Large data global regularity for the classical equivariant Skyrme model

This article is concerned with the large data global regularity for the equivariant case of the classical Skyrme model and proves that this is valid for initial data in $H^s\times H^{s-1}(\mathbb{R}^3)$ with $s>7/2$.

1. Introduction 1.1. Statement of the problem and main result. One of the fundamental models in classical field theory is the Gell-Mann-Lévy model [9], also known as the classical nonlinear σ model, which is described by the action where S µν = h AB ∂ µ U A ∂ ν U B is the pullback metric associated to a map U : R 3+1 → S 3 , g = diag(−1, 1, 1, 1) is the Minkowski metric, and h is the induced Riemannian metric on S 3 from the Euclidean one on R 4 . An important feature of this theory is that it does not admit static topological solitons and this is precisely the motivation that led Tony Skyrme to introduce, in a series of seminal papers [19,20,21], a physically relevant modification of the Gell-Mann-Lévy model, which no longer has this limitation. The static topological solitons of the Skyrme theory are known as skyrmions and, from a historical perspective, they represent the first of their type to model a particle. The action corresponding to the Skyrme model is given by where α is a constant having the dimension of length. As the value of α does not play an important role in our arguments other than being positive, from here on in, we set it to α = 1 in order to simplify the exposition. For a more comprehensive discussion of the physical descriptions and motivations for both the Gell-Mann-Lévy and Skyrme models, we refer the reader to our recent monograph [7] and references therein. Our focus in this article is on the Euler-Lagrange equations associated to the equivariant case of the Skyrme model; i.e., we work with formal critical points for (1) under the ansatz U (t, r, ω) = (u(t, r), ω), where g = −dt 2 + dr 2 + r 2 dω 2 , h = du 2 + sin 2 u dω 2 , are the previous metrics written in polar form. The relevant variational equation is the one for the azimuthal angle u, (2) 1 + 2 sin 2 u r 2 (u tt − u rr ) − 2 r u r + sin 2u r 2 1 + u 2 t − u 2 r + sin 2 u r 2 = 0, which is of quasilinear type. A formal computation shows that solutions to this equation have the energy-type quantity The integer u(t, ∞) − u(t, 0) π is called the topological charge of the map U and, like the energy, it is also a conserved quantity. In what follows, we assume that (4) u(t, 0) = N 1 π, N 1 ∈ N, u(t, ∞) = 0.
The subsequent theorem is the main result of this paper.

Remark 1.2. It is important to compare this result with what is known for the corresponding equation of the Gell-Mann-Lévy model, i.e.,
u tt − u rr − 2 r u r + sin 2u r 2 = 0, which is of semilinear type and looks considerably simpler than (2). Surprisingly, Shatah [18] showed that there are smooth data that lead to solutions of this equation that blow up in finite time, with Turok and Spergel [23] later finding such solutions in closed form.

1.2.
Comments on previous relevant works and comparison to main result. Since it first appeared, the Skyrme model has received considerable interest from both the mathematics and physics communities, with comprehensive lists of references being available in the book by Manton and Sutcliffe [15] and in our monograph [7]. In here, we strictly focus on a number of works that better correlate to ours.
For the static problem, we start by mentioning the proof for the existence and uniqueness of skyrmions with arbitrary topologic charges, which is due to Kapitanskiȋ and Ladyzhenskaya [12] and is based on variational techniques. An alternative approach relying strictly on ODE type methods was formulated later by McLeod and Troy [16]. The asymptotic stability of the skyrmion with unit topologic charge was numerically investigated by Bizoń, Chmaj, and Rostworowski [2], while, just recently, Creek, Donninger, Schlag, and Snelson [6] proved rigorously its linear stability.
In what concerns the non-static problem, the Euler-Lagrange equations, for both the general and equivariant cases of the Skyrme model, have been the focus of quite a few studies in recent years. As these are evolution equations, one natural question to study about them is the well-posedness of the associated initial value problem. This is a very challenging task, mainly due to the quasilinear nature of the equation (displayed above by (2)) and the fact that scaling heuristics in a small energy scenario show that the Cauchy problem is supercritical with respect to the energy (for more details, see [8]).
For the general case, Wong [24] displayed regimes in which the problem is regularly hyperbolic and, consequently, is locally well-posed for almost stationary initial data. The same paper also showcased frameworks which lead to an ultrahyperbolictype breakdown of hyperbolicity and, thus, to ill-posedness. In the equivariant case, Geba, Nakanishi, and Rajeev [8] proved global well-posedness and scattering for the Cauchy problem associated to (2), when N 1 = 0 and the initial data have a small Besov-Sobolev norm at the level of H 5/2 (R 3 ).
However, the most relevant work to the current paper, which also served as one of its sources of inspiration, is due to Li [13]. His main result is a proof of Theorem 1.1 in the more restrictive setting s ≥ 4. Our argument follows the general framework of [13]; nevertheless, certain key parts have been reworked and streamlined and the main argument is now transparent (e.g., Propositions 5.1 and 6.1). Additionally, we have a comprehensive appendix that discusses the regularity of the initial data, which is an important element of the proof. A difference of the present work with [13] is in the last step of our argument, where our approach is able to handle fractional derivatives. A final remark has to do with our belief that Theorem 1.1 is optimal with regard to the spaces used in its proof. However, we do not pursue this issue in the present work.
A parallel literature exists on other Skyrme-like theories, like the Faddeev and Adkins-Nappi models, for which we ask the interested reader to consult, yet again, our book [7].
1.3. Outline of the paper. In section 2, we reformulate our result in terms of a newly introduced function v and reduce the argument for Theorem 1.1 to the verification of a continuation criterion for v. Subsequently, we construct another auxiliary function Φ, which is intimately tied to v, but is more amenable to the methods we have in mind. Also here, we reduce once more the argument to the proof of the finiteness of certain Sobolev norms for derivatives of Φ. Section 3 is devoted to setting up the necessary notational conventions and to collecting the analytic tools used throughout the paper. In section 4, we start in earnest our proof and show that both v and Φ satisfy energy-type estimates, which lead to fixed-time decay bounds. Sections 5 and 6 are dedicated to upgrading these inequalities to match Sobolev regularities at the level of H 2 and H 3 , respectively. In section 7, we conclude the analysis by showing that Φ is regular enough to imply that the continuation criterion for v is valid. The article finishes with an appendix which certifies that the Sobolev regularity of the initial data assumed in every part of the main argument is indeed the right one.
Acknowledgements. The first author was supported in part by a grant from the Simons Foundation # 359727.

2.1.
Introducing the function v and initial reductions. We start by writing the equation (2) in the form is the radial wave operator in R 3+1 . We perform the substitution (6) u(t, r) = r v(t, r) + ϕ(r), with ϕ : R + → R + being a smooth, decreasing function, verifying ϕ ≡ N 1 π on [0, 1] and ϕ ≡ 0 on [2, ∞). We also need to introduce a finer version of ϕ, labelled ϕ <1 , which shares the same smoothness and monotonicity with ϕ, but now satisfies ϕ <1 ≡ 1 on [0, 1/2] and ϕ <1 ≡ 0 on [1, ∞). Furthermore, we write ϕ >1 to denote the function 1 − ϕ <1 . As a consequence, we obtain A careful analysis shows that with all N i = N i (x) being even, analytic, and satisfying We refer the reader to [13] for the precise formulae of these functions. The previous substitution is motivated by the fact that it reduces the proof of Theorem 1.1 to the one of the subsequent result concerning (7), which could be verified through a fairly straightforward argument (e.g., see Subsection 2.3 in Creek [5]).
Then there exists a global radial solution v to the Cauchy problem associated to (7) with The approach in proving this theorem relies on a classical result (e.g., see Theorem 6.4.11 in Hörmander [11])) that allows us to derive global solutions from local ones, which additionally satisfy a continuation criterion. The entire argument is then reduced to demonstrating the following theorem.

2.2.
The construction of the auxiliary function Φ and further reductions. The proof of Theorem 2.2 is somewhat indirect, in the sense that we argue for (10) by constructing a new function Φ, which satisfies an equation that is easier to study than the one for v (i.e., (7)). The first step in this construction aims to eliminate the derivative terms on the right-hand side of (7) and, for that purpose, we take The wave equation satisfied by Φ 1 is given by To deal with the 1/r 2 singularity, we introduce next It seems that we took care of the singularity only to introduce a new one in front of the integral. However, this can be seen to be removable by writing 1 and then making the change of variable w = N 1 π + ry in the integral multiplied by ϕ <1 . Even if the equation is fixed, a formal argument shows that one might have which does not fit the expected results of our approach.
To address this final issue, we take After careful computations, we deduce that with the associated wave equation being Ã =Ã(r, y) := 1 + 2 sin 2 (ry + ϕ(r)) r 2 and ϕ ≥1/2 = ϕ ≥1/2 (r) is a generic smooth function, with bounded derivatives of all orders and supported in the domain {r ≥ 1/2}, which may change from line to line.
In order to prove Theorem 2.2, it is obvious from its formulation that we can additionally assume that s is sufficiently close to 7/2. In fact, we show that if 7/2 < s < 18/5, then which, coupled with Sobolev embeddings and radial Sobolev inequalities, implies (10).

Notations and analytic toolbox
3.1. Notational conventions. First, we write A B to denote A ≤ CB, where C is a constant depending only upon parameters which are considered fixed throughout the paper. Two such important parameters are the conserved energy (3), written in terms of the initial data (u 0 , u 1 ) in Theorem 1.1 as Secondly, as is the custom for w = w(t, x), we work with ∇w = (∂ t w, ∇ x w) and where X(R n ) is a normed/semi-normed space (e.g., X = L q or H σ orḢ σ ) and I ⊆ R is an arbitrary time interval. Furthermore, for ease of notation, in the case when I × R n = [0, T ) × R 5 , we drop from the previous notation the dependence on the domain and simply write This is because the majority of the norms we are dealing with from here on out refers to this particular situation.
3.2. Analytic toolbox. Here, we list a number of analytic facts that we will use throughout the argument. First, we recall the classical and general Sobolev embeddings and the radial Sobolev estimates ( [22], [4]) which are valid for radial functions defined on R n . Related to these, we write down Hardy's inequality ( [17]) , both of which hold true for general functions on R n .
Next, using the Riesz potential D σ = (−∆) σ/2 , we record the fractional Leibniz estimate ( [10], [3]) and the Kato-Ponce type inequalities ([14]) We also mention the well-known Moser bound where F ∈ C ∞ (R k ; R), F (0) = 0, and γ = γ(σ) ∈ C(R; R). Following this, we recall the Bernstein estimates where P >λ is a Fourier multiplier localizing the spatial frequencies to the region {|ξ| > λ}. Finally, we recount the classical Strichartz inequalities for the 5 + 1-dimensional linear wave equation, which take the form with I being a time interval and A straightforward consequence of the previous bound is the following generalized energy estimate:

Energy-type arguments
In this section, we start in earnest our analysis and prove that and, subsequently, Next, we use this information and the radial Sobolev inequalities (18)- (19) to derive preliminary decay estimates for both v and Φ, which, in turn, imply valuable asymptotics for Φ and Φ.

4.1.
Energy-type arguments for v. Based on the formula (6) and Hardy's inequality (20), we infer that Hence, by the fundamental theorem of calculus, we also have which finishes the proof of (29).

4.2.
Energy-type arguments for Φ. We proceed by using the formula (11) to deduce Moreover, a direct argument relying on the same formula yields which, coupled with the Sobolev embeddings (17), further implies 1.
If we argue like we did for v, then we obtain Thus, in order to conclude the argument for (30), we need to obtain a favorable estimate for Φ r L ∞ L 2 , which is quite technical in nature. First, we prove the following fixed-time inequality.
Next, we show that another related fixed-time estimate is true.
holds true uniformly in time on [0, T ).
Proof. The first step in the argument is to prove where ϕ <r0 = ϕ <r0 (r) = ϕ <1 (r/r 0 ) and r 0 ≤ 1 is a scale to be further calibrated.
On the other hand, with the help of (11), we derive If we also factor in (34) and (35), then this estimate implies It is clear now that (39) holds as a result of this bound, (41), and (42). The second step in the proof of (38) consists in rewriting the second term on the right-hand side of (39) in a friendlier format. For this purpose, we introduce If we take advantage of (11) and r 0 ≤ 1, then we obtain Hence, we can restate (39) as d dt The third step of this argument involves integrating the previous estimate over the interval [0, t] ⊂ [0, T ), which leads to (43) We address first the integral term on the right-hand side, for which a direct computation based on (11) and (13) reveals that with (44)Ã r = −4 sin 2 (ry + ϕ) r 3 + 2 sin 2(ry + ϕ) · (y + ϕ r ) r 2 .
Using the formula (6), it is easy to derive Moreover, a direct analysis using Maclaurin series shows that, when r ≥ 1, we have Hence, by collecting the last five mathematical statements and applying the Sobolev embeddings (17), we conclude that 1.
Using the basic estimate | sin x| ≤ |x|, we easily infer that |H(r, w)| w 4 + w 6 and, as a consequence, we have 1.
Together with (43) and (47), this bound yields In the last step of this proof, we show that if we choose r 0 to be sufficiently small, then the following estimate holds: It is clear that, jointly with (48), this inequality implies (38). We claim that a calculus-level analysis finds that is true if r 1 is sufficiently small. Therefore, by choosing and also using (11), we infer which finishes the argument for (49).
We can finally now invoke the basic inequality which, coupled to (36) and (38), yields the desired control over Φ r L ∞ L 2 in the form of (50) Φ r L ∞ L 2 1.

4.3.
Preliminary decay estimates and asymptotics. First, we take advantage of (29) and (30) and derive decay estimates for both Φ and v.
Proof. The radial Sobolev inequalities (18) and (19) easily imply thus proving (51) and half of (52) on the basis of (29) and (30). For the other half of (52), we work in the regime when r ≪ 1. If we choose r < 1/2 and use (11), then we obtain dy and, consequently, Relying on (51), we deduce r|v| 0 | sin z|dz r 1/2 , which shows that r|v| ≤ π/2 if r is sufficiently small. Hence, we can argue that in this scenario we have thus proving the desired bound.
Next, we use these decay estimates and obtain asymptotics for Φ and Φ in terms of v, which turn out to be very important in further arguments.
In the complementary case when r 1, we easily have 1 ≤Ã ≤ 1 + 2 r 2 ∼ 1 and the derivation of (54) follows exactly like above.

H 2 -type analysis
In this section, we take the next step in our analysis and show that This is done by first deriving a wave equation for Φ t , which is then investigated by using the Strichartz estimates (27). As a result, we obtain the desired Sobolev regularity for both Φ t and Φ tt . Jointly with the main equation satisfied by Φ (i.e., (12)), this information yields that Φ ∈ L ∞Ḣ 2 . Following this, we improve the decay estimates (51) and (52).

5.1.
Argument for theḢ 1 and L 2 regularities of Φ t and Φ tt . We start by noticing that a simple differentiation with respect to t of both (11) and (12) produces From these equations, it is clear that the expressionÃ(r, v) will play an important role moving forward. For this purpose, we rely on the decay estimate (52) to easily infer (58) | sin u| min 1 r , r 1/4 , which leads to and, subsequently, Moreover, the conservation of energy (3) and (58)  We now have all that is needed to prove that Φ t and Φ tt haveḢ 1 and L 2 regularities, respectively.
holds true for all pairs (p, q) satisfying Proof. We start by applying the Strichartz estimates (27) to the nonlinear wave equation (57) for the case when σ = 1 and (p ′ ,q ′ ) = (1, 2). We derive that is valid for all intervals I = [a, b] ⊂ [0, T ) and pairs (p, q) satisfying (63). For the last term on the right-hand side, we use (60) and (61) to obtain It is easy to see that we are allowed to have (p, q) = (2, 5) in (64) and, as a result, we deduce Recalling our notational conventions, it follows that for |I| ∼ 1, yet sufficiently small, we have Therefore, if one chooses T 1 to be the maximal length of an interval for which the previous bound holds true, then holds true for as long as 0 ≤ kT 1 < (k + 2)T 1 < T, with k being a nonnegative integer. Given that the assumptions of Theorem 2.2 guarantee, through the Appendix, that it is clear that the previous facts lead to Now, we can go back to (65) and claim Coupled to (64), this estimate implies that Φ t L p L q 1 also holds true for (p, q) = (2, 5) satisfying (63) and, thus, finishes the proof.
5.2.Ḣ 2 regularity for Φ and improved decay estimates. Using the newfound regularities for Φ t and Φ tt in conjunction with the wave equation (12), we show that, at this stage of the analysis, Φ hasḢ 2 regularity.
Next, a direct application of the radial Sobolev inequalities (18) and (19) and of the asymptotic equation (53), in the context of theḢ 2 regularity for Φ, leads to the following upgrade for the previous decay estimates satisfied by Φ, v, and A(r, v) − 1.

H 3 -type analysis
Here, we show that The approach is similar to the one used in the previous section, in the sense that we start by writing a wave equation for Φ tt and we analyze it through the Strichartz estimates (27). This yields the expected regularity for both Φ tt and Φ ttt . Next, we tie these regularities to equations satisfied by Φ t and Φ r to deduce that Φ t ∈ L ∞Ḣ 2 and Φ ∈ L ∞Ḣ 3 , respectively. Finally, we continue to improve the decay rates for Φ, v, andÃ(r, v) − 1.
6.1. Derivation ofḢ 1 and L 2 regularities for Φ tt and Φ ttt . By differentiating (56) and (57) with respect to t, we obtain It is important to notice that (56) and (59) imply These are all the necessary prerequisites to argue for the desired regularities for Φ tt and Φ ttt .
Proof. The argument is a carbon copy of the one for Proposition 5.1, in the sense that we start by writing down the Strichartz estimates (27) for the equation (78), i.e., which are valid in the same context as the one for which (64) holds true. Following this, we apply (80), the Sobolev embeddings (17), (62), and (61) to deduce

This leads to
where |I| ∼ 1, yet small enough. Subsequently, by also invoking the Appendix, we derive which suffices to claim that Φ tt L p L q 1 holds true for pairs (p, q) = (2, 5) satisfying (63).
6.2.Ḣ 3 andḢ 2 regularities for Φ and Φ t and further improvement of the decay information. As an immediate consequence of the previous proposition, we obtain the corresponding Sobolev regularity for Φ t .

Proposition 6.3. Under the assumptions of Theorem 2.2, it is true that
Proof. We start by arguing that, due to (81), A direct computation using (12) yields and, taking into account the formula (13), we deduce that It is easy to check that and, as a result, we turn our attention to the other two terms on the right-hand side of the previous bound. For the integral term, we rely on (13) Finally, for the term in (85) involving v r , we obtain from (11) that It is straightforward to argue that ϕ ≥1/2 r 3 L ∞ L 10/3 1 and, using (87) and (88), we also have that Furthermore, due to the Sobolev embeddings (17) and (66), we derive Therefore, as the combined result of the last four mathematical statements, we infer that Ã 1/2 (r, v)v r L ∞ L 10/3 1.
As a corollary of this result, we can argue as in the section devoted to the H 2 -type analysis and further improve the decay estimates for Φ, v, andÃ(r, v) − 1.

Proposition 6.4. Under the assumptions of Theorem 2.2, we have
Remark 6.5. We notice that the decay bound (92) easily implies the portion of the main estimate to be proved about v (i.e., (10)), which doesn't involve its gradient: 1.

Remark 6.6.
In what concerns ∇v, we can show with the facts obtained so far that holds true. First, we can rewrite (90) in the form and we can infer from (56) that Thus, in the regime when r ≥ 1, we deduce based on (44), (45), (59), and (52) that Finally, with the help of (18) and (76), we derive which yields (95).

Final estimates and conclusion of the argument
On the basis of the previous two remarks, specifically the estimates (94) and (95), in order to conclude the argument for (10) (and thus finish the proof of Theorem 2.2), we are left to show that 1 is valid. As in the derivation of (95), we start by relying on (96), (97), (44), (46), (59), and (92) to infer that and, with the help of (30) and the classical Sobolev embedding (16), we obtain the desired bound if we show that The strategy is to prove first the finiteness of the norm involving Φ t by using energy estimates applied to equation (57). As a byproduct of this argument, we also get to control Φ tt L ∞Ḣs−2 , which, coupled to equation (12) satisfied by Φ, allows us to deduce the finiteness of Φ L ∞Ḣs and thus finish the proof of (99). 7.1. New qualitative bounds for v andÃ(r, v). As one can imagine from the structure of the equations involved in this step of the main argument (i.e., (12) and (57)), it is important to derive more qualitative information on v andÃ(r, v), in addition to (29), (92), and (93). First, we prove the following result.
Next, due to the presence ofÃ −2 on the right-hand side of (57), we also need estimates for derivatives ofÃ −1 .
Proof. We start by applying the Kato-Ponce type inequalities (23) and (24) to deduce and Following this, due toÃÃ −1 (r, v) ≡ 1 and σ > 0, we notice that Moreover, it is easily seen that (13) implies Using these two observations jointly with the previous two bounds and (93), we derive and Consequently, we infer that Finally, combining this estimate with an application of the interpolation inequality (21) that yields we reach the desired conclusion.
7.2. Improved Sobolev regularities for Φ t and Φ. Now, we have all the prerequisites needed to upgrade the H 2 and H 3 regularities for Φ t and Φ, respectively, to the level of the ones featured in (99). As outlined at the start of this section, we first focus our analysis on Φ t . Proposition 7.3. Under the assumptions of Theorem 2.2, with s > 7/2 replaced by 7/2 < s < 18/5, Proof. We commence by relying on the energy-type estimate (28) applied to (57) to argue that The Appendix ensures that and, hence, in order to deduce (111), it is enough to show that (113) Φ t L 1Ḣs−2 1.
Using (57) and the fractional Leibniz bound (22), we derive that First, we deal with the norms involving Φ t , for which an application of the Sobolev embeddings (17) produces since s < 18/5. This also implies Therefore, due to (34) and (82), we obtain Next, we work on the norms depending onÃ andÃ −1 . On account of (22), (109), and (110), we infer that Using the interpolation inequality (21) and (101), we deduce Finally, by invoking (93) and (110), we also control the L ∞ t,x norms in (114) and, thus, the argument for (113) is concluded.
Following this, we can finish the proof of (99) and, consequently, the proof of our main result by deriving the expected Sobolev regularity for Φ. Proof. We start the argument by using (111) and 7/2 < s < 18/5 to argue that On one hand, a joint application of (53), (54), and (92) yields On the other hand, we observe that and, subsequently, a tedious but direct computation based on (12) leads to From previous calculations, we already know that the last term on the right-hand side has a finite L ∞ L 2 norm, while (93), (100), (92), (101), and together imply 1, and Therefore, we are left to analyze the two integral terms in the expression for ∆ Φ. For the second one, we can use the obvious boundÃ ≥ 1 and (44)-(46) to infer that , r ≥ 1, r 2 |v| 9 , r < 1.
Therefore, we deduce If we rely yet again on (92), we derive which, jointly with the estimates for the other terms in the formula of ∆ Φ, implies Together with (116), (117), and (118), this bound shows that (115) is valid and the argument is finished.
which is analyzed separately in the r ≤ 1 and r > 1 regimes. In the former, we easily have while a Maclaurin analysis yields When r > 1, it follows directly that Based on these findings, (124), and (125), we infer that 1.
Due to this estimate and the ones used in the argument for Φ t (0), we deduce It may seem right now that we could just invoke (105) and, consequently, (130) would follow. However, this is not the case as (119) is required in the main argument at a point that precedes and most likely influences the proof of (105), making this approach circular. However, parts of the asymptotics used in proving (105) can still be employed here, as they were argued for with facts found prior to (119). A term-by-term analysis of the right-hand side of (7) evaluated at t = 0 yields: 1, and 1 r ϕ >1 N (r, rv + ϕ, ∇(rv + ϕ)) t=0 1.
Collectively, these three estimates and (7) show that which, on the basis of (132), implies that (130) holds true. This finishes the proof of this proposition.
Next, we can build off of this result and prove that (120) is valid.
For proving (134), we adopt a similar approach to the one leading to (133), in the sense that we estimate the gradient ∇ t,r = (∂ t , ∂ r ) at t = 0 for each of the terms on the right-hand side of (7). In fact, one can see relatively easily that most of the corresponding terms in the two analyses share a generic core and, thus, we can just investigate the slight differences appearing in this new framework. First, it is straightforward to argue that ∇ t,r 1 r ∆ 3 ϕ L 2 (R 5 ) 1 + v(0) L 2 (R 5 ) + ∇ t,r v(0) L 2 (R 5 )

1.
Secondly, when we deal with terms involving the cutoff ϕ <1 , we notice that differentiation of the expressions having the generic profileÑ (rv)v k is easy to manage. This is due to the control (9) we have onÑ and that the gradient of such an expression is made of terms likẽ N (rv)v k−1 ∇ t,r v,Ñ ′ (rv)v k r∇ t,r v,Ñ ′ (rv)v k+1 .
Therefore, by comparison to the analysis for (133), we either replace v(0) by ∇ t,r v(0) or we have an extra factor of r∇ t,r v(0) or v(0), For the former case, we estimate the gradient in the same L p space (i.e., L ∞ (R 5 ) or L 2 (R 5 )) as we did v(0), while for the latter one we can place both extra factors in L ∞ (R 5 ) due to (125) and to the presence of ϕ <1 , which forces r ≤ 1. Similar arguments can be done for the terms N 3 (rv)v(v 2 t − v 2 r ) and N 4 (rv)rv 4 v r , with slight modifications for when the gradient falls on the derivative terms. In this case, we need to estimate N 3 (rv(0))v(0)(v t (0)∇ t,r v t (0) − v r (0)∇ t,r v r (0)) and N 4 (rv(0))rv 4 (0)∇ t,r v r (0), and we place all factors in L ∞ (R 5 ), with the exception of the second order derivatives, which are bounded in L 2 (R 5 ). We control v tt (0) through (136) and ∂ t v r (0) L 2 (R 5 ) = ∂ r v t (0)) L 2 (R 5 ) ∼ v t (0)) Ḣ1 (R 5 ) 1.

1.
This finishes the discussion of terms localized by ϕ <1 .
Finally, we address the gradient for the terms on the right-hand side of (7) involving N (r, rv + ϕ, ∇(rv + ϕ)). We claim the analysis is almost equivalent to the one just above, with one exception. In this case, factors of r introduced by differentiation are not friendly due to the localization induced by ϕ >1 . However, we claim that in the structure of N (r, rv+ϕ, ∇(rv+ϕ)), there are sufficient negative powers of r to offset this issue and we ask the careful reader to verify this.
Next, we use the Moser inequality (25) for the C ∞ function