p1)/γ−1 s−2a1ds < CM2T 1−(r1−γ−3 +n/π2−n/p1)/γ−1 −a1 , where, by Proposition 2.3, the last inequality holds if γ−1 > r1 − γ−3 + n p0 − n p1 + 2γ−1 a1 = 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 . (3.16) Here is a summary of the inequalities needed to obtain the required bounds on J2 and K2: r1 ≥ 1 (by (3.12)), r1 ≥ 1− n p0 + n p1 (by(3.15)), 2r1 ≥ 1 + γ−3 − n p0 + 2n p1 (by (3.15)), γ−1 > 2r1 − 2r0 + 2n p0 − 2n p1 (by (3.13)), γ−1 > 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 (by (3.16)). By noting that( 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 ) ︸ ︷︷ ︸ RHS of (3.16) − ( 2r1 − 2r0 + 2n p0 − 2n p1 ) ︸ ︷︷ ︸ RHS of (3.13) = r1 − γ−3 + n p0 − n p1 ≥ 0 we conclude that (3.16) implies (3.13), and so the list reduces to 2r1 ≥ max { 2, 1 + γ−3 − n p0 + 2n p1 } , γ−1 > 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 . EJDE-2020/54 GMHD-α SYSTEMS 13 3.4. J3 and K3. Provided that r2−1+γ−3 ≤ r0 and 1 π3 ≥ 1 p0 , Proposition 2.4 gives us J3 = sup (0,T ) ∫ t 0 ‖e(t−s)L1W1(B,B)‖r0,p0ds ≤ sup (0,T ) ∫ t 0 (t− s)−(r0−(r2−1+γ−3 )+n/π3−n/p0)/γ−1 ‖W1(B,B)‖r2−1+γ−3 ,π3 ds. Once again, applying Propositions 2.1 and 2.5 gives us ‖W1(B,B)‖r2−1+γ−3 ,π3 = ‖P (1− L3)−1 div(B ⊗B)‖r2−1+γ−3 ,π3 ≤ C‖B ⊗B‖r2,π3 ≤ C‖B‖r2,p2‖B‖p′ , where the product estimate requires that r2 ≥ 0 and 1 π3 = 1 p2 + 1 p′ . Provided that r2 < n p2 and 1 p′ = 1 p2 − r2 n , (3.17) which combines with the previous equation to give 1 π3 = 2 p2 − r2 n , we can bound ‖B‖p′ by ‖B‖r2,p2 thanks to Proposition 2.2. Thus, ‖W1(B,B)‖r2−1+γ−3 ,π3 ≤ C‖B‖2r2,p2 . Moreover, Proposition (2.4) requires that 1 p0 ≤ 1 π3 = 2 p2 − r2 n , which can be restated as r2 ≤ 2n p2 − n p0 . (3.18) Plugging the bound for W1(B,B) back into the integral gives us J3 ≤ sup (0,T ) ∫ t 0 (t− s)−(r0−(r2−1+γ−3 )+n/π3−n/p0)/γ−1 ‖W1(B,B)‖r2−1+γ−3 ,π3 ds ≤ C sup (0,T ) ∫ t 0 (t− s)−(r0−(r2−1+γ−3 )+n/π3−n/p0)/γ−1 ‖B‖2r2,p2ds ≤ C‖B‖20;r2,p2 sup (0,T ) ∫ t 0 (t− s)−(r0−(r2−1+γ−3 )+n/π3−n/p0)/γ−1 ds < CM2T 1−(r0−(r2−1+γ−3 )+n/π3−n/p0)/γ−1 , where once again the last inequality holds if γ−1 > r0 − (r2 − 1 + γ−3 ) + n π3 − n p0 = r0 − 2r2 − γ−3 + 1 + 2n p2 − n p0 . (3.19) 14 L. RIVA, N. PENNINGTON EJDE-2020/54 The same bounds for W1(B,B) work in the case of K3, so that K3 = sup (0,T ) ta1 ∫ t 0 ‖e(t−s)L1W1(B,B)‖r1,p1ds ≤ sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r2−1+γ−3 )+n/π3−n/p1)/γ−1 ‖W1(B,B)‖r2−1+γ−3 ,π3 ds ≤ C sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r2−1+γ−3 )+n/π3−n/p1)/γ−1 ‖B‖2r2,p2ds ≤ C‖B‖20;r2,p2 sup (0,T ) ta1 ∫ t 0 (t− s)−(r1−(r2−1+γ−3 )+n/π3−n/p1)/γ−1 ds < CM2T 1−(r1−(r2−1+γ−3 )+n/π3−n/p1)/γ−1 +a1 , which holds provided that γ−1 > r1 − (r2 − 1 + γ−3 ) + n π3 − n p1 = r1 − 2r2 − γ−3 + 1− n p1 + 2n p2 . (3.20) What follows is the list of inequalities needed to bound J3 and K3 as desired: r2 − 1 + γ−3 ≤ r0 (semigroup estimate for J3), r2 − 1 + γ−3 ≤ r1 (semigroup estimate for K3), r2 ≥ 0 (product estimate), r2 < min { n p2 , 2n p2 − n p0 } (by (3.17)-(3.18)), γ−1 > r0 − 2r2 − γ−3 + 1− n p0 + 2n p2 (by (3.19)), γ−1 > r1 − 2r2 − γ−3 + 1 + 2n p2 − n p1 (by (3.20)). We see that (3.20) suffices for (3.19) since( r1 − 2r2 − γ−3 + 1 + 2n p2 − n p1 ) ︸ ︷︷ ︸ RHS of (3.20) − ( r0 − 2r2 − γ−3 + 1− n p0 + 2n p2 ) ︸ ︷︷ ︸ RHS of (3.19) = r1 − r0 + n p0 − n p1 ≥ 0, and so the list reduces to r2 − 1 + γ−3 ≤ r0, 0 ≤ r2 < min { n p2 , 2n p2 − n p0 } , γ−1 > r1 − 2r2 − γ−3 + 1 + 2n p2 − n p1 . 3.5. Bounding I4. Applying Minkowski’s inequality to I4 gives I4 ≤ L1 + L2, EJDE-2020/54 GMHD-α SYSTEMS 15 where L1 := sup (0,T ) ∫ t 0 ‖e(t−s)L2W3(u,B)‖r2,p2ds L2 := sup (0,T ) ∫ t 0 ‖e(t−s)L2W3(B, u)‖r2,p2ds We can immediately note that, since W3 is not symmetric, L1 6= L2, but our techniques will give the same bound for each. So, we set L := L1 and proceed to bound only L1. Proposition 2.4 gives us L ≤ sup (0,T ) ∫ t 0 (t− s)−(r2−(r2−1)+n/π4−n/p2)/γ−2 ‖W3(u,B)‖r2−1,π4 ds, provided that 1/π4 ≥ 1/p2. Continuing with W3(u,B), we obtain ‖W3(u,B)‖r2−1,π4 = ‖P div(u⊗B)‖r2−1,π4 ≤ C‖u⊗B‖r2,π4 ; an application of Proposition 2.1 gives us ‖u⊗B‖r2,π4 ≤ C ( ‖u‖r2,p′‖B‖p′′ + ‖B‖r2,p2‖u‖q′′ ) as long as 1 π4 = 1 p′ + 1 p′′ = 1 p2 + 1 q′′ . We want to bound ‖u⊗B‖r2,π4 by ‖u‖r0,p0‖B‖r2,p2 , which requires three applica- tions of Proposition 2.2. First, we obtain ‖u‖r2,p′ ≤ ‖u‖r0,p0 if 0 ≤ r0 − r2 < n p0 and 1 p′ − r2 n = 1 p0 − r0 n . (3.21) We further obtain ‖u‖q′′ ≤ ‖u‖r0,p0 provided that r0 < n p0 and 1 q′′ = 1 p0 − r0 n . (3.22) The last embedding, ‖B‖p′′ ≤ ‖B‖r2,p2 , requires r2 < n p2 and 1 p′′ = 1 p2 − r2 n . (3.23) Combining Equations (3.21)-(3.23) gives us 1 π4 = 1 p0 + 1 p2 − r0 n , which is required to satisfy 1 p2 ≤ 1 π4 = 1 p0 + 1 p2 − r0 n =⇒ r0 ≤ n p0 . (3.24) This is the bound we were looking for: ‖W3(u,B)‖r2−1,π4 ≤ C ( ‖u‖r2,p′‖B‖p′′ + ‖B‖r2,q′‖u‖q′′ ) ≤ C‖u‖r0,p0‖B‖r2,p2 . We can plug the above into L and obtain L ≤ sup (0,T ) ∫ t 0 (t− s)−(1+n/π4−n/p2)/γ−2 ‖W3(u,B)‖r2−1,π4 ds 16 L. RIVA, N. PENNINGTON EJDE-2020/54 ≤ C sup (0,T ) ∫ t 0 (t− s)−(1+n/π4−n/p2)/γ−2 ‖u‖r0,p0‖B‖r2,p2ds ≤ C‖u‖0;r0,p0‖B‖0;r2,p2 sup (0,T ) ∫ t 0 (t− s)−(1+n/π4−n/p2)/γ−2 ds ≤ CM2T 1−(1+n/π4−n/p2)/γ−2 , which holds if γ−2 > 1 + n π4 − n p2 = 1− r0 + n p0 . (3.25) The list of inequalities necessary to bound L is thus 0 ≤ r0 − r2 < n p0 (by (3.21)), r2 < n p2 (by (3.23)), r0 ≤ n p0 (by (3.24)), γ−2 > 1− r0 + n p0 (by (3.25)). 3.6. Wrapping up. On one final note, we point out that since( 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 ) ︸ ︷︷ ︸ RHS of (3.16) − ( 1− 2r0 + r1 + 2n p0 − n p1 ) ︸ ︷︷ ︸ RHS of (3.10) = 2r1 − 1− γ−3 + n p0 − 2n p1 ≥ 0 we have that (3.16) implies (3.10), and so the following is the definitive list con- taining all the inequalities needed for I2, I3, and I4: γ−3 − 1 ≤ r0 ≤ γ−3 ≤ r1, r2 − 1 + γ−3 ≤ r0, r2 ≤ r0 < n p1 , 2r1 ≥ max { 2, 1 + γ−3 − n p0 + 2n p1 } , r2 < min { n p2 , 2n p2 − n p0 } , γ−1 > 3r1 − 2r0 − γ−3 + 3n p0 − 3n p1 , γ−1 > 1− 2r2 + r1 − γ−3 − n p1 + 2n p2 , γ−2 > 1− r0 + n p0 . The above inequalities coincide with those in Theorem 3.1, and so the proof is complete. � EJDE-2020/54 GMHD-α SYSTEMS 17 References [1] Gallagher, I.; Planchon, F.; On global infinite energy solutions to the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal., 161(4) (2002), 307–337. [2] Kato, T.; Ponce, G.; The Navier-Stokes equation with weak initial data. Internat. Math. Res. 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Lorenzo Riva Creighton University, 2500 California Plaza, Omaha, NE 68178, USA Email address: LorenzoRiva@creighton.edu Nathan Pennington Creighton University, 2500 California Plaza, Omaha, NE 68178, USA Email address: NathanPennington@creighton.edu 1. Introduction 2. Notation and supporting facts 3. Main result and its proof 3.1. I2 and I3. 3.2. J1 and K1 3.3. J2 and K2. 3.4. J3 and K3. 3.5. Bounding I4. 3.6. Wrapping up References