Global solutions of aggregation equations and other flows with random diffusion

Aggregation equations, such as the parabolic-elliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier–Lebesgue spaces with quantifiable high probability.

Here, g(x) = 1 2π ln |x| is the Newtonian potential on R 2 and ν ≥ 0 is the diffusion strength.It is natural to assume that θ 0 ≥ 0 and consider solutions θ ≥ 0, as θ is supposed to represent a density.If ν > 0, then equation (1.1) is known as the parabolic-elliptic Patlak-Keller-Segel (PKS) equation, which is a model for the aggregation of cells by chemotaxis [Pat53,KS70,Nan73].If ν = 0, then the equation is the gradient flow of the Newtonian energy with respect to the 2-Wasserstein metric.The equation has been studied as a model for the evolution of vortex densities in superconductors [E94,CRS96] and as a model for adhesion dynamics [NPS01,Pou02].
It is a straightforward calculation that any smooth solution to (1.1) conserves mass, so we can unambiguously write M = R 2 θ(x)dx = R 2 θ 0 (x)dx.Suppose that θ is a solution to (1.1) with finite second moment R 2 |x| 2 θ t (x)dx.Evidently this quantity is strictly positive if θ t is not identically zero.Using integration by parts, one computes Thus, if M > 8πν, then the second moment is strictly decreasing at a linear rate.Since the second moment is nonnegative, this implies that the maximal time of existence for θ is finite.In particular, we see that if ν = 0, so there is no diffusion, then any nonzero, sufficiently localized classical solution to (1.1) must have finite lifespan [JL92].In fact, for initial datum in L 1 , one has a unique, global mild solution to (1.1) if and only if M ≤ 8πν [Wei18].For the asymptotic behavior of solutions, we refer to [BDP06,CD14] (M < 8πν), [BCM08,GM18] (M = 8πν), and [Vel02,Vel04a,Vel04b] (M > 8πν), and references therein.In the case ν = 0, one has a sharp bound for the time of existence for compactly supported L ∞ weak solutions to (1.1), which are necessarily unique, as well as exact solutions that provide an explicit example of finite-time collapse to a nontrivial measure [BLL12].
For the deterministic dynamics of equation (1.1), we see that global existence is a nonstarter for classical solutions if the diffusion is too weak relative to the size of the initial data.But in the past two decades, there has been intense research activity on understanding how adding some noise structure (in varying forms) to deterministic equations can impact the behavior of solutions.A small, non-exhaustive sample of this research is given by [dBD02, dBD05, FGP10, FGP11, DT11, Fla11, GHV14, CG15, BFGM19, BNSW20, FL21] and references therein.Concerning equations of the form (1.1), Flandoli et al. [FGL21] have shown that blow-up is delayed in a 3D version of (1.1) with positive ν on T d by adding a suitable multiplicative noise of transport type.Misiats et al. [MST21] have shown that some choices of random perturbations of equation (1.1) with ν > 0 lead to global solutions for small-mass initial data, while other choices lead to finite-time blow-up with positive probability for all initial data .
To the best of our knowledge, prior works have not shown that noise prevents finite-time blow-up, in particular for the case ν = 0 when all smooth, sufficiently localized solutions necessarily blow up in finite time. 1 Recently, the second author together with Buckmaster et al. [BNSW20] showed that adding random diffusion leads to global solutions with positive probability for the invsicid surface quasi-geostrophic (SQG) equation with Gevrey-type initial data.Unlike the equation (1.1), inviscid SQG is a Hamiltonian flow, and the long-time dynamics of classical solutions is still unresolved.In light of this result, it is natural to ask if random diffusion may somehow improve the existence theory for the equation (1.1), for which solutions a priori behave very differently.Thus, we pose the following question, which the present article seeks to answer.
Question 1.1.Can one restore global existence of sufficiently regular solutions to (1.1) by adding a suitable random diffusion?1.2.Problem formulation.In order to investigate the regularizing effect of random diffusion and answer Question 1.1, let us start from more general deterministic equations of the form One could also include a diffusion term −|∇| λ θ, for λ > 0, in the right-hand side (see Remark 1.8 below), but we will not do so here.Above, M is a d × d constant matrix.There are several meaningful choices for M. For instance, if we choose M to be −I, then we get gradient flows.While if we choose M to be antisymmetric,2 then we obtain conservative/Hamiltonian flows.We assume that g ∈ S ′ (R d ) is a tempered distribution, such that the Fourier transform of ∇g is locally integrable and satisfies the bound |ξĝ(ξ)| |ξ| 1−γ for some 0 < γ < d + 1.The model case is when g is a log or Riesz potential according to the rule The choice of sign determines whether the potential is repulsive (+) or attractive (−).When γ = 2, (1.4) is a constant multiple of the Coulomb/Newtonian potential.We refer to the ranges γ > 2 and γ < 2 as sub-Coulombic and super-Coulombic, respectively.
The general equation (1.3) encompasses a wide class of physical models.Focusing first on the conservative case, in which M is antisymmetric, the most notable examples are in dimension 2. If M is rotation by π 2 and g(x) = − 1 2π log |x| is the Coulomb potential, then (1.3) becomes the incompressible Euler vorticity equation (for instance, see [MP12, Section 1.2] or [MB02, Chapter 2]).If g(x) = C|x| −1 , then equation (1.3) becomes the inviscid SQG equation, which models the motion of a rotating stratified fluid with small Rosby and Froude numbers in which potential vorticity is conserved [Ped13,CMT94,HPGS95,Res95].More generally, choosing g(x) = C γ |x| γ−2 , for 0 < γ < 2, leads to the generalized SQG (gSQG) family of equations [PHS94, CCC + 12], for which the Euler vorticity equation is the γ → 2 − limit.While global-well posedness is known for classical [Wol33,H 33] and weak [Yud63] solutions to the Euler case, the global existence of smooth solutions to the gSQG equation is a major open problem-it is only known if one adds suitably strong diffusion to (1.3) (e.g.see [CW99, KNV07, CV10, CV12]).We refer the reader to [CMT94, Res95, CF02, Gan08, BSV19, CGSI19, BvCK20, GP21, HK21] and references therein for more information on the well-posedness and long-time dynamics of the gSQG equation.
In the gradient-flow case, in which M = −I, equation (1.3) has been studied for several applications in addition to the aforementioned ones of adhesion dynamics, chemotaxis, and vortices in superconductors.To name a few: materials science [HP06], cooperative control [GP03], granular flow [BCP97, BCCP98, Tos00, CMV06], phase segregation in lattice matter models [GL97, GL98, GLM00], and swarming models [MEK99, MEKBS03, TB04, TBL06].Several works have focused on the well-posedness and long-time dynamics.We recount some of the results for the model interaction (1.4), which is sometimes called a fractional porous medium equation.In particular, in the repulsive case γ = 2, global existence, uniqueness, and asymptotic behavior of nonnegative classical and L ∞ weak solutions are known [LZ00, AS08, BLL12, SV14].The case 2 < γ < d + 1 is easier and follows by the same arguments [CCH14, Section 4] (see also [BLR11] for an L p wellposedness result).For 0 < γ < 2, local well-posedness of nonnegative classical solutions is known [CJ21] and global existence, regularity, and asymptotic behavior of certain nonnegative weak solutions are known [CV11a, CV11b, CSV13, CV15, BIK15, CHSV15, LMS18].To our knowledge, these weak solutions are only known to be unique if d = 1 [BKM10].It is also an open problem whether classical solutions are global if 0 < γ < 2. If one allows for mixed-sign solutions, then the repulsive and attractive equations are equivalent by multiplication by −1.[BLL12] has established well-posedness, in particular maximal time of existence, for compactly supported classical and L ∞ weak solutions in the γ = 2 case.In particular, nonnegative classical and L ∞ weak solutions in the γ = 2 case are known to blow up in finite time, as remarked at the beginning of the introduction.[MZ05] has shown the existence of global renormalized solutions in the sense of DiPerna-Lions [DL89b,DL89a].We also mention the works [MZ05,AMS11,Mai12] for an equation arising in vortex superconductivity, which reduces to the repulsive γ = 2 case of equation (1.3) when one considers nonnegative solutions.
As a unifying perspective, the equation (1.3) may be viewed as an effective description of firstorder mean-field dynamics of the form Here, s ≥ 0 (we will determine further restrictions later), W is a standard real Brownian motion, and the stochastic differential should be interpreted in the Itô sense.The ˙superscript formally denotes differentiation with respect to time.We note that our choice of random diffusion differs from that of [BNSW20], which used the fractional Laplacian |∇| s = (−∆) s 2 .In that article, the authors work in the periodic setting of T 2 , and after modding out by the mass of the solution, which is conserved, homogeneous and inhomogeneous Sobolev spaces are equivalent.This equivalence fails on R d , and therefore the fractional Laplacian creates problems at low frequency, as will become clear to the reader in Sections 3 and 4 (see Remark 1.7 for further comments).Accordingly, we opt to add an inhomogeneity to rectify this issue.We emphasize that our choice of random perturbation differs from the aforementioned prior works [FGL21, MST21] on stochastic PKS equations, which did not consider random diffusion as in (1.6).
A priori, it is not clear how to interpret the SPDE (1.6).Moreover, it is not clear that the stochastic term in the right-hand side is regularizing since W t does not have definite sign.Formally, suppose that we have a solution θ to (1.6), and let us set µ t := e −νW t (1+|∇| s ) θ t , where for each realization of W , Γ t := e −νW t (1+|∇| s ) is the Fourier multiplier with symbol e −νW t (1+|ξ| s ) .As in [BNSW20], to compute the equation satisfied by µ, we formally use the Fourier transform together with Itô's lemma to obtain Above, [Γ, θ] denotes the quadratic covariation of the processes Γ and θ.Also, we have implicitly used that Γ t and |∇| s commute, both being Fourier multipliers.Observe that (1.7) is a random PDE which may be interpreted pathwise (i.e. for fixed realization of W , which almost surely is a locally continuous path on [0, ∞)).Additionally, thanks to the nontrivial quadratic variation of Brownian motion, we have gained a diffusion term in this equation.Rather than deal with the original equation (1.6), we shall base our mathematical interpretation on (1.7).
Remark 1.1.One may wonder why we choose the Itô formulation in (1.6) as opposed to the Stratonovich formulation which is formally equivalent to the Itô equation (1.9) Suppose we define µ t := e −νW t (1+|∇| s ) θ t as before.Then again using Itô's lemma, we find No longer do we gain a fractional diffusion term, which, as we shall see below, is fatal to our arguments.The preceding conclusion is to be expected.Indeed, if W is a C 1 path, then by ordinary calculus, ∂ t µ = −ν(1 + |∇| s )Γθ Ẇ + Γ∂ t θ; and the Stratonovich formulation is precisely chosen to preserve the ordinary rules of calculus.
To the best our knowledge, our result is the first demonstration that a random diffusion term can lead to global solutions for equations which, without any diffusion, necessarily blow up in finite time.This provides an affirmative answer to Question 1.1.Furthermore, Theorem 1.2 substantially generalizes the prior work of Buckmaster et al. [BNSW20, Theorem 1.1], which was limited to the SQG case M equals rotation by π 2 and ĝ(ξ) = |ξ| −1 , corresponding to a conservative/Hamiltonian flow.In particular, our result covers the full range of interactions in the model case (1.4) and also allows for interactions (e.g.d < γ < d + 1) which may not be singular in physical space near the origin but have very slow decay or even growth at ∞.
We do not say anything here about the asymptotic behavior of the solutions we construct, only that they are global.It would be interesting to give an asymptotic description of the solution as t → ∞, valid at least with positive probability.Indeed, the reader will recall from the beginning of the introduction that such a description is known for the deterministic PKS equation.We hope to address this question in future work.
Before transitioning to discuss the proof of Theorem 1.2, let us record a few remarks on the statement of and assumptions behind the theorem.
Remark 1.4.So as to make the result as accessible as possible, we have opted not to include in the statement of Theorem 1.2 the explicit relations the parameters, such as d, γ, s, r 0 , σ, have to satisfy in order for the theorem to apply.These relations are explicitly worked out in Sections 3 and 4 during the proofs of Propositions 3.1 and 4.1.Here and throughout this article, the reader should keep in mind that the most favorable choices for s, r are s = 1 and r = 1.
Remark 1.5.The conditions (1.12), (1.14) allows for initial data of arbitrarily large mass.Indeed, focusing on the γ ≤ 1 case, suppose that μ0 ∈ C ∞ c and μ0 (0 Taking β = ε ν 2 2 , for given ε ∈ (0, 1), and requiring (1.20) we see that (1.12) holds.For fixed ν, we can make the left-hand side of the preceding inequality arbitrarily small by letting L → 0 + .While for given L, we can take ν arbitrarily large so that (1.20) holds.The latter case is reminiscent of the mass-diffusion threshold we saw earlier for the PKS equation.
Additionally, one might think that by increasing ν, the diffusion becomes stronger and therefore one should get a "better" result.But P(Ω α,β,ν ) evidently decreases to zero as ν → ∞, assuming β is fixed.The reason has to deal with the resulting growing variance of νW t appearing in the definition of Γ t , which requires a large value of β to be absorbed by the exponential weight in our function spaces.
Remark 1.7.Theorem 1.2 is also valid if R d is replaced by T d .In fact, since Fourier space is discrete on the torus, we do not have the same issues at low frequency as in the setting of R d , and therefore one can replace equation (1.6) with (1.21) An elementary computation reveals that solutions conserve mass and therefore one may quotient out the mass by assuming it is zero.As a result, the zero Fourier mode vanishes and one has an equivalence of homogeneous and inhomogeneous Sobolev norms.Working with (1.21) simplifies the proof greatly, as the two-tiered norm for γ > 1 becomes unnecessary.
Remark 1.8.Theorem 1.2 is still valid if one adds a deterministic diffusion term −χ|∇| λ θ to the right-hand side of (1.6), for χ, λ > 0, which leads to (1.7) being replaced by Since a deterministic diffusion term only makes the circumstances for global existence more favorable, we have opted not to include this term.
1.4.Comments on proof.We briefly comment on the proof of Theorem 1.2.In light of the success of [BNSW20] in showing that adding random diffusion to the inviscid SQG equation leads, with high probability, to global solutions, and that the SQG equation is a special case of (1.3), we are guided by the approach of the cited work.There are two main steps: (1) Local well-posedness via contraction mapping argument, (2) Monotonicity of the Gevrey norm via energy estimate.As discussed below, repeating the proof of [BNSW20] in our more general context would fail due to issues at low frequency related to working on R d , as opposed to T d , and issues at high frequency stemming from the singularity of our interactions.Several new ideas are consequently needed.
Step (1), carried out in Section 3, proceeds by rewriting the equation (1.7) in mild form (see (3.3)) which is amenable to a contraction mapping argument for short times.The main difficulty is estimating the nonlinear term in the scale of Gevrey-type spaces defined in Section 3.1-the exponential weights of which are used to absorb the Γ operators-in which we want to construct solutions.In particular, the velocity field M∇g * µ can be singular compared to the regularity of the scalar µ, as opposed to of the same order in the SQG case of [BNSW20], which requires carefully balancing the derivatives in the nonlinearity against the diffusion.
It turns out that using L 2 -based function spaces, as in [BNSW20], leads to a restriction on γ that scales linearly in the dimension d, which would then limit us to strictly sub-Coulombic interactions g in dimensions d ≥ 4. One of our new insights is to instead consider Gevrey-Fourier-Lebesgue hybrid spaces (see (3.4)), which of course include the function spaces of [BNSW20] as a special case.In particular, our new spaces behave well with respect to Sobolev embedding when the integrability exponent r → 1 + , becoming an algebra at r = 1.
Another challenge in the local well-posedness step is the singularity near the origin of the Fourier transform ĝ when γ is large.In particular, for γ > 1, |ξĝ(ξ)| may blow up as |ξ| → 0. Dealing with this issue requires using a two-tiered function space, compared to the case γ ≤ 1.More precisely, at high frequency, we need our functions to be in an exponentially-weighted Fourier-Lebesgue space with high regularity index and low integrability exponent; while at low frequency, we need our functions to be in a similarly weighted space with low regularity index and high integrability exponent.This leads us to the multi-parameter scale of spaces X σ,σ,r,r, φ,γ introduced in (3.7) (more generally, see Section 3).
After some paraproduct analysis and a fair amount of algebra to determine what conditions all the various parameters have to satisfy, we prove Proposition 3.1, which asserts local well-posedness in the class of solutions satisfying for some σ lwp < σ.Here, 1 (•) denotes the indicator function for the condition (•).Although the Sobolev index σ lwp is strictly less than that of the initial datum, we will later improve it to σ through a bootstrap argument.
Step (2), carried out in Section 4, consists of upgrading the local solution from step (1) to a global solution and also upgrading the Sobolev index from σ lwp to σ.The original idea of [BNSW20, Proposition 4.1], modified to our setting and presented for the r = 2 case, is to prove an inequality for the time derivative of the "energy" e φ t (1+|∇| s ) µ t 2 H σs , which shows that this quantity is strictly decreasing on an interval [0, T ], provided it is not too large at initial time and that e φ t (1+|∇| s ) µ t H (σ+1)s remains finite on the same interval.With this type of conditional monotonicity result, the authors of that work could exploit the fact that the initial datum belongs to a space with higher Gevrey index α + ǫ in order to iteratively extend the lifespan of the solution, losing a decreasing fraction of ǫ along each step of the iteration.Note that in their work the Sobolev index from the local well-posedness does not change.
Since we deal with γ that are more singular than in [BNSW20] and step (1) only gives local solutions in a rougher space than that claimed in the statement of Theorem 1.2, we need a more sophisticated argument.Moreover, we need to work in our scale of Fourier-Lebesgue spaces, with the auxiliary space if γ > 1.We prove a similar conditional monotonicity result for the energy for any 0 ≤ ǫ ′ ≤ ǫ, assuming σ r , σ q are sufficiently large depending on d, γ, s, r, q.Similar to step (1), the bulk of the labor consists of paraproduct analysis for the nonlinearity and determining the set of conditions that the parameters d, γ, s, r, q, σ r , σ q have to satisfy in order for the paraproduct analysis to be valid.In order to access the monotonicity result, since σ r > σ lwp and σ q > 0, we exploit the higher Gevrey index of the initial datum together with an embedding lemma (see Lemma 3.5) to conclude that if (1.23) holds, then for any 0 ≤ ǫ ′ < ǫ and σ r , σ q ∈ R, We then obtain global existence by a lemma (see Lemma 4.5) which quantifies the improvement in the lifespan of the solution as we decrease ǫ ′ .Finally, we conclude global existence and monotonicity also hold with ǫ ′ = ǫ by essentially Fatou's lemma.
1.5.Organization of article.We close the introduction by outlining the remaining body of the article.In Section 2, we introduce the basic notation of the article and review some frequently used facts from Fourier analysis.In Section 3, we begin (Section 3.1) with our class of Gevrey-Fourier-Lebesgue spaces and their properties and then (Section 3.2) show the local well-posedness of the Cauchy problem for equation (1.7).In Section 4, we first (Section 4.1) show the monotonicity property of the Gevrey norm.We then (Section 4.2) use this property together with the local theory from Section 3 in order to prove our main result, Theorem 1.2. (2.1) In the case m > 1, the notation should be understood component-wise.Given a function m : R d → C m , we use the notation m(∇) to denote the C m -valued Fourier multiplier with symbol m(ξ).In the particular, the notation |∇| = (−∆) 1 2 denotes the Fourier multiplier with symbol |ξ| and ∇ := (1 + |∇| 2 ) 1/2 denotes the multiplier with Japanese bracket symbol (1 + |ξ| 2 ) 1/2 .2.2.Sobolev embedding.For the reader's convenience, we state and prove an elementary Sobolev embedding tailored to the Fourier analysis of Sections 3 and 4. To the state the lemma, we recall that the Bessel potential space W s,p is defined by and the Fourier-Lebesgue space Ŵ s,p is defined by For p = 2, these two spaces coincide by Plancherel's theorem, and, following standard notation, we shall write H s .When s = 0, we shall also adopt the notation Ŵ 0,r = Lr .

Local well-posedness
We investigate the local well-posedness of the Cauchy problem Set A := (1 + |∇| s ) 2 .It will be convenient to introduce the bilinear operator The reader should note that B itself depends on time through Γ, and, when necessary, we shall make explicit this time dependence by writing B t (f, g).We rewrite (3.1) in mild form In order to perform a contraction mapping argument based on the mild formulation (3.3), we use a generalization of the scale of Gevrey function spaces from [BNSW20] (see also [FT89]).Given a ≥ 0 and κ ∈ R, we define We refer to a as the exponential weight or Gevrey index, κ as the Sobolev index, and r as the integrability exponent.If r < ∞, then the completion with respect to this norm of functions with compactly supported Fourier transforms in L r defines a Banach space, as the reader may check.For 0 < T < ∞ and a continuous function φ : [0, T ] → [0, ∞), we define We write C 0 ∞ when sup 0≤t≤T is replaced by sup 0≤t<∞ .Set We also allow for T = ∞, replacing [0, T ] in the preceding line with [0, ∞).Evidently, this defines a Banach space.
To deal with possible issues at low frequencies when γ is large, we also have need for the Banach spaces The main result of this section is the following proposition.
There exists r 0 ≥ 1 depending on d, γ, s, such that the following holds.For any 1 ≤ r ≤ r 0 , there exists σ 0 ∈ (0, 2s−1 s ) depending on d, γ, r, s, such that for any σ ∈ (σ 0 , 2s−1 s ) with 1 − γ ≤ σs, there exists a constant C > 0 depending on d, γ, s, σ, r, q, β, ν, such that for µ 0 to the Cauchy problem (3.1), with T ≥ C(|M|R) Remark 3.2.Compared to statement of Theorem 1.2, where the Sobolev indices σ r , σ q can be arbitrarily large, Proposition 3.1 contains the restriction σ < 2s−1 s and the second Sobolev index is set to zero.These restrictions are temporary: we only need them to first obtain the existence of a solution.Using a monotonicity argument in the next section, which is in the spirit of persistence of regularity arguments, we then allow for larger values of σ.
Remark 3.3.A lower bound for for r 0 is explicitly worked out in the proof of Proposition 3.1.See condition (LWP2) below and the ensuing analysis.
Remark 3.4.The solutions constructed by Proposition 3.1 do not a priori conserve mass. 3To see this, note that by using equation (3.1) and the fundamental theorem of calculus, (3.12) where the ultimate equality follows from expanding the square and using the Fourier transform.Solving the ODE (3.12), we find So if µ 0 has zero mass, then µ t has zero mass for all times t.Otherwise, the magnitude of the mass is exponentially decreasing as t → ∞.Recalling the mass/diffusion threshold for the PKS equation, this decreasing of the mass of our solutions may provide some intuition why global existence is ultimately possible.
3.1.Gevrey embeddings.Before proceeding to the contraction mapping step, we record some elementary embeddings satisfied by the spaces G κ,r a .
Proof.First, observe that for any a ′ ≥ a ≥ 0, , since e (a−a ′ )(1+|ξ| s ) ≤ e a−a ′ .Also, for any κ ′ ≥ κ, we trivially have from Observe from the power series for z → e z that (3.19) where ⌈•⌉ is the usual ceiling function.Implicitly, we have used 3.2.Contraction mapping argument.Next, we define the map We check that this map is well-defined on for φ t = α + βt, with α, β, σ, r, q, γ > 0 satisfying the conditions in the statement of Proposition 3.1.To this end, we assume here and throughout this subsection that we have a realization of W belonging to Ω α,β,ν .
Lemma 3.6.If β < ν 2 2 , then for any 1 ≤ r ≤ ∞, 0 < s ≤ 1, σ ∈ R, and α > 0, it holds that Next, we observe from the bilinearity of B that and therefore , also assume that we are given 1 ≤ q < d γ−1 .Additionally, suppose that β < ν 2 2 .There exists an r 0 ∈ [1, ∞], depending on d, γ, s, such that the following holds.For any 1 ≤ r ≤ r 0 , there exists σ 0 ∈ (0, 2s−1 s ) depending on d, γ, s, r, such that for any σ ∈ (σ 0 , 2s−1 s ) with 1 − γ ≤ σs, there exists a constant C depending only on d, γ, r, q, σ, s, β, ν, such that for any T > 0, Remark 3.8.We give bounds on the size of the threshold r 0 from the statement of Lemma 3.7 during the proof of the lemma.We have omitted them from the statement in order to simplify the presentation.There is an extensive amount of algebra to determine the final conditions on the parameters, but if the reader is not interested in this optimization, they can just consider r = s = 1, which is straightforward to check.
Proof of Lemma 3.7.We make the change of unknown µ t j := e −φ t A 1/2 ∇ −σs ρ t j , so that (3.27) ρ t j Lr = µ t j G σ,r φ t .By Minkowski's inequality, we see that and by definition of the Ŵ σs,r norm, the preceding right-hand side equals Using φ t − φ τ = β(t − τ ), the preceding expression is controlled by Since 0 < s ≤ 1, and therefore by • ℓ 1 ≤ • ℓ s , and φ τ − νW τ ≥ 0 for all 0 ≤ τ ≤ t by assumption, it follows from the triangle inequality that By assumption that β < ν 2 2 , we have With these observations, we reduce to estimate the expression To deal with the inhomogeneity of (1 + |ξ| s ) 2 , we split the integral with respect to ξ into the low-frequency piece |ξ| ≤ 1 and the high-frequency piece |ξ| > 1.At low frequency, we can crudely estimate everything directly to find .
Above, we have used our assumption that |ηĝ(η) If 1 − γ < 0, then we have to be careful about singularities at low frequency.More precisely, by Hölder's inequality, we can control the L r ξ norm by the L ∞ ξ norm.For q(1 − γ) > −d, Hölder's inequality gives The remaining expression is handled by estimate (3.41) below.

2s
. Hence, As before, we have to be careful about singularities in η at low frequency if 1 − γ < 0. Observe from Young's inequality that q−1 , (3.40) for any q < d γ−1 .For η at high frequency, we have by Young's inequality followed by Lemma 2.1 that any 1 ≤ p ≤ r, In order to obtain estimates that close, we need the top Sobolev index appearing in (3.41) to be ≤ σs.This leads us to the following conditions: Putting together the estimates (3.40) and (3.41), we have shown that Combining the estimates (3.35) and (3.43), we have shown that (3.33) is d,γ,s,σ,r,q In order to complete the proof of the lemma, it is important to list all the conditions we imposed on the parameters d, γ, σ, s, r during the course of the above analysis: and since we require σ > 0, we need s > 1 2 .Given any value of 0 < γ < d + 1, (LWP2)a can be satisfied by choosing r = 1 and σs ≥ min{0, 1 − γ}.More generally, we can satisfy all three conditions by arguing as follows.Given 0 < γ < d + 1 and 1 2 < s ≤ 1, condition (LWP3) implies that for any choice r ≥ 1, (3.45) σs < 2s − 1.
According to (LWP2)b, it is possible to find such a σ ≥ 1−γ s and r > 1 if and only if According to (LWP2)c, it is possible to find such a σ > 0 and r > 1 if and only if According to (LWP2)d, it is possible to find such a σ > 0 and r > 1 if and only if for such choice of r, there exists p ∈ (1, r) such that Since the preceding constraint is equivalent to such a p exists if and only if With this case analysis, the proof of Lemma 3.7 is now complete.
Lemma 3.9.Let d ≥ 1, 1 < γ < d + 1. Suppose that γ, r, s, σ satisfy the constraints of Lemma 3.7 and that Then for any 1 ≤ q < d γ−1 , there exists a constant C > 0 depending on d, γ, r, q, s, σ, β, ν, such that for any T > 0, (3.52) Proof.Set q ′ := q q−1 .The proof follows the exact same lines of Lemma 3.7 with r replaced by 2q ′ .Using the estimates (3.35), (3.40), (3.41), we find that for any choice 1 ≤ p ≤ 2q ′ .We want all the norms appearing in the right-hand side to be controlled by Ŵ 0,2q ′ and Ŵ σs,r .Using Lemma 2.1, we see that we need to choose p ≤ r so that For any choice of p, we have another use of Lemma 2.1 tells us that we need Since we also needed σs + d r > d p , the existence of a p satisfying both the upper and lower bounds is true if and only if Note that if γ ≥ 1, then for any σs > 0, we have Under the constraints of the preceding paragraph, the right-hand side of (3.53) is d,γ,r,q,σ,s Taking the supremum over τ ∈ [0, T ] in the right-hand side and using fundamental theorem of calculus leads to the desired conclusion.
An immediate corollary of Lemmas 3.7 and 3.9 is the following estimate for the Duhamel term.
Corollary 3.10.Under the assumptions of Lemmas 3.7 and 3.9, there exists a constant C > 0 depending on d, γ, r, q, σ, s, β, ν, such that for any T > 0 and Proof of Proposition 3.1.Putting together the estimates of Lemma 3.6 and Corollary 3.10, we have shown that there exists a constant C > 0 depending on d, γ, r, q, σ, s, β, ν, such that We now want to show that for any appropriate choice of T , the map T is a contraction on the Indeed, from the estimates (3.61) and (3.62), we see that if then T is a contraction on B R (0).So by the contraction mapping theorem, there exists a unique . We note that T ≥ C ′ (|M|R) − 2s 2s−σs−1 , where C ′ > 0 is a possibly different constant than C but depending on the same parameters.

Global existence
We now show that with quantifiable high probability, there exists a global solution µ ∈ C 0 ∞ X σr,σq,r, 2q to the Cauchy problem (3.1), provided σ r , σ q , r, q are appropriately chosen.Moreover, the function is strictly decreasing on [0, ∞), provided that µ 0 X σr ,σq ,r,q α,γ is sufficiently small.This then proves Theorem 1.2.4.1.Monotonicity of Gevrey norm.The goal of this subsection is to show the following.Suppose we have a solution µ ∈ C 0 T X κr,κq,r, 2q q−1 φ,γ to (3.1), where φ t := α + βt, such that µ also belongs , for sufficiently larger κ ′ r > κ r and κ ′ q > κ q , and such that µ 0 X κr ,κq ,r, 2q q−1 α,γ is sufficiently small depending on d, γ, r, q, κ r , κ q , s, β, ν, |M|.If κ r , κ q are sufficiently large depending on d, s, γ, then the quantity µ t X κr ,κq ,r, 2q q−1 φ t ,γ must be strictly decreasing on the interval [0, T ].In other words, the Gevrey norm of µ t is strictly decreasing on an interval, provided that we know a Gevrey norm with higher Sobolev index (but the same Gevrey index) remains finite on the same interval.
If γ ≤ 1, then there is a threshold κ 0,r ∈ R depending on r, d, s, γ, such that for any κ r > κ 0,r , the following holds.There is a constant C r > 0, depending only on d, γ, r, s, κ r , such that if is a solution to (3.1), for some T > 0, satisfying If γ > 1, then there is a threshold κ 0,q ∈ R depending on q, d, s, γ, such that for any κ q > κ 0,q , the following holds.There is a constant C q > 0, depending only on d, γ, q, s, κ q , such that if q−1 φ is a solution to (3.1), for some T > 0, satisfying , where κ r > κ 0,r , such that 3) also holds.
The proof of Proposition 4.1 consists of several lemmas.To begin, we observe from the chain rule and using equation (3.1) (there is an approximation step we omit), Replacing the first and second terms by their magnitudes, we see that the right-hand side is ≤ Using the elementary inequality (remember that s ≤ 1) together with our assumption that β < ν 2 2 , we arrive at the inequality It now follows from this identity, the chain rule, and differentiating inside the integral that for any 1 ≤ r < ∞, We need to show that the second term in (4.11) is not so large that it cannot be absorbed by the first term, which is negative.This is a problem in Fourier analysis, which we address with the next two lemmas.Lemma 4.3.For any t > 0 with φ t − νW t ≥ 0, it holds for any test functions f, g that (4.12) Proof.We observe from the definition (3.2) of B t that for any test functions f, g, Writing 1 = e φ t (1+|η| s ) e −φ t (1+|η| s ) = e φ t (1+|ξ−η| s ) e −φ t (1+|ξ−η| s ) , we see that the magnitude of the preceding right-hand side is controlled by where we have used our assumption that |ηĝ(η)| γ |η| 1−γ .Since φ t − νW t ≥ 0 by assumption and , the desired inequality now follows.
Next, we observe that by applying Lemma 4.3 to the second term in the right-hand side of (4.11), we need to estimate expressions of the form (4.15) R d e φ t (1+|ξ| s ) ξ κs ĥ(ξ) where f, g, h are test functions.We take care of such expressions with the next lemma.
So by Young's inequality followed by application of Lemma 2.1, it holds for any 1 Again using Young's inequality and Lemma 2.1, it holds for any 1 So we can evenly distribute the derivatives between f and g and use Young's inequality together with Lemma 2.1 to obtain Combining the estimates (4.18), (4.19), (4.20), (4.21), we see that Above, we have limited ourselves to the cases r = 1 and r > 1, 1 < p, p < ∞ so as to simplify the exposition (the cost is an insignificant ε loss at the endpoint exponents).In order to obtain the desired estimate (4.16), we need the maximum Sobolev index of the norms in (4.22) to be < (κ + 2 r )s.This leads us to make the following assumptions on the parameters d, s, κ, γ, r, p, p.Let us analyze the preceding assumptions.
For any 1 2 < s ≤ 1, (L1)a always holds if we assume 2(1 − γ) ≤ κ.For (L1)b, (4.34) which is ensured by taking κ sufficiently large depending on given γ, s.For (L2)a, we need , the first inequality is valid if d(r−1) r < κs, which holds by taking κ sufficiently large depending on given d, r, s.For the second inequality, we see The left-hand side of the second inequality is maximized by choosing p = 1, so it would suffice to assume d, γ, s, r satisfy For (L2)b, we need The first inequality is equivalent to The left-hand side is maximized by choosing p = r, therefore it suffices to assume (4.40) The second inequality is equivalent to the left-hand side of which is maximized by choosing p = 1.So, it would suffice to assume which is seen to hold by choosing κ sufficiently large depending on given d, γ, r, s.For (L2)c, we observe that the inequality is equivalent to which is valid provided that κ is sufficiently large depending on given d, γ, r, s.
The preceding sets of assumptions tell us that given d, γ, r, s, there is a threshold κ 0 depending on d, γ, r, s, such that for all κ > κ 0 , the right-hand side of (4.22) is controlled by r )s,r e φ t A 1/2 g Ŵ κs,r + e φ t A 1/2 f Ŵ κs,r e φ t A 1/2 g Ŵ (κ+ 2 r )s,r .Recalling the starting inequality (4.17), we see that the proof is complete.
We are now prepared to prove Proposition 4.1.
We next show that this monotonicity property of the norm e φ t A 1/2 µ t Ŵ κs,2q ′ also implies a monotonicity property for the norm e φ t A 1/2 µ t Ŵ κs,r , for appropriate κ, provided that e αA 1/2 µ 0 Ŵ κs,r is sufficiently small.If γ ≤ 1, then this step is unnecessary and the argument given above suffices with 2q ′ replaced by r.Let κ 0,r denote the regularity threshold given by Lemma 4.4, and let κ r > κ 0,r .Again using Hölder's inequality and Lemmas 4.3 and 4.4, we see that where the constant C r,q > 0 depends only on d, γ, r, q, s, κ r .We use the subscript r to emphasize the dependence on r, as we shall momentarily invoke another constant and regularity parameter depending on q.Applying the preceding bound to the differential identity (4.11) it follows that Since 1 − γ < 0 if γ > 1, we know that there is a constant C q depending on d, γ, q, s, κ q , for κ q > κ 0,2q ′ , such that if (4.54) Therefore, suppose that Under these assumptions, it follows by repeating the continuity argument from above that the quantity e φ t A 1/2 µ t Ŵ κr s,r is strictly decreasing on [0, T ].With this last bit, the proof of Proposition 4.1 is complete.Let us first present the case 0 < γ ≤ 1, which is simpler due to not needing a two-tiered norm.Fix ǫ > 0 and suppose that µ 0 ∈ G σ 0 ,r α+ǫ for σ 0 above the regularity threshold κ 0,r given by Proposition 4.1.Throughout this subsection, we assume that the parameters d, γ, r, s, σ 0 , α, β, ν satisfy all the constraints of Theorem 1.2.We also assume that (4.58) where C mon = C r > 0 is the constant from Proposition 4.1.Assuming a realization of W from Ω α,β,ν and given r ≥ 1 sufficiently small depending on d, γ, s, Proposition 3.1 implies that for any 0 < σ < 2s−1 s , with 1 − γ ≤ σs, sufficiently large depending on d, γ, s, r, there is a maximal solution µ to the Cauchy problem (3.1) with lifespan [0, T max,σ,ǫ ), such that µ belongs to C 0 T G σ,r φ+ǫ for any 0 ≤ T < T max,σ,ǫ .Our main lemma to conclude global existence is the following result relating the lifespan of µ t in G σ,r φ t +ǫ to the lifespan of µ in the larger space G σ,r φ t +ǫ ′ , for any ǫ ′ ∈ [0, ǫ).Lemma 4.5.Let µ be as above.There exists a constant C > 0 depending on d, γ, r, s, σ, β, ν such that for any 0 ≤ ǫ 2 < ǫ 1 ≤ ǫ, the maximal times of existence T max,σ,ǫ 1 , T max,σ,ǫ 2 of µ t as taking values in G σ,r φ t +ǫ 1 , G σ,r φ t +ǫ 2 , respectively, satisfy the inequality Proof.Fix 0 < ǫ 2 < ǫ 1 ≤ ǫ.For any σ ′ ≥ σ, it follows from Lemma 3.5 that µ is also a solution in C 0 T G σ ′ ,r φ+ǫ 2 for any 0 ≤ T < T max,σ,ǫ 1 .Furthermore, we have the quantitative bound where κ 0,r is the regularity threshold of Proposition 4.1.Using the assumption (4.58), we can then apply Proposition 4.1 to conclude that the function t → µ t G σ ′ ,r φ t +ǫ 2 is strictly decreasing on the interval [0, T ] for any T < T max,σ,ǫ 1 .In particular, since we can ensure that σ ≤ σ ′ , it follows from this monotonicity and Lemma 3.5 that (4.62) Let C lwp,σ be the constant from Proposition 3.1, and choose T * < T max,σ,ǫ 1 so that (4.63) Thus by relabeling time, we can apply Proposition 3.1 once more, but with initial datum µ T * , to find that µ belongs to C 0 T 0 G σ,r φ+ǫ 2 , where which is a contradiction.Thus, T max,σ,ǫ ′ = ∞.
We have shown that for any 0 < ǫ ′ < ǫ and any 0 < σ < 2s−1 s sufficiently large depending on d, γ, s, r, it holds that µ C 0 T G σ,r φ+ǫ ′ < ∞ for all T > 0. Using the arbitrariness of ǫ ′ , we see from Lemma 3.5 that for any T > 0, µ Finally, we show that µ actually belongs to C 0 ∞ G σ 0 ,r φ+ǫ and that the decreasing property holds on [0, ∞).Note that there is no longer a loss in the Gevrey index value (i.e.ǫ ′ = ǫ).To this end, we observe from the result of the preceding paragraph and Fatou's lemma that for any t ≥ 0, Similarly, for any t 2 ≥ t 1 ≥ 0, (4.69) , where the inequality is strict if t 2 > t 1 .This completes the proof of Theorem 1.2 in the case γ ≤ 1.

4. 2 .
Proof of Theorem 1.2.We now use the local well-posedness established by Proposition 3.1 together with the monotonicity of the Gevrey norm established by Proposition 4.1 in order to show that with high probability, solutions in the class we consider are global.Moreover, their Gevrey norm strictly decreases as time t → ∞.This then proves Theorem 1.2.To show the desired result, we use a refined reformulation of the iterative argument from [BNSW20, Section 5].
1.6.Acknowledgments.The first author thanks Sylvia Serfaty for helpful comments on the relevance of equation (1.3) for vortices in superconductors.Both authors gratefully acknowledge the hospitality of the Institute for Computational and Experimental Research in Mathematics (ICERM) where the manuscript for this project was completed during the "Hamiltonian Methods in Dispersive and Wave Evolution Equations" semester program.
2. Preliminaries2.1.Notation.Given nonnegative quantities A and B, we write A B if there exists a constant C > 0, independent of A and B, such that A ≤ CB.If A B and B A, we write A ∼ B. To emphasize the dependence of the constant C on some parameter p, we sometimes write A p B or A ∼ p B. We denote the natural numbers excluding zero by N and including zero by N 0 .Similarly, we denote the positive real numbers by R + .The Fourier and inverse transform of a function f : R d → C m are defined according to the convention f ≤ ⌈σ r − σ⌉! (ǫ 1 − ǫ 2 ) ⌈σr−σ⌉ µ t