Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge

We prove that the Yang-Mills equation in Lorenz gauge in the (3+1)-dimensional case is locally well-posed for data of the gauge potential in \begin{document} $H^s$ \end{document} and the curvature in \begin{document} $H^r$ \end{document} , where \begin{document} $s >\frac{5}{7}$ \end{document} and \begin{document} $r > -\frac{1}{7}$ \end{document} , respectively. This improves a result by Tesfahun [ 16 ]. The proof is based on the fundamental results of Klainerman-Selberg [ 6 ] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [ 14 ] and Tesfahun [ 16 ].

Setting β = 0 in (2) we obtain the Gauss-law constraint The total energy for YM, at time t, is given by and is conserved for a smooth solution decaying sufficiently fast at spatial infinity, i.e., E(t) = E(0).
Hence we may impose a gauge condition. We exlusively study the Lorenz gauge ∂ α A α = 0. Other convenient gauges are the Coulomb gauge ∂ j A j = 0 and the temporal gauge A 0 = 0. It is well-known that for the low regularity well-posedness problem for the Yang-Mills equation a null structure for some of the nonlinear terms plays a crucial role. This was first detected by Klainerman and Machedon [4], who proved global well-posedness in the case of three space dimensions in temporal gauge in energy space. The corresponding result in Lorenz gauge, where the Yang-Mills equation can be formulated as a system of nonlinear wave equations, was shown by Selberg and Tesfahun [14], who discovered that also in this case some of the nonlinearities have a null structure. This allows to rely on some of the methods that were previously used for the Maxwell-Dirac equation in [2] and the Maxwell-Klein-Gordon equation in [13]. Tesfahun [16] improved the local well-posedness result to data without finite energy, namely for (A(0), (∂ t A)(0)) ∈ H s × H s−1 and (F (0), (∂ t F )(0)) ∈ H r ×H r−1 with s > 6 7 and r > − 1 14 , by discovering an additional partial null structure. Local well-posedness in energy space was also given by Oh [10] using a new gauge, namely the Yang-Mills heat flow. He was also able to show that this solution can be globally extended [11]. Tao [15] showed local wellposedness in H s × H s−1 for s > 3 4 in temporal gauge, but limited to small data. Tao's result was generalized to space dimensions n ≥ 3 by the author [12]. In space dimension n the critical regularity with respect to scaling is s = n 2 − 1 . In the case n = 4 where the energy space is critical Klainerman and Tataru [7] proved small data local well-posedness for a closely related model problem in Coulomb gauge for s > 1. Klainerman and Selberg [6] treated the local well-posedness problem with minimal regularity for some systems of nonlinear wave equations. Especially, they showed local well-posedness for a model problem related to the Yang-Mills system in the almost critical region, where s > n 2 − 1 and n ≥ 4. Recently the result [7] was significantly improved by Krieger and Tataru [9], who were able to show global well-posedness for data with small energy. In high space dimension n ≥ 6 (and n even) Krieger and Sterbenz [8] proved global well-posedness for small data in the critical Sobolev space.
In the present paper we consider the local well-posedness problem for large data without finite energy for the Yang-Mills system in Lorenz gauge and space dimension n = 3 . Our main result is local well-posedness for s > 5 7 and , H r−1 ) and (existence and) uniqueness in a certain subspace (Theorem 2.1 and Corollary 2.1). This is an improvement of Tesfahun's result [16]. It is the first local well-posedness result for s < 3 4 and holds even for large data. Crucial for this result are on one hand the methods developed in the papers by Selberg-Tesfahun [14] and Tesfahun [16], especially their detection of the null structure in most -unfortunately not all -critical nonlinear terms. On the other hand we rely on the methods by Klainerman and Selberg [6] for a model problem for Yang-Mills, which ignores the gauge condition. We have to modify their solution space appropriately and show that its main features are preserved. We were unable to come down to the critical value s = 1 2 , which is prevented mainly by one of the nonlinear terms, for which no null structure is known and which leads to the estimate (29). [14] and [16] used solution spaces of wave-Sobolev type H s,b , which are closely related to the Bourgain-Klainerman-Machedon spaces X s,b , for which a convenient atlas of bilinear estimates was proven by [1] in dimension n = 3 . If one uses solution spaces of H s,b -type it seems to be impossible to obtain our results, because some of the bilinear estimates which we need simply fail. For details we refer to the remark at the end of the paper. Therefore it is necessary to modify the solution spaces appropriately.
In chapter 2 we recall the reformulation of the Yang-Mills equation as a system of nonlinear wave equations and state our main theorem (Theorem 2.1 and Corollary 2.1). We also fix some notation. Chapter 3 contains the bilinear estimates in wave-Sobolev spaces. Moreover we define the solution spaces and state its fundamental properties. We reduce the local well-posedness problem to a suitable set of nonlinearities in Proposition 3.10, where we completely rely on [6]. In chapter 4 we formulate the Yang-Mills equations in final form -using the whole null structure -and the necessary nonlinear estimates as in [16]. We also review some well-known properties of the standard null forms and the additional one detected in [16]. In (the most voluminous) chapter 4 we prove the multilinear estimates for the nonlinearities.
2. Main results. Expanding (2) in terms of the gauge potentials {A α }, we obtain: If we now impose the Lorenz gauge condition, the system (3) reduces to the nonlinear wave equation In addition, regardless of the choice of gauge, F satisfies the wave equation Indeed, this will follow if we apply D α to the Bianchi identity and simplify the resulting expression using the commutation identity and (2) ( [14]). Expanding the second and fourth terms in (5), and also imposing the Lorenz gauge, yields Note on the other hand by expanding the last term in the right hand side of (4), we obtain We want to solve the system (6)-(7) simultaneously for A and F . So to pose the Cauchy problem for this system, we consider initial data for (A, F ) at t = 0: In fact, the initial data for F can be determined from (a,ȧ) as follows: where the first three expressions come from (1) whereas the last one comes from (2) with β = i. Note that the Lorenz gauge condition ∂ α A α = 0 and (2) with β = 0 impose the constraintsȧ (10) Now we formulate our main theorem.
Theorem 2.1. Let n = 3 and assume that s and r satisfy the following conditions: Given initial data (a,ȧ) ∈ H s × H s−1 , (f,ḟ ) ∈ H r × H r−1 , there exists a time T > 0, T = T ( a H s , ȧ H s−1 , f H r , ḟ H r−1 ), such that the Cauchy problem (6), (7), (8) has a unique solution A ∈ F s T , F ∈ G r T (these spaces are defined in Def.

3.1). This solution has the regularity
Remark 2.1. 1. The most natural relation between s and r is r = s − 1 .This is not allowed in Theorem 2.1. In this case the condition 2r − s > −1 would force s > 1, which would exclude the most interesting range 5 7 < s ≤ 1 . 2. The assumptions on s and r imply 4s − 3 > r > s 2 − 1 2 , which can only be fulfilled, if s > 5 7 , and therefore r > − 1 7 . One easily checks that the choice s = 5 7 + , r = − 1 7 + satisfies our assumptions, if > 0 is small enough. 3. The following conditions are automatically fulfilled 1. Let s, r fulfill the assumptions of Theorem 2.1. Moreover assume that the initial data fulfill (9) and (10). Given any (a,ȧ) ∈ H r+1 ×H r , there exists a time T > 0, T = T ( a H s , ȧ H s−1 , f H r , ḟ H r−1 ), such that the solution (A, F ) of Theorem 2.1 satisfies the Yang-Mills system (1), (2) with Cauchy data (a,ȧ) and the Lorenz gauge condition ∂ α A α = 0 .
Proof of the Corollary. The solution (A, F ) does not necessarily fulfill the Lorenz gauge condition and (1), i.e. F = F [A] . If however the conditions (9) and (10) are assumed then these properties are satisfied and (A, F ) is a solution of the Yang-Mills system (1), (2) with Cauchy data (a,ȧ). This was shown in [14], Remark 2. Remark 2.2. 1. Because s < r + 1 by assumption the potential A possibly looses some regularity compared to its data, whereas this is not the case for F , which is the decisive factor, whereas the regularity of A is of minor interest.
2. If (a,ȧ) ∈ H r+1 × H r , then (f,ḟ ), defined by (9), fulfill (f,ḟ ) ∈ H r × H r−1 , as one easily checks. Let us fix some notation. We denote the Fourier transform with respect to space and time by . = ∂ 2 t −∆ is the d'Alembert operator, a± := a± for a sufficiently small > 0, and · := (1 + | · | 2 ) 1 2 . H s,r x denotes the L r -based Sobolev space with respect to the space variables and H s = H s,2 x . The standard wave-Sobolev spaces H s,b of Bourgain-Klainerman-Machedon type are the completion of the Schwarz space S(R 1+3 ) with norm We also define H s,b T as the space of the restrictions of functions in

respectively.
Let ∂ denote the collection of space and time derivatives. If u, v ∈ S and u, v are tempered functions, we write u v iff | u| ≤ v, and means up to a constant. If u = (u 1 , . . . , u N ) and v = (v 1 , . . . , v N ), then u v (resp. u v) means u I v I (resp. u I v I ) for I = 1, . . . , N .
3. Preliminaries. The Strichartz type estimates for the wave equation are given in the next proposition.
then the following estimates hold Proof. This is the Strichartz type estimate, which can be found for e.g. in [3], Prop. 2.1, combined with the transfer principle.

HARTMUT PECHER
Proof. We use the following special case of Prop. 3.1: The following proposition follows from [7], Theorem 5 and the transfer principle.
Proposition 3.2. Let n ≥ 2, and let (q, r) satisfy: The following product estimates for wave-Sobolev spaces were proven in [1].
holds, provided the following conditions are satisfied: The following multiplication law is well-known: Then the following product estimate holds: We now come to the definition of the solution spaces, which are similar to the spaces introduced by [6]. We prepare this by defining the following modification of the standard L q t L r x -spaces. Definition 3.1. If 1 ≤ q, r ≤ ∞, u ∈ S and u is a tempered function, set x be the corresponding subspace of S . This is a translation invariant norm and it only depends on the size of the Fourier transform. Observe that Our solution space is defined as follows: This is a Banach space ( [6], Prop. 4.2). Next we recall some fundamental properties of the L q t L r x -spaces, which were given by [6], starting with a Hölder-type estimate.
The following duality argument holds.
for all G, then x . for all F .
The next immediate consequence shows that a Sobolev type embedding also carries over to the L q t L r x -spaces.
Proof. We adapt the proof of [6], Prop. 4.7 in space dimension n ≥ 4 to the case . Then the left hand side of (13) is bounded by Under the assumptions of Prop. 3.8 : Proof. This follows from Prop. 3.8 by use of Prop. 3.6.
Proof. In the special case p = q Prop. 3.8 gives x gives the result. Proposition 3.9. If 1 < q ≤ 2, 1 q + 1 q = 1 and s = 2 q − 1 the following estimate holds Proof. This follows from Corollary 3.3 by use of Prop. 3.6.
Finally, we formulate the fundamental theorem which allows to reduce the local well-posedness for a system of nonlinear wave equations to suitable estimates for the nonlinearities. It is also essentially contained in the paper by [6].

HARTMUT PECHER
Proof. This is proved by the contraction mapping principle provided the solution space fulfills suitable assumptions. The case of a single equation u = M(u, ∂u) and the solution space X s given by the norm u X s = Λ + u x , γ > 0 small, was proven by [6], Theorems 5.4 and 5.5, Propositions 5.6 and 5.7. Our case is a straightforward modification of their results. We just remark that the only modification in the case of our solution space is the following estimate in the proof of [6], Prop. 5.6: x . The first estimate follows from Corollary 3.2, and the last estimate holds by our assumption s > 5 7 .
4. Reformulation of the problem and null structure. The reformulation of the Yang-Mills equations and the reduction of our main theorem to nonlinear estimates is completely taken over from Tesfahun [16] (cf. also the fundamental paper by Selberg and Tesfahun [14]). The standard null forms are given by For g-valued u, v, define a commutator version of null forms by Note the identity Define where ε ijk is the antisymmetric symbol with ε 123 = 1 and R i = Λ −1 ∂ i are the Riesz transforms. Now we refer to Tesfahun [16], who showed that the system (6), (7) in Lorenz gauge can be written in the following form where Here Here especially the splitting of the spatial part A = (A 1 , A 2 , A 3 ) of the potential into divergence-free and curl-free parts and a smoother part is used where Now, looking at the terms in M and N and noting the fact that the Riesz transforms R i are bounded in the spaces involved, the estimates in Proposition 3.10 reduce to proving (we remark, that due to the multilinear character of the nonlinearity the estimates for the difference can be treated exactly like the other estimates): 1. the corresponding estimates for the null forms Q ij , Q 0 and Q ∈ {Q 0i , Q ij } : the following estimate forΓ 1 β and other bilinear terms and 2. the following trilinear and quadrilinear estimates: where Π(· · · ) denotes a multilinear operator in its arguments. The matrix commutator null forms are linear combinations of the ordinary ones, in view of (15). Since the matrix structure plays no role in the estimates under consideration, we reduce (21)-(25) to estimates of the ordinary null forms for Cvalued functions u and v (as in (14)).
The null forms above satisfy the following estimates.
Next we consider the termΓ 1 β . We may ignore its matrix form and treat ) for k = 1, 2, 3 and where we used the Lorenz gauge ∂ 0 A 0 = ∂ i A i in the last line in order to eliminate one time derivative. Thus we have to consider The proof of the following theorem was essentially given by Tesfahun [16]. In fact the detection of this null structure was the main progress of his paper over Selberg-Tesfahun [14].

5.
Proof of the nonlinear estimates. Important remark: We assume in the following that the Fourier transforms of u and v are nonnegative. This means no loss of the generality, because the norms involved in the desired estimates do only depend on the size of the Fourier transforms.
Proof of (25). We recall (39) for α = : . Thus we have to show the following estimates and remark that we only have to consider the first and third term, because the last two terms are equivalent by symmetry. 1. For the first term it suffices to show which is a consequence of Prop. 3.3 under our conditions s ≥ r and 2s − r > 3 2 . 2. For the second term we show Proof of (24). We use (41). Thus we have to show the following estimates and remark that we only have to consider the first two terms, because the last two terms are equivalent by symmetry. 1. For the first term it suffices to show 3 2 , and s 0 + s 1 + s 2 + s 1 + s 2 > 3 2 , if 4s − r > 3, which holds under our assumptions. 2. For the second term we show which is a consequence of Prop. 3.3 as in 1., if 2s − r > 3 2 and 3s − 2r > 2, which holds under our assumptions.
Proof of (29). A. We start with the first part of the F s -norm. As before it is easy to see that we can reduce to + . This is a consequence of Prop. 3.3. One easily checks that it can be applied under the conditions s ≤ r + 1, 2r − s > −1, 4r − s > −2 and 3r − 2s > −2, all of which are satisfied under our assumptions. B. For the second part of the F s -norm we reduce to and further to Next we show for a suitable r 2 the estimate which by duality is equivalent to Using the fractional Leibniz rule we have to consider two terms: where we choose 1 q = 1 2 − 1 7 + 7 3 and 1 r2 = 1 7 − 7 3 , so that by Sobolev H 3 7 −7 x → L q x , and uΛ x . Thus we obtain by Cor. 3.1: for r > − 1 7 . Proof of (26). A. For the first part of the F s -norm it is sufficient to show for the minimal value s = 5 7 +, because the estimate for any s > 5 7 follows immediately. We use Lemma 4.2.
a. We first consider Γ 1 2 (u, v) . By (45) it suffices to show the following estimates, all of which are consequences of Proposition 3.3.
b. Assume that u and v have frequencies ≥ 1, so that Λ α + u ∼ D α + u . In this case we use (42) and consider Γ 1 1 (u, v) . By (44) we may reduce the estimates for the first and third term on the right hand side to Both estimates follow from Proposition 3.3. The second term is reduced to the following estimate −2 , 1 2 + . By the fractional Leibniz rule we have to show the following two estimates: b1. ( where we used Cor. 3.7 and Prop. 3.5 in order to replace L p t L q x -norms by L p t L q xnorms. c. Consider (Λ −2 u)v and u(Λ −2 v) . It suffices to show a. We first consider Γ 1 2 (u, v) and use (45). 1. The estimate for the first term on the right hand side reduces to and therefore we only have to prove This follows from Proposition 3.2 with parameters q = 28 13 , r = 28, σ = 11 14 − 3 , s 1 = 6 7 + 3 and s 2 = 5 7 − 1 2 . The claimed estimate follows, because s 1 < s + 1 2 − 2 and s 2 < s − 1 2 − 2 under our assumption s > 5 7 . 2. The second term is modified as follows:

We have to prove
We obtain by Proposition 3.9 We now show that uv We start with Proposition 3.9 By the fractional Leibniz rule we have to consider two terms. 2.1. 3. The last term on the right hand side of (44) requires The left hand side is estimated using Prop. 3.9 by Λ We crudely estimate the left hand side by by the choice s 1 = 5 7 < s, s 2 = − 1 7 < s + 1 . The proof of (26) is now complete. Proof of (21) and (22). We have to prove Using the fractional Leibniz rule we obtain: 3.1. By Sobolev we have This follows by Prop. 3.2 with parameters q = 28 13 , r = 28, σ = 11 14 − 3 . This requires s 1 , s 2 < 13 14 and s 1 + s 2 ≥ 15 14 . We choose s 1 = 13 14 − < s + 1 2 − 2 , s 2 = 1 7 + < s − 1 2 − 2 . 2. The estimate for the second term reduces to where we used Λ 2 − u Λ 2 + u . By Prop. 3.9 we obtain the following bound for the left hand side: Λ