LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA

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1 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA DELPHINE CÔTE AND RAPHAËL CÔTE ABSTRACT. We study a class of solutions to the parabolic Ginzburg-Landau equation in dimension or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke s weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the works of Bethuel, Orlandi and Smets [8, 9] to infinite energy data; they allow to consider the point vortices on a lattice (in dimension ), or filament vortices of infinite length (in dimension 3).. INTRODUCTION We consider the parabolic Ginzburg-Landau equation for complex functions u ɛ : d [0, + ) (PGL ɛ ) and its associated energy E ɛ (w) = e ɛ (w)(x)d x = d uɛ t u ɛ = ɛ u ɛ( u ɛ ) on d (0, + ), u ɛ (x, 0) = u 0 ɛ (x) for x d, d w(x) + V ɛ (w(x)) d x for w : d, where V ɛ denotes the non-convex double well potential and e ɛ is the energy density: V ɛ (w) = ( w ), and e 4ɛ ɛ (w) = w + V ɛ (w). It is a classical result that an initial data u 0 L Ḣ yields a global in time solution u(t) ([0, ), L Ḣ ). The Ginzburg-Landau equation (PGL ) in the plane (d = ) admits vortex solutions of the form Ψ(x, t) = Ψ(x) = U l (r) exp(ilθ), l, U l (0) = 0, U l (+ ) = where (r, θ) corresponds to the polar coordinates in (by scaling, (PGL ɛ ) admits stationary vortex solutions as well). Such functions Ψ define complex planar vector fields whose zeros are called vortices (of order l, also called l-vortices). Vortices solutions arise naturally in Physics applications, and it is an important question to study the asymptotic analysis, as the parameter ɛ goes to zero, of solutions to (PGL ɛ ). We must stress out the fact that a single vortex does not belong to Ḣ ( d ) (for d =, Ψ (r, θ) d/r, Ψ / L ( ) either). To overcome this problem, an easy way out is to consider configurations of multiple vortices where the sum of degrees of the vortices is equal to zero. In that case, the initial Date: December 3, Mathematics Subject Classification. 35K45, 35C5, 35Q56. Key words and phrases. Ginzburg-Landau, vortex, mean curvature flow, infinite energy. D.C. would like to thank Fabrice Bethuel for introducing her to this problem, and his constant support and encouragement. R.C. gratefully acknowledges support from the Agence Nationale de la Recherche under the contract MAToS ANR-4-CE

2 DELPHINE CÔTE AND RAPHAËL CÔTE data belongs to the space of energy L Ḣ, and we talk about well-prepared data, see for example Jerrard and Soner [0], Lin [3], Sandier and Serfaty [9], and Spirn [33]. One way to relax this condition was done in the seminal works by Bethuel, Orlandi and Smets [8, 9, 0, ]: they consider (PGL ɛ ) with u ɛ : d [0, + ) and assume that the initial data u 0 ɛ is in the energy space and verifies the bound (.) E ɛ (u 0 ɛ ) M 0 where M 0 is a fixed positive constant. Observe that this condition encompasses large data, and almost gets rid of any well preparedness assumption. This only limitation can be seen as follows in dimension d = (where vortices are points): (.) allows general sum of vortices, which are balanced by adding vortices at infinity (where the center of the vortices goes to spatial infinity as ɛ 0): in that case, for each ɛ > 0, the initial data is of finite energy, but the limiting configuration can be any configuration of finitely many vortices. The main emphasis of [8], valid in any dimension, is placed on the asymptotic limits of the Radon measures µ ɛ defined on d [0, + ), and their time slices µ t ɛ defined on d {t}, by (.) µ ɛ (x, t) = e ɛ(u ɛ )(x, t) d xd t, and µ t ɛ (x) = e ɛ(u ɛ )(x, t) so that µ ɛ = µ t ɛ d t. The bound on the energy gives that, up to a subsequence ɛ m 0, there exists a Radon measure µ = µ t d t defined on d [0, + ) such that µ ɛ µ, and µ t ɛ µt as measures on d [0, + ) and d {t} for all t 0, respectively (see [8, Lemma ] and [9]). The purpose of [8] is to describe the properties of the measures µ t : the main result is that asymptotically, the vorticity µ t evolves according to motion by mean curvature in Brakke s weak formulation. In dimension d =, though, the vorticity µ t is supported on a finite set of points (the vortices). One can actually compute that the energy of a l-vortex is roughly πl : the above bound (.) implies that only a finite number of vortices can be created (at most M 0 /π). However the mean curvature flow for discrete points is trivial, they do not move. Therefore, in order to see the vortices evolve, one needs to consider a different regime, where time is adequately rescaled by a factor. This is done by Bethuel, Orlandi and Smets in [9, 0, ]: they describe completely the asymptotics, and analyze precisely the dissipation times where collision or splitting of vortices occur. Again, the only assumption is the bound (.) on the initial data u 0 ɛ (and thus u0 ɛ is in the energy space). Our goal in this paper is to extend the results in [8], by relaxing the global energy bound (.) to a local one. More precisely, we study families of solutions to (PGL ɛ ) whose initial data u 0 ɛ satisfy the following assumptions, for some constant M 0 > 0: ɛ > 0, u 0 ɛ L ( d ), (H (M 0 )) ɛ > 0, x d, e ɛ (u 0 ɛ )(y)d y M 0. B(x,) Observe that [5, Theorem ] shows the existence of a unique solution u ɛ (t) to (PGL ɛ ) with initial data u 0 ɛ which is globally well defined for positive times. The crucial point is that (H (M 0 )) is a property which propagates along time, if one allows M 0 to depend on time. We also refer to [] where the Ginzburg-Landau functional is studied under a local energy bound in (and a Γ -convergence result is obtained). Let us emphasize that the analysis here is done in the original time scale and not in the accelerated time scale (which is relevant for dimension d = only). d x,

3 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 3 Our main results are the following. We define limiting energy µ t and construct the vorticity set Σt µ (and prove some regularity properties). Then we consider the concentrated energy ν t on Σt µ and show that it evolves under the mean curvature flow in a weak formulation. Finally we focus in dimension d =, and show that in this case Σ t µ is made of a finite set of points which do not move. We start with the description of the vorticity set and the decomposition of the asymptotic energy density. Theorem.. Let (u ɛ ) ɛ (0,) be a family of solutions of (PGL ɛ ) such that their initial conditions u 0 ɛ satisfy (H (M 0 )). Then there exist a subset Σ µ in d (0, + ), and a smooth real-valued function Φ defined on d (0, + ) such that the following properties hold. () Σ µ is closed in d (0, + ) and for any compact subset K d (0, + ) \ Σ µ, u ɛ (x, t) uniformly on K as ɛ 0. () For any t > 0 and x d, Σ t µ = Σ µ d {t} verifies d (Σ t µ B(x, )) C M 0. (3) The function Φ verifies the heat equation on d (0, + ). (4) For each t > 0, the measure µ t can be exactly decomposed as (.3) µ t = Φ (., t) d + Θ (x, t) d Σ t µ µ t (B(x, r)) where Θ (., t) := lim is a bounded function. r 0 ω d r d (5) There exists a positive function η defined on (0, + ) such that, for almost every t > 0, the set Σ t µ is (d )-rectifiable and Θ (x, t) η(t), for d a.e. x Σ t µ. In view of the decomposition (.3), µ t can be split into two parts: a diffuse part Φ, and a concentrated part (.4) ν t := Θ (x, t) d Σ t µ. By (3), the diffuse part is governed by the heat equation. Our next theorem focuses on the evolution of the concentrated part ν t. Theorem.. The family (ν t ) t>0 is a mean curvature flow in the sense of Brakke (see Section 4. for definitions). For our last result, we focus on dimension d =, where vortices are points. We show that these vortex points do not move in the original time scale. Theorem.3. Let d =. Then Σ t µ is a (countable) discrete set of, which we can enumerate Σ t µ = {b i (t) i }. Also, for all x, and the points b i do not move, i.e. Card(Σ t µ B(x, )) C M 0, t > 0, b i (t) = b i, + and ν (t) = σ i (t)δ bi, where the functions σ i (t) are non-increasing. i=

4 4 DELPHINE CÔTE AND RAPHAËL CÔTE Local bounds on the energy, that is, assumption (H (M 0 )), make the set of admissible initial data more natural. We can consider general vortex configurations in dimension, without adding a vortex at infinity to balance it (so as make the total sum of the vortices degrees equal to 0 for each ɛ > 0); important physically relevant examples encompass point vortices on a infinite lattice (dimension ) or general vortex filament families (with possibly infinite length) in dimension 3. Another striking difference betwen the global bound (.) and (H (M 0 )) is the following. In dimension d 3, (.) implies that after some finite time, the vorticity vanishes, that is Σ t µ d [0, T] for some T depending on M 0 (see [8, Proposition 3]). This is now longer the case under (H (M 0 )), which we believe is a more physically accurate phenomenon. The assumption that u 0 ɛ L ( d ) seems technical (because it comes without bounds in term of ɛ), but uneasy to get rid of: the main reason being the lack of a suitable local well posedness in the space of functions with uniformly locally finite energy. Indeed, the closest result in this respect (besides [5] in the L setting), is the work by Ginibre and Velo [7], whose results do not apply to the Ginzburg-Landau nonlinearity, and and even then, their control of the solution at time 0 seems too weak. These results are an extension of the works of Bethuel, Orlandi and Smets [8, 9], and the proofs are strongly inspired by these: Theorems. and. by Theorems A and B in [8] and Theorem.3 by Theorem 3. in [9]. Our main contribution will be to systematically improve their estimates, in order to solve the new problems raised by our dealing with infinite energy solutions of (PGL ɛ ); especially to make sense of a monotonicity property, which is at the heart of the proofs in [8, 9]. We will also need to derive pointwise estimates on u ɛ and L space time estimates on t u ɛ : in the finite energy setting, it appears as the flux of the energy, but this is no longer the case in our context. A leitmotiv of this paper is that, although many of the bounds in [8, 9] are global in time and/or space, their arguments are in fact local in nature, and so can be adapted under the hypothesis (H (M 0 )). In the proofs, we will focus on the differences brought by our change of context, and only sketch the arguments when they are similar to that of [8, 9]. A natural question is now to focus on dimension d = and to study the dynamics of vortices in the accelerated time frame, as it is done in [9, 0, ]. We believe that the arguments in these works could be extended under the hypothesis (H (M 0 )). However one has to make a meaningful sense of the limiting equation, (a pseudo gradient flow of the Kirchoff-Onsager functional involved), as it is not obviously well posed for a countable infinite number of points. We leave these perspectives to subsequent research. This paper is organized as follows. In Section, we study (PGL ɛ ) and prove our main PDE tool, namely the clearing-out (stated in Theorem.). In Section 3, we define the limiting measure and the vorticity set Σ µ : we prove in particular regularity properties of Σ t µ and complete the proof of Theorem.. In Section 4, we show Theorem., that is, the singular part ν t follows the mean curvature flow in Brakke s weak formulation. Finally, in Section 5, we focus on dimension d = and prove Theorem.3.. PDE ANALYSIS OF (PGL ɛ ).. Statement of the main results on (PGL ɛ ). In this section, we work on (PGL ɛ ), that is with smooth solutions u ɛ, where the parameter ɛ, although small, is positive. We derive a number of properties on u ɛ, which enter directly in the proof of the clearing- out Theorem 3.4 at the limit ɛ 0. Heuristically, the clearing-out means that if there is not enough energy in some region of space, then at a later time, vortices can not be created in that region. Let us first state the main results which will be proved in this section.

5 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 5... Clearing-out and annihilation for vorticity. The two main ingredients in the proof of the clearingout are a clearing-out theorem for vorticity, as well as some precise pointwise energy bounds. Throughout this section, we suppose that 0 < ɛ <. We define the vorticity set ɛ as ɛ = Here is the precise statement. (x, t) d (0, + ) : u ɛ (x, t) Theorem.. Let 0 < ɛ <, u 0 ɛ L ( d ) and u ɛ be the associated solution of (PGL ɛ ). Let σ > 0 be given. There exists η = η (σ) > 0 depending only on the dimension n and on σ such that if (.) e ɛ (u 0 ɛ ) exp x η, d 4 then u ɛ (0, ) σ. Notice that we only assume that u 0 ɛ L, whereas in [8, Theorem ] the assumption was E ɛ (u ɛ ) < + ; of course this latter bound will not be available for Theorem.. Observe that L prevents us to use a density argument, and even more so as we are interested in non zero degree initial data. Also, the asumption (.) is not enough by itself to ensure existence and uniqueness of the solution to (PGL ɛ ), but L is suitable (see [5]). Nonetheless the proof follows closely that of [8, Part I], and we will only emphasizes the differences. The proof of Theorem. requires a number of tools, in particular the monotonicity formula, first derived by Struwe [35] in the case of the heat-flow for harmonic maps a localizing property for the energy inspired by Lin and Rivière [6] refined Jacobian estimates due to Jerrard and Soner [] techniques first developed for the stationary equation (for example [4, 5, 6]). Equation (PGL ɛ ) has standard scaling properties. If u ɛ is a solution to (PGL ɛ ), then for R > 0 the function (x, t) u ɛ (Rx, R t) is a solution to (PGL) R ɛ, to which we may then apply Theorem.. As an immediate consequence of Theorem. and scaling, we have the following result. Proposition.. Let T > 0, x T d, and set z T = (x T, T). Let u 0 ɛ L ( d ) and u ɛ be the associated solution of (PGL ɛ ). Let R > ɛ. Assume moreover e R d ɛ (u ɛ )(x, T) exp x x T d x η d 4R (σ), then u ɛ (x T, T + R ) σ. The condition in Proposition. involves an integral on the whole of d. In some situations, it will be convenient to integrate on finite domains. Here is how one should localize in space the conditions. Proposition.3. Let u ɛ be a solution of (PGL ɛ ) satisfying the initial data (H (M 0 )). Let σ > 0 be given. Let T > 0, x T d, and R [ ɛ, ]. There exists a positive continuous function λ defined on (0, + ) such that if then η(x T, T, R) R d B(x T,λ(T)R). e ɛ (u ɛ )(x, T)d x η (σ), u ɛ (x, t) σ for t [T + T 0, T + T ] and x B(x T, R ), where T 0 = max ɛ, ( η η (σ) ) d R (T 0 = ɛ in dimension d = ), and T = R.

6 6 DELPHINE CÔTE AND RAPHAËL CÔTE Furthermore, λ is non increasing on (0, ], and non decreasing on [, + ), and there exists an absolute constant C (not depending on T) such that T > 0, τ (T/, T), λ(τ) Cλ(T). Remark. Recall that in dimension d 3, a bound on the initial energy on the whole space (.) implies that in finite time, the vorticity vanishes (i.e Σ t µ d [0, T]). It is an easy consequence of the monotonicity formula combined with Theorem. ( w,ɛ (x, t, t) 0 uniformly in x). In the case of uniform local bound on the energy H (M 0 ), this result does not persist, because the monotonity formula does not imply the vanishing of w,ɛ for large times. This is one striking difference with the finite energy case.... Improved pointwise energy bounds. The following result reminds of a result of Chen and Struwe [4] developped in the context of the heat flow for harmonic maps. Theorem.4. Let u ɛ be a solution of (PGL ɛ ) whose initial data satisfies (H (M 0 )). Let B(x 0, R) be a ball in d and T > 0, T > 0 be given. Consider the cylinder Λ = B(x 0, R) [T, T + T]. There exist two constants 0 < σ and β > 0 depending only on d such that the following holds. Assume that u ɛ σ on Λ. Then (.) e ɛ (u ɛ )(x, t) C(Λ) e ɛ (u ɛ ), Λ for any (x, t) Λ := B(x 0, R T ) [T +, T + T]. Moreover, 4 e ɛ (u ɛ ) = Φ ɛ + κ ɛ in Λ, where the functions Φ ɛ and κ ɛ are defined on Λ, and verify (.3) Φ ɛ Φ ɛ = 0 in Λ, t κ ɛ L (Λ ) C(Λ)M 0 ɛ β, Φ ɛ L (Λ ) C(Λ)M 0. On Λ one can write u ɛ = ρ ɛ e iϕ ɛ where ϕɛ is smooth (and ρ ɛ = u ɛ ), and we have the bound (.4) ϕ ɛ Φ ɛ L (Λ ) C(Λ)ɛ β. Combining Proposition.3 and Theorem.4, we obtain the following immediate consequence. Proposition.5. Let u ɛ be a solution of (PGL ɛ ) satisfying whose initial data satisfies (H (M 0 )). There exist an absolute constant η > 0 and a positive function λ defined on (0, + ) such that, if for x d, t > 0 and r [ ɛ, ], we havḙ e ɛ (u ɛ ) η r d, B(x,λ(t)r) then e ɛ (u ɛ ) = Φ ɛ + κ ɛ in Λ (x, t, r) B(x, r ) [t + 6 r, t + r ], where Φ ɛ and κ ɛ are as in Theorem.4. In particular, µ ɛ = e ɛ(u ɛ ) C(t, r) on Λ (x, t, r). 4 (The constant η is actually defined as η = η (σ) where σ is the constant in Theorem.4 and η is the function defined in Proposition.3).

7 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA Identifying the sources of non compactness. We identified in the previous arguments a possible source of non compactness, due to oscillations in the phase. But this analysis was carried out on the complement of the vorticity set. Now u ɛ is likely to vanish on ɛ, which leads to a new contribution to the energy: however, this new contribution does not correspond to a source of non compactness, as it is stated in the following theorem. Theorem.6. Let u ɛ be a solution of (PGL ɛ ) whose initial data satisfies (H (M 0 )). Let K d (0, + ) be any compact set. There exist a real-valued function Φ ɛ and a complex-valued function w ɛ, both defined on a neighborhood of K, such that () u ɛ = w ɛ exp(iφ ɛ ) on K, () Φ ɛ verifies the heat equation on K, (3) Φ ɛ (x, t) C( ) M 0 for all (x, t) K, (4) w ɛ Lp ( ) C(p, ), for any p < d + d. Here, C(K) and C(p, K) are constants depending only on K, and K, p respectively. The proof relies on the refined Jacobian estimates of [0]. We stress out the fact that Theorem.6 provides an exact splitting of the energy in two different modes, that is the topological mode (the energy related to w ɛ ), and the linear mode (the energy of Φ ɛ ): in some sense, the lack of compactness is completely locked in Φ ɛ. The remainder of this section is to provide proofs for the results described above, which will be done in section.4; we need some preliminary considerations before... Pointwise estimates. In this section, we provide pointwise parabolic estimates for u ɛ solution of (PGL ɛ ), which rely ultimately on a supersolution argument, i.e a variant of the maximum principle. Proposition.7. Let u 0 ɛ L ( d ) and u ɛ be the associated solution of (PGL ɛ ). Then for all t > 0, u ɛ (t), u ɛ (t) and t u ɛ (t) are in L ( d ). More precisely, there exists a (universal) constant K 0 > 0 such that for all t > ɛ and x d, (.5) u ɛ (x, t), u ɛ (x, t) K 0 ɛ, tu ɛ (x, t) K 0 ɛ. Also, for all t > 0 and x d, u ɛ (x, t) max(, u 0 ɛ L ). Remark. We emphasize that, past the time layer t ɛ, u ɛ (t) L is bounded independently of u 0 ɛ. Proof. We make a change of variable, setting so that the function v satisfies v(x, t) = u ɛ (ɛx, ɛ t), (.6) t v v = v( v ) on d [0, + ). We have to prove that, for t and x d, v(x, t), v(x, t) K 0, t v(x, t) K 0, and that for t > 0 and x d, v(x, t) max( u 0 ɛ L, ). Recall that v b ((0, + ), L ( d )), and that lim sup t 0 + v(t) L u 0 ɛ L (see [5]). We begin with the L estimates for v. Set σ(x, t) = v(x, t). Multiplying equation (.6) by U, we are led to the equation for σ, (.7) σ t σ + v + σ( + σ) = 0.

8 8 DELPHINE CÔTE AND RAPHAËL CÔTE Consider next the EDO (.8) y (t) + y(t)(y(t) + ) = 0, and notice that (.8) possesses the explicit solution defined for t > 0 by (.9) y t0 (t) = exp( (t t 0)/) exp( (t t 0 )/), with t 0 = ln max(, u 0 ɛ, L ) so that y t0 (0) = max(, u 0 ɛ L ) and as consequence We claim that sup x d σ(x, 0) y t0 (0). (.0) t > 0, x d, σ(x, t) y t0 (t). Indeed, set σ(x, t) = y 0 (t). Then (.) t σ σ + σ( + σ) = 0, and therefore by (.7), (.) t ( σ σ) ( σ σ) + ( σ σ)( + σ + σ) 0. Note that + σ + σ = v + σ 0 and σ(0) σ(0) > 0. The maximum principle implies that t > 0, x d, σ(x, t) σ(x, t) 0, which proves the claim (.0). Then observe that t 0 < 0 and that y 0 is decreasing on (0, + ), so that t > 0, x d, σ(t, x) y t0 (t) y 0 (t). Observe that the first bound give v(x, t) max(, u 0 ɛ L for all t > 0 and x d. Al,so for t and x d, v(x, t) + y 0 (). We next turn to the space and time derivatives. Since v(x, t) + y 0 (/) for t /, there exists K (independent of ɛ) such that (.3) t, x d, v(x, t) 3 + v(x, t) K. Let t. Now, differentiating in space the Duhamel formula between times t / / and t gives t v(t) = ( G)(/) v(t /) + ( G)(t s) (v(s)( v(s) )ds, t / where G(x, t) = (4π) d/ e x /4t is the heat kernel. Recall that x G(t) L C/ t. Also, as t / /, there holds v(t /) L and (.3) for all s [t /, t]: hence v(t) L G(/) L v(t /) L + t 0 C + CK t G(t s) L v(s)( v(s) )ds L ds t / Similarly, we can differentiate the Duhamel formula twice: v(t) = ( G)(/) v(t /) + t ds t s CK. ( G)(t s) ((v(s)( v(s) ))ds, t /

9 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 9 Using that (v(s, x)( v(s, x) ) CK v(s, x) and G(t) C/t, we can differentiate once Finally, v(t) L G(/) L v(t /) L t + G(t s) L (v(s)( v(s) ))ds L ds t / C t CK + ds CK. t s t / t v = v + v( v ) v + K CK. We have the following variant of Proposition.7. Proposition.8. Let u 0 ɛ L ( d ) and u ɛ be the associated solution of (PGL ɛ ). Assume that for some constants C 0, C 0 and C 0, x d, u 0 ɛ (x) C 0, u 0 ɛ (x) C ɛ, u 0 ɛ (x) C ɛ. Then for any t > 0 and x d, we have where C depends only on C 0, C and C. u ɛ (x, t) C 0, u ɛ (x, t) C ɛ, tu ɛ (x, t) C ɛ, Proposition.8 provides an upper bound for u ɛ. The next lemma provides a local lower bound on u ɛ, when we know it is away from zero on some region. Since we have to deal with parabolic problems, it is natural to consider parabolic cylinders of the type (.4) Λ α (x 0, T, R, T) = B(x 0, αr) [T + ( α ) T, T + T]. Sometimes, it will be convenient to choose T = R and write Λ α (x 0, T, R). Finally if there is no ambiguity, we will simply write Λ α, and even Λ if α =. Lemma.9 ([8]). Let u 0 ɛ L ( d ) satisfying (H (M 0 )) and u ɛ be the associated solution of (PGL ɛ ). Let x 0 d, R > 0, T 0 and T > 0 be given. Assume that u ɛ on Λ(x 0, T, R, T), then u ɛ C(α, Λ)ɛ ( φ ɛ L (Λ) + ) on Λ α, where φ ɛ is defined on Λ, up to a multiple of π, by u ɛ = u ɛ exp(iφ ɛ ). Proof. We refer to [8, Lemma., p. 5]..3. The monotonicity formula and some consequences. In this section, we provide various tools which will be required in the proof of Theorem The monotonicity formula. For (x, t ) d [0, + ) we set z = (x, t ). For t > 0 and 0 < R t we defined the weighted energy, scaled and time shifted, by E w,ɛ (u ɛ, z, R) = E w (z, R) := e R d ɛ (u ɛ )(x, t R ) exp x x (.5) d x. d 4R Also it will be convenient to use the multiplier (.6) Ξ(u ɛ, z )(x, t) = 4 t t [(x x ). u ɛ (x, t) + (t t ) t u ɛ (x, t)].

10 0 DELPHINE CÔTE AND RAPHAËL CÔTE We stress out that in the integral defining E w, we introduced a time shift δt = R. The following monotonicity formula was first derived by Struwe [35], and used in his study of the heat flow for harmonic maps. Proposition.0. Let u 0 ɛ L ( d ) satisfying (H (M 0 )) and denote u ɛ the associated solution of (PGL ɛ ). We have, for 0 < r < t, d E w dr (z, r) = r d d r ((x x ) u ɛ (x, t r ) r t u ɛ (x, t r )) exp x x d x 4r + V r d ɛ (u ɛ )(x, t r ) exp x x (.7) d x d 4r = (4π)d/ t t Ξ(z )(x, t)g(x x, t t )d xδ t r r (t) d+ + (4π) d/ r V ɛ (u ɛ )(x, t)g(x x, t t )d xδ t r (t), d+ where G(x, t) denotes the heat kernel exp x for t > 0, (.8) G(t, x) = (4πt) d 4t 0 for t 0. In particular, (.9) de w dr (z, r) 0. As a consequence, R E w (z, R) can be extended to a non-decreasing, continuous function of R on [0, t ], with E w (z, 0) = 0. Proof. For 0 < R < t, the map (R, x) e ɛ (u ɛ )(x, t R ) exp x x is smooth (due to 4R parabolic regularization), and satisfies domination bounds due to (.5) and the gaussian weight. Therefore R E w (z, R) is smooth on (0, t ) and we can perform the same computations as in Proposition. in [8]; integrations by parts are allowed for the same reasons. This proves formula (.7), and the monotonicity property follows immediately. It remains to study the continuity at the endpoints. For the limit R 0, the bounds (.5) show that e ɛ (u ɛ )(x, t R ) C(t )/ɛ uniformly for R t /, so that in that range E w (z, R) = R e ɛ (u ɛ )(x, t R ) R exp x x d x d 4R R C(t )/ɛ (4π) d/ 0 as R 0. For the limit R t, let us recall that for any p < +, u ɛ (, t) u 0 ɛ strongly in Lp loc (d ) (see [5, Theorem ]), and as u ɛ (x, t) max( u 0 ɛ L, ), we infer that V (u ɛ )(x, t R ) exp x x d x V (u 0 4R ɛ )(x) exp x x d x. 4t For the derivative term, we use the Duhamel formula: t u ɛ (t) = G(t) u 0 ɛ + G(t s) (u ɛ ( u ɛ ))(s)ds = G(t) u 0 ɛ + D(x, t). 0

11 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA The Duhamel term D(x, t) is harmless, indeed Therefore D(x, t) t 0 G(t s) L u ɛ ( u ɛ ) L ds C max(, u 0 ɛ L )3 t D(x, t R ) exp x x 4R 0 ds t s C max(, u 0 ɛ L )3 t. d x C(t R ) 0 as R t. The linear term requires to recall Claim 3 of [5]. Due to assumption H (M 0 ), for any α > 0, u 0 ɛ L (e α x d x). From [5, Claim 3], we infer that for β > α, (where τ h φ(x) = φ(x h)), and satisfies τ h u 0 ɛ u0 ɛ L (e β x d x) 0 as h 0 τ h u 0 ɛ u0 ɛ L (e β x d x) CeC(β) h. As a consequence, we can apply Lebesgue s dominated convergence theorem and conclude that G(t) u 0 ɛ u0 ɛ L (e β x d x) 0 as t 0. Choose β = /(8t ) and α = /(7t ). Then it follows, using Cauchy-Schwarz inequality, that (G(t R ) u 0 ɛ )(x) exp x x u 0 4R ɛ (x) exp x x d x 4t G(t R ) u 0 ɛ )(x) u0 ɛ (x) ( G(t R ) u 0 ɛ )(x) + u0 ɛ (x) ) exp x x d x 4R + u 0 ɛ exp (x) exp d x x x 4R G(t R ) u 0 ɛ )(x) u0 ɛ (x) L (e β x d x) x x 4t G(t R ) u 0 ɛ )(x) + u0 ɛ (x) L (e β x d x) + o() 0 as R t. (The o() on the second last line comes from u 0 ɛ L (e β x d x) and Lebesgue s dominated convergence theorem.) Hence, summing up, we proved that R E w (z, R) is (left-)continuous at R = t..3.. Bounds on the energy. Lemma.. There exists a constant C (not depending on the dimension) such that the following holds. Let (v ɛ ) ɛ>0 be a family of functions satisfying H (M 0 ), then for any R > 0, ɛ > 0, y d x y, e ɛ (v ɛ )(x) exp d x C(d)( + R) d M 0. Reciprocally, if (w ɛ ) ɛ>0 is a family of functions such that for some R > 0, ɛ > 0, y d x y, e ɛ (w ɛ )(x) exp d x M 0, then (w ɛ ) ɛ>0 satisfies H (C( + R ) d M 0 ). R R

12 DELPHINE CÔTE AND RAPHAËL CÔTE Proof. By translation invariance, we can assume y = 0. We consider the case R (the case R is dealt with R = ). For k in d, denote Q k the cube in d, of length R and centered at Rk d. Then x Q k, x R k R d. Also there exists a constant C(d) such that any cube Q k is covered by C(d)R d balls of radius. Therefore, e ɛ (v ɛ )(x)e x R d x exp (R k R d) e R ɛ (v ɛ )(x)d x k d Q k C(d)R d M 0 d). k d e ( k The series is clearly convergent, which gives the first result claimed. For the second, we clearly have x y e ɛ (w ɛ )(x) exp d x e R ɛ (w ɛ )(x)d x. e B(y,R) This means that the energy on balls of radius R is at most em 0. If R, then this is enough. If R, then any ball of radius can be covered by at most C(d)/R d balls of radius R, so that for all y d, e ɛ (w ɛ )(x)d x C(d) em B(y,R) R d 0. The first consequence of the monotonicity formula is that (H ) is a condition which propagates in time in the following way. Proposition.. Let u ɛ be a solution of (PGL ɛ ) satisfying the initial condition (H (M 0 )). Then for any T > 0, (x, t) u ɛ (x, T + t) is still a solution of (PGL ɛ ), whose initial condition satisfies H (C(d)( + T)M 0 ). More precisely there holds (.0) ɛ > 0, y d, R > 0, e ɛ (u ɛ )(x, t)d x C(d)( + R) d ( + t)m 0. B(y,R) y, t + between 4 and Proof. Let R =, then applying the monotonicity formula at the point t +, we get 4 d e ɛ (u ɛ )(x, t) exp x y d d x e d (4t + ) d ɛ (u 0 ɛ ) exp x y d x. d 4t + Hence, d e ɛ (u ɛ )(x, t) exp x y d x ( + 4t)) C(d)( + t)m 0. d C(d) ( + 4t) d/ M 0 Using Lemma. we have the result for R = and so for R. Finally, for R, we cover B(y, R) by balls of radius, which can be done with at most C(d)R d balls. As an immediate consequence, we infer an upper bound on the energy on compact sets. The bound (.) below will be very useful in order to prove Theorem.6 in the same way as in [8].

13 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 3 Corollary.3. Let u ɛ be a solution of (PGL ɛ ) satisfying the initial data (H (M 0 )). Then for any compact K d [0, + ), we have (.) e ɛ (u ɛ )(x, t)d xd t C(K)M 0. K Proof. Being compact, K is bounded: K B(0, R) [0, T]. Therefore, e ɛ (u ɛ )(x, t)d xd t e ɛ (u ɛ )(x, t)d xd t K B(0,R) [0,T] T C(d)( + R) d ( + t)m 0 d t C(d)( + R) d ( + T)T M 0. 0 Whence (.) Space-time estimates. One crucial lack in relaxing (H 0 ) to (H ) is that the energy no longer provides a bound on t u ɛ L x,t E ɛ (u ɛ ). To remedy this, we make use of the Ξ multiplier. Lemma.4. Let u 0 ɛ L ( d ) and u u ɛ be the associated solution of (PGL ɛ ). For any z = (x, t ) d [0, + ), the following equality holds, for R = t. (.) (V ɛ (u ɛ ) + Ξ(u ɛ, z ))(x, t) G(x x, t t ) d xd t d [0,t ] where Ξ is defined in (.6). = (4π) d/ t d e ɛ (u)(x, 0) exp d {0} x x 4t d x = E w (z, R ), Proof. Integrating equality (.7) from 0 to R (recall that E w (z, 0) = 0), we obtain R (.3) (4π) d/ E w (z, R ) = rdr V ɛ (u(x, t))g(x x, t t ) d x 0 d {t r } R + rd r (x x ) u 4r ɛ r t u G(x x, t t ) d x. 0 d {t r } Expressing the integral on the right-hand side of (.3) in the variable t = t r (so that d t = rd r) yields 0 (4π) d/ E w (z, R ) = d t V ɛ (u(x, t))g(x x, t t ) d x t d {t} 0 rd r 4 t t ((x x ). u r t u) G(x x, t t ) d x. t d {t} Proposition.5. For any compact K d [0, + ), there exist a constant C(K) such that (.4) t u ɛ (x, t) d xd t C(K)M 0 K Proof. It suffices to prove the bound on the compacts K T = B(0, T) [0, T] for all T. Let t = T and x = 0, then for (x, t) B(0, T) [0, T], we have t t T x x so that G(x x, t t ) exp x x e /4 (4π(t t )) d/ 4(t t ) (4πT) d/

14 4 DELPHINE CÔTE AND RAPHAËL CÔTE and t u ɛ (x, t) T 4 t t ((t t ) t u ɛ (x, t)) T Ξ(u ɛ, z )(x, t) + t t ((x x ). u ɛ (x, t)) T Ξ(u ɛ, z )(x, t) + u ɛ (x, t). Therefore, using V ɛ (u ɛ ) 0, (.) and (.), we get T 0 B(0, t u ɛ (x, t) d xd t C(T) T) Ξ(u ɛ, z )(x, t)g(x x, t t )d xd t d [0,t ] + u B(0, ɛ (x, t) d xd t t ) [0,t ] C(T) (V ɛ (u ɛ ) + Ξ(u ɛ, z ))(x, t)g(x x, t t )d xd t d [0,t ] + e B(0, ɛ (u ɛ )(x, t)d xd t t ) [0,t ] C(T)E w (z, t ) + C(t )M 0 C(T)M Localizing the energy. In some of the proofs of the main results, it will be convenient to work on bounded domains for fixed time slices. But since the integral in the definition of E w is computed on the whole space, we will have to use two kinds of localization methods. The first one results from the monotonicity formula. Proposition.6. Let u ɛ be a solution of (PGL ɛ ) satisfying the initial bound (H (M 0 )). Let T > 0 x T d and r, λ > 0. There holds (.5) d e ɛ (u ɛ )(x, T) exp x x T 4r + C(d) d x e ɛ (u ɛ )(x, T)d x B(x T,λr) r T + r d exp λ 8 ( + T + r ) d M 0. Proof. We split the integral between B(x T, λr) and its complement. Now for x such that x x T λr, we have exp x x T exp λ exp x x T. 8r 8 8r Therefore e ɛ (u ɛ )(x, T) exp d x x T 4r d x e ɛ (u ɛ )(x, T)d x + e λ /8 B(x T,λr) d e ɛ (u ɛ )(x, T) exp x x T 8r d x.

15 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 5 Now, we apply the monotonicity formula to the second integral term of the right hand side, at the point (x T, T + r ), and between r and T + r. We get ( e ɛ (u ɛ )(x, T) exp x x T d x r) d d 8r e (T + r ) d ɛ (u 0 ɛ )(x) exp x x T d x d 4(T + r ) and the conclusion follows. C(d) ( + T + r ) d (T + r ) d M 0, The second localization method, inspired by Lin and Rivière [6] is based on a Pohozaev type inequality. Proposition.7. Let u 0 ɛ L ( d ), u ɛ be the associated solution of (PGL ɛ ) satisfying the initial bound (H (M 0 )), and Ξ as in (.6). Let 0 < t < T. The following inequality holds, for any x T d : d e ɛ (u ɛ )(x, t) x x T 4(T t) exp x x T d x 4(T t) d e ɛ (u ɛ )(x, t) exp x x T d x d 4(t t) + (V ɛ (u ɛ ) + 3Ξ(u ɛ, z T )) exp d x x T 4(T t) As a consequence, e ɛ (u ɛ ) exp( x x T d {t} 4(T t) ) d x e ɛ (u ɛ ) exp( x x T B(x T,r T ) {T} 4(T t) ) d x + (V ɛ (u) + 3Ξ(u, z T )) exp( x x T d d {t} 4(T t) ) d x where r T = d(t t)..4. End of the proof of clearing-out. In this paragraph, we complete the proofs of Theorems.,.4 and.6. Outline of the proof of Theorem.. The proof follows word for word that of Theorem in [8, Sections 3, p ]. Indeed, either calculations are made on bounded domains K [T 0, T ] (where K is a compact of d and T > T 0 > 0), on which we have the same kind of bounds for the energy: for any t [T 0, T ]: e ɛ (u ɛ ) (x, t)d x C(K, T )M 0 (obtained in K (.0)), for the kinetic energy, in L (d xd t): t u ɛ (x, t)d x C(K, T)M 0 (obtained in K [0,T ] (.4)) pointwise: u ɛ + ɛ u ɛ + ɛ t u ɛ (x, t) C for (x, t) K [ɛ, T ] (obtained in (.5) and useful as soon as ɛ is so small that T > ɛ ), or we multiply the energy by a weight of the form exp x R, for which we have the monotonicity formula and bounds (.5). For the convenience of the reader, we remind the main steps of the argument. d x.

16 6 DELPHINE CÔTE AND RAPHAËL CÔTE σ Let r ɛ = K ɛ and t ɛ = r ɛ 0. As tu ɛ K/ɛ (in view of (.5)), (.6) u ɛ (0, ) u ɛ (0, t ɛ ) + K ɛ ( t ɛ) u ɛ (0, t ɛ ) + σ. Now, as x u ɛ K/ɛ (again (.5)), one easily deduces (see [5, Lemma 3.3 p.458]) that u ɛ (0, t ɛ ) C ( u ɛ d ɛ (x, t ɛ ) ) d+ (.7) d x. B(0,ɛ) This bound can be related to the weighted energy as follows: ( u ɛ d ɛ (x, t ɛ ) ) d x C (.8) V rɛ d ɛ (u ɛ )(x, t ɛ ) exp x d x E rɛ w,ɛ ((0, ), r ɛ ). B(0,ɛ) B(0,ɛ) Let δ (0, /) to be fixed at the end of the proof. By considering the variations of the weighted energy on time intervals of the form [ δ k, δ (k+) ] for 0 k /4 (so that δ k r ɛ and E w,ɛ ((0, ), r ɛ ) E w,ɛ ((0, ), δ k ) by monotonicity), at least one of these intervals (say for k 0 ) shows a decay less than 8η ln δ (we gained a factor). One can actually perform a time shift and rescaling (as k /4), so that we can furthermore assume k 0 = 0: (.9) E w,ɛ (u, (0, ), ) E w,ɛ (u, (0, ), δ) 8η ln δ, E w,ɛ ((0, ), r ɛ ) E w,ɛ ((0, ), ). Gathering (.6), (.7), and (.8), u ɛ (0, ) σ + C E w,ɛ((0, ), ) d+. The crux of the argument is therefore to bound E w,ɛ ((0, ), ) solely in terms of η and more precisely, (.30) E w,ɛ ((0, ), ) C(δ) η. To prove (.30), the starting point is the observation that u ɛ u ɛ = u ɛ u ɛ + u ɛ u ɛ, (which can be derived easily writing u ɛ = u ɛ e iϕ ɛ ). As uɛ C/ɛ from (.5), and we get the pointwise bound ( u ɛ ) u ɛ u ɛ + CV ɛ (u ɛ ), e ɛ (u ɛ ) u ɛ u ɛ + CV ɛ (u ɛ ) + u ɛ u ɛ. It is not so hard to obtain improved bounds on u ɛ and its derivative. Indeed, write the parabolic equation for u ɛ : using (.) (in particular to treat the t u ɛ terms) and an averaging in time argument, one can infer that the set of times t such that uɛ + V ɛ (u ɛ ) (x, t)d x C(δ) η(e w ((0, ), ) + ) B(0,) is of large relative measure in [ 4δ, δ ] (for some explicit C(δ)). It follows that the crucial term to estimate is u ɛ u ɛ. For this term, one has the following Hodge-de Rham decomposition u ɛ u ɛ = dφ(t) + d ψ(t) + ξ(t) on B(0, 3/) {t} where d is the exterior derivative on d (d is its adjoint). It is constructed as follows: define the Jacobian Ju ɛ := d(u ɛ du ɛ ), and let ψ(t) be such that ψ(t) = χju ɛ where χ is a cut-off function on B(0, ). Observe that d dψ(t) is closed on B(0, 3/) {t}; invoking the Poincaré lemma, there exists ξ(t) such that dξ(t) = d dψ(t), d ξ(t) = 0. Then φ(t) is obtained by invoking once again the Poincaré lemma. Direct elliptic estimates show that ξ(t) L (B(0,3/)) C ψ(t) L (B(0,)).

17 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 7 Noticing that u ɛ t u ɛ = d (u ɛ du ɛ ), we have the elliptic equation for φ(t): for any small δ > 0, + x x φ(t) = u δ ɛ δ u ɛ t u ɛ (d x ψ(t) + ξ(t)) δ. One can obtain weighted elliptic estimates for the operator + x x δ = exp exp x 4δ 4δ and from there obtain the bound φ(t) exp x d x Cδ d E 4δ w ((0, ), δ) + C(δ) R(t) + R(t)E w ((0, ), δ), where R(t) only involves ξ(t) (already bounded), Ξ(u ɛ ) and V ɛ (u ɛ ) (taken care of by (.)) and ψ(t): R(t) = (Ξ(u ɛ, (0, ))(x, t) + V ɛ (u ɛ )) exp x d x d 4δ + ψ(x, t) + ξ(x, t) ) exp x d x. B(0,3/) 4δ The remaining task is to estimate ψ(t), which is the most involved. A first ingredient is a refined estimate on the Jacobian due to Jerrard and Soner []: there exist β, C > 0 such that for any smooth w and test function ϕ, (.3) Jw, ϕ d x d C ϕ L + Cɛ β ϕ W, e ɛ (w)d x Supp ϕ + e ɛ (w)d x Supp ϕ ( + d (Supp ϕ) ). (Observe the gain). A second ingredient is to compare ψ with the solution ψ to the analoguous heat equation t ψ ψ(t) = χjuɛ. One can get bounds on ψ by the use of the monotonicity formula on u ɛ, and then relate to ψ by treating t ψ as a perturbation. After an averaging in time argument, one can choose a right time slice t [ 4δ, δ ] such that, using the bounds from (.) and (.4), (Ξ(u ( t) d/ ɛ, (0, ))(x, t) + V ɛ (u ɛ ))(x, t) exp x d 4( t) ln( t) d x C η ln ɛ, t (a bound suitable for ψ(t) and ξ(t)) and ψ(t) d x C(δ)ɛ /6 E w ((0, ), δ) B(0,) + C(δ)(E w ((0, ), δ) + ) V ɛ (u ɛ )(t) exp x d x. d 4δ Combining all the above (and the monotonicity formula), one can choose δ > 0 small enough (independent of η or u ɛ ) so that E w ((0, ), δ) E w((0, ), ) + C η. Then using the first part of (.9), this proves (.30) and the proof is complete. From Theorem., Proposition. follows immediately. We now provide a proof of Proposition.3 which also is a consequence of Theorem..

18 8 DELPHINE CÔTE AND RAPHAËL CÔTE Proof of Proposition.3: Let x 0 be any given point in B x T, R. The crux of the argument is the following claim. Claim.8. We can find λ(t) > 0 such that, for every T 0 < r < T, e r d ɛ (u ɛ )(x, T) exp x x 0 d x η d 4r, provided that η η (recall that λ enters in the definition of η). To prove the claim, we use (.5). Let λ > 0 and T 0 < r < T = R, we have e r d ɛ (u ɛ )(x, T) exp x x 0 d x e d 4r r d ɛ (u ɛ )(x, T)d x (.3) As r, it follows that B(x 0,λr) d r + C(d) exp λ ( + T + r ) d M 0. T + r 8 d r ( + T + r ) d ( + T)d C(d). T + r We first choose a function λ 0 continuous such that for T > 0, λ 0 (T) is so large that ( + T)d (.33) C(d) exp λ 0 (T) M 0 η (σ). 8 T d Observe this can be done with furthermore requiring that there exists an absolute constant c > 0 such that T > 0, c ln( + /T) λ 0 (T) ln( + /T). c Then it follows that Finally define T > 0, τ (T/, T), λ(t) = T d λ 0 (τ) λ 0 (T). supt [T,] λ 0 (t) if T, sup t [,T] λ 0 (t) if T. Then the function λ is positive, continuous and satisfies the conditions of Proposition.3. Furthermore, since x 0 belongs to B(x, R/) and r < R, it follows that Therefore, r d e ɛ (u ɛ )(x, T)d x B(x 0,λ 0 R) d Choosing T 0 = η d R, we obtain η (.34) r d B(x 0, λ 0 (T)r) B(x T, λ(t)r). B(x 0,λ 0 (T)R) R d r R d R d η r e ɛ (u ɛ )(x, T)d x η. e ɛ (u ɛ )(x, T)d x B(x T,λ(T)R) R T0 d η. We finally combine (.3), (.33) and (.34), and the claim is proved. The conclusion then follows from Proposition..

19 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 9 The proofs of both Theorems.4 and.6 are now exactly the same as in [8] (see sections 5. to 5.4) since computations are made on compact domains Ω d (0, + ), where the bounds (.0), (.4) and (.5) hold. As in the proof of Theorem. (and explained at its beginning), these are the only bounds which are made use of, along with the monotonicity formula and the identity (.). For the convenience of the reader, we remind a few elements of the proofs. Outline of the proof of Theorem.4. One writes u ɛ = u ɛ e iϕ ɛ, and starts with the equation for the phase ϕ ɛ. As u ɛ /, it is a uniformly parabolic equation. After performing a spatial cut-off in order to get rid of boundary conditions on Λ, one splits ϕ ɛ = ϕ ɛ,0 + ϕ ɛ,, where ϕ ɛ,0 is a solution to the linear heat equation (which turns out to be the desired phase Φ ɛ ) and ϕ ɛ, is a solution to a nonlinear equation with 0 initial data. ϕ ɛ,0 admits improved bounds due to parabolic regularity ϕ ɛ,0 L L (Λ 3/4 ) + ϕ ɛ,0 L (Λ 3/4 ) C(Λ) e ɛ (u ɛ ). ϕ ɛ, is shown to be essentially a perturbation (via a fixed point argument), and so ϕ ɛ enjoys integrability bounds in L L q (Λ 3/4 ) for some q > (depending on d). Plugging this information in the equation for u ɛ, one infers that e ɛ (u ɛ ) (.35) ϕ ɛ,0 + κ ɛ, where κ ɛ C(Λ)M 0 ɛ α, Λ 3/4 for some α depending only on d. An extra ingredient is provided by a result by Chen and Struwe [4] where the bound (.) is proved under an extra assumption of small energy. Together with (.35) and a scaling argument, one can relax the small energy assumption, and prove (.). Finally, inserting (.) in the equation for u ɛ, Lemma.9 allows to improve the estimates to u ɛ + V ɛ (u ɛ ) C(λ)ɛ α, and from there, obtain L bounds on ϕ ɛ, : this yields the bounds (.3) and (.4), and completes the proof of Theorem.4. Outline of the proof of Theorem.6. It suffices to prove the estimates on any cylinder of the form Λ = B [T 0, T ] where B is a ball and T > T 0 > 0. Up to increasing slightly the cylinder (and due to (.)), one can assume that (.36) e ɛ (u ɛ )d xd t + Λ e ɛ (u ɛ )dσ(x, t) M ln ɛ. Λ Denote now δ and δ the (space time) exterior derivatives on d+. The main step is a Hodge-de Rham type decomposition with estimates: (.37) u ɛ δu ɛ = δφ + δ Ψ + ζ, where a key feature is to prove that in addition to the expected bound Φ L (Λ) + Ψ L (Λ) C(Λ), one also has for any p, + d (.38) Ψ Lp (Λ) C(p, Λ)M, and ζ Lp (Λ) C(p, Λ) ɛ. In other words, Φ completely accounts for the lack of compactness. The argument for (.37) is as follows: one starts with the usual Hodge-de Rham decomposition on Λ (which is simply connected: here we use that (d + ) ) so that u ɛ δu ɛ = d Λ Φ Λ + d Λ Ψ Λ. Λ

20 0 DELPHINE CÔTE AND RAPHAËL CÔTE Then on Λ, Λ Ψ Λ = J Λ u ɛ : the Jerrard and Soner estimate on the Jacobian (.3) gives the / ln ɛ gain J Λ u ɛ C 0,α ( Λ) C(Λ), from where a similar gain is derived on Ψ Λ. Let Φ 0 be the harmonic extension of Φ Λ on Λ: we gauge away by considering on Λ v ɛ := u ɛ e iφ 0. One now considers on Λ the Hodge-de Rham decomposition v ɛ δv ɛ = δφ + δ Ψ, so that u ɛ δu ɛ = δφ + δ Ψ + δφ 0 + ( u ɛ )δφ 0. Define ζ := ( u ɛ )δφ 0 : it satisfies (.38) using the bound on the energy (.36), and ζ Λ as well. For Ψ, one has again Ψ = Jw ɛ, now with boundary conditions involving Ψ Λ and ζ Λ. A Jacobian estimate similar to (.3) (and proven in [7, Proposition II.]) thus yields the bound (.38) on Ψ. Finally defining Φ := Φ 0 + Φ completes the decomposition. With (.37) and (.38) at hand, we now define Φ ɛ by writing Φ = Φ ɛ + Φ, where t Φ Φ = t Φ Φ on Λ, with the boundary condition Φ = 0 on B {T 0 } B (T 0, T ). In particular, Φ ɛ solves the (homogeneous) heat equation on Λ. From parabolic estimates, Φ ɛ L (Λ) C x,t Φ L (B {T 0 } B (T 0,T )) C(Λ) ln ɛ, which is the first estimate (iii). It remains to bound w ɛ := u ɛ e iφ ɛ in Ẇ,p (Λ). First we separate between A Λ = {((x, t) Λ w ɛ (x, t) ɛ /4 } and B Λ = Λ \ A Λ. On B Λ, the rough estimate w ɛ (x, t) C/ɛ inherited from (.5) yields w ɛ p L p (B Λ ) C(Λ) ( w ɛ (x, t) ) d xd t ɛ p Λ ɛ / C(Λ)ɛ p / ( u ɛ (x, t) ) d xd t C(Λ)ɛ 3/ p ln ɛ 0 ɛ Λ (observe that u ɛ = w ɛ and p < d + 3 d ). We now work on A Λ. Then (with the same computation as in the proof of Theorem.), we have w ɛ w ɛ w ɛ = w ɛ w ɛ + w ɛ w ɛ ( u ɛ ) + w ɛ w ɛ. It follows from (.38) (writing w ɛ in terms of u ɛ ) that w ɛ w ɛ Lp (Λ) C(Λ). Then one actually bounds ( u ɛ ) Lp (Λ): for this, one writes the (uniformly) parabolic equation for ρ = u ɛ, from which we get ρ χ ( ρ) u ɛ χ + ( ρ χ + t ρ χ) ρ, (where χ is a suitable non negative cut-off function, χ = on Λ). To bound the right-hand side, we split again between A Λ and B Λ : B Λ has small measure (arguing as before) and on A Λ one uses smallness of ρ. Gathering all gets a final bound ( u ɛ ) p L p (Λ) C(Λ)ɛ/4 p/8 ln ɛ 0. This gives estimate (iv).

21 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 3. DESCRIPTION OF THE LIMITING MEASURE AND CONCENTRATION SET Let u ɛ be a solution of (PGL ɛ ) satisfying the initial data (H (M 0 )). Our goal in this section is to study the asymptotic limit, as ɛ 0, of the Radon measures µ ɛ defined on d [0, + ) by µ ɛ (x, t) = e ɛ(u ɛ )(x, t) d xd t. To that purpose, we will study their time slices µ t ɛ defined on d {t} by µ t ɛ (x) = e ɛ(u ɛ )(x, t) When ɛ 0, these measures converge to µ and µ t respectively (up to a subsequence), as it is shown in the next paragraph. Then we will study the evolution of µ t, and show that it follows Brakke s weak formulation of the mean curvature flow. For this, we will of course heavily rely on the properties of µ t ɛ obtained in the previous section. 3.. Absolute continuity with respect to time of the limiting measure. According to inequality (.0), we have for all R > 0 and T > 0, (3.) dµ ɛ (x, t) C(d)T( + T)( + R) d M 0. B(0,R) [0,T] The bound (3.) yields a limiting measure via a diagonal extraction argument. This can also be done simultaneously for the time sliced measures: more precisely, following the proof in Brakke [3] word for word, we have the following. Theorem 3.. There exist a sequence ɛ m 0, a Radon measure µ defined on d [0, + ), bounded on compact sets and, for each t 0, a Radon measure µ t on d {t} such that: d x. (3.) µ ɛm µ as m, and for all t 0, µ t ɛ m µ t as m. Moreover, the (µ s ) s enjoy the bound (3.3) R > 0, t > 0, µ t (B(0, R)) C(d)( + t)( + R)d M 0. and µ = µ t d t. For the proof of Theorem 3., we will need a few classical identites for the evolution of µ t ɛ. Before we dive into the study of the singular measure µ t, let us state two useful following identities. Lemma 3.. Let u 0 ɛ L ( d ) and u ɛ be the associated solution of (PGL ɛ ). Then, for all χ ( d ) and for all t 0, we have d (3.4) χ(x)dµ t ɛ d t = χ(x) tu ɛ d x + χ(x) tu ɛ. u ɛ d x. d d {t} d {t} We usually choose χ 0, and this choice makes the first term of the right-hand side non positive. To handle the second term, we provide another identity which involves the stress-energy tensor. Lemma 3.3. Let X ( d, d ). Then for all t 0, (3.5) eɛ (u ɛ )δ i j i u ɛ j u ɛ j X i = d {t} d {t} X. tu ɛ. u ɛ (Here we use Einstein s convention of implicit sommation over repeated indices.) d x.

22 DELPHINE CÔTE AND RAPHAËL CÔTE The proof of Lemma 3.3 is given in [9], and involves the stress-energy matrix A ɛ given by (3.6) A ɛ = A ɛ (u ɛ ) := e ɛ (u ɛ ) Id u ɛ u ɛ = T(u ɛ ) + V ɛ (u ɛ ) Id, where the matrix T(u) and the potential V ɛ are given by (3.7) T(u) = u Id u u, V ɛ (u) = ( u ) 4ɛ. Combining Lemma 3. and Lemma 3.3 with the choice X = χ, we get rid of the time derivative of the right hand side of (3.4). More precisely Lemma 3.4. χ(x)dµ t ɛ t = χ(x) tu ɛ d d {t} Proof of Theorem 3.. It boils down to the following claim. D χ u d x + ɛ. u ɛ χe ɛ (u ɛ ) d x. d {t} Claim 3.5. Let T > 0. We consider a sequence ɛ m 0. Then there exists a subsequence ɛ σ(m) such that for every s [0, T], µ s ɛ σ(m) µ s as m, in the sense of measures. Indeed let us proof Theorem 3. assuming this claim holds. With T n = n, the claim let us dispose of sequences ɛ n,m 0 as m + such that µ s ɛ enjoys the desired convergence on s [0, T n,m n ]. By an argument of diagonal extraction, we deduce that there exist a sequence ɛ m 0 and, for each s 0, a measure µ s on d {s} such that for every s 0, µ s ɛ µ s m as m. We now prove the claim. We use the following lemma, which is an easy variant of Helly s selection principle. Lemma 3.6. Let I be an at most countable set, and let (f m i ) m,i I be a collection of real-valued functions defined on some interval (a, b). Assume that for each i I, the family (f m i ) m is equibounded and satisfies the following semi-decreasing property (3.8) For all δ > 0, there exist τ > 0 and m i such that, if s, s (a, b) and s τ s s, then for all m m i, f i m (s ) f i m (s ) + δ. Then there exist a subsequence σ(m) and a family (f i ) i I of real-valued functions on (a, b) such that for all s (a, b) and i I, f i σ(m) (s) f i (s). Let (χ i ) i I be a countable family of compactly supported non-negative smooth functions on d ; assume that for all i I, 0 χ i, and that Span(χ i ) i I is dense in 0 c (d ). Let m 0 be such that if m m 0, then ɛ m. We define for m, i I the function f i m defined on [0, T] by f i m (s) = d χ i dµ s ɛ m. Step. We first show that (f m i ) mm0 satisfies (3.8). Let i I. Recalling Lemma 3.4, we have d ds χ i dµ s ɛ d m = χ i (x) tu ɛm d {s} ln ɛ m d x + (D χ i u ɛm. u ɛm χ i e ɛm (u ɛm )) d x. ln ɛ m d {s}

23 LIMITING MOTION FOR THE PARABOLIC GINZBURG-LANDAU EQUATION WITH INFINITE ENERGY DATA 3 Therefore d χ i dµ s ɛ (D χ ds d m i u ɛm. u ɛm χ i e ɛm (u ɛm )) d x. d {s} Let R > 0 such that Supp(χ i ) B(0, R). We have, for s [0, T] d ds f i m (s) = d χ i dµ s ɛ 3 χ ds d m i (3.9) 3 χ i M 0 C(T, R), B(0,R) {s} by Proposition.3. δ Let δ > 0. We set τ = 3 χ i C M 0 C(T, R). If s, s [0, T] and s τ s s, the inequality (3.9) leads to Therefore e ɛm (u ɛm ) d x ln ɛ m f i m (s ) f i m (s ) 3 χ i (s s )M 0 C(T, R) (s s ) δ δ. τ m m 0, f i m (s ) f i m (s ) + δ. Now, we prove that i I, the family (f i m ) mm 0 is equibounded. Let i I. Let R > 0 such that Supp(χ i ) B(0, R) and t [0, T]. We have χ i dµ t ɛ dµ t d m ɛ M m 0 C(T, R). B(0,R) Therefore, m m 0, f i m L C(T, R), i.e. the family (f i m ) mm 0 is equibounded. According to Lemma 3.6, there exist a subsequence σ(m) and for all i I, a Radon measure f i such that for all i I, and s (a, b) f i σ(m) (s) f i (s). Hence, for all s [0, T], µ s ɛ σ(m) (χ i ) converges as m +. Step. Let s 0 [0, T] be arbitrary but fixed, and χ C 0 c (d ). Let us show that (µ s 0 ɛσ(m) (χ)) m is a Cauchy sequence in. Let α > 0. Since Span(χ i ) is dense in C 0 c (d ), there exist a finite subset J I, and real numbers (λ j ) j J such that λ j χ j χ j J L α 3M 0 C(T, R). Let m k be two integers, we have µ s 0 ɛσ(m) (χ) µ s 0 ɛσ(k) (χ) µs 0 ɛσ(k) (χ) λ j µ s 0 ɛσ(k) (χ j ) + λ j (µ s 0 ɛσ(m) (χ j ) µ s 0 ɛσ(k) (χ j )) j J j J + λ j µ s 0 ɛσ(m) (χ j ) µ s 0 ɛσ(k) (χ). Recall that µ s 0 ɛσ(k) (χ) j J j J λ j µ s 0 ɛσ(k) (χ j ) = (χ d j J λ j χ j )dµ s 0 ɛσ(k).