Norm inflation for a non-linear heat equation with Gaussian initial conditions

We consider a non-linear heat equation $\partial_t u = \Delta u + B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We show that, if the initial condition $u_0$ is taken from a sequence of smooth Gaussian fields with a specified covariance, then $u$ exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which complements recent well-posedness results in arXiv:2111.10652 and arXiv:2201.03487. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $B^{-1/2}_{\infty,\infty}$; the space $B^{-1/2}_{\infty,\infty}$ is an endpoint since the equation is locally well-posed for $B^{\eta}_{\infty,\infty}$ for every $\eta>-\frac12$.


Introduction
We consider the initial value problem for a non-linear heat equation of the form where where B i : E × E → E is bilinear, then we assume that B i is not symmetric for some 1 ≤ i ≤ d.We note that B i is symmetric if and only if there exists a bilinear map Bi : It is easy to show that, for η > −1 2 , a solution u to (1.1) exists for T > 0 sufficiently small depending (polynomially) on |u 0 | C η , where C η = B η ∞,∞ is the Hölder-Besov space of regularity η (see Section 2.1 for a definition of B α p,q ).Furthermore, the map C η ∋ u 0 → u is locally Lipschitz once the target space is equipped with a suitable norm and T is taken sufficiently small on each ball in C η .We give a proof of these facts in Appendix B. (If each B i were symmetric, these facts would hold for η > −1.) The goal of this article is to prove a corresponding local ill-posedness result for a family of function spaces that carry sufficiently irregular Gaussian fields.Leaving precise definitions for later, the main result of this article can be stated as follows.
Theorem 1.1.Suppose that X is an E-valued Gaussian Fourier series (GFS) on T d whose Fourier truncations {X N } N ≥1 satisfy Assumption 2.6 below.Then there exists another E-valued GFS Y , defined on a larger probability space, such that Y law = X and, for every δ > 0, lim where u is the solution to (1.1) with u 0 = X N + Y N and if u blows up at or before time δ, then we treat sup t∈[0,δ] | T d u t (x) dx| = ∞.
We formulate a more general and precise statement in Theorem 2.13 below.See the end of Section 2 for a description of the proof and of the structure of the article.The definition of an E-valued GFS is given in Section 2.3.
Remark 1.2.The Gaussian free field X on T d with E|X k | 2 = |k| −2 for k = 0 (see e.g.[She07]) is a GFS (for any d ≥ 1) and X satisfies Assumption 2.6 for d = 3 (but not other values of d).
There is considerable interest in studying partial differential equations (PDEs) with random initial conditions in connection to local and global well-posedness; for dispersive PDEs, see for instance [Bou94,BT08,OP16,Poc17] and the review article [BOP19]; for recent work on parabolic PDEs, see [CC23,CCHS22b,HLR23]. Furthermore, there have been many developments in the past decade in the study of singular stochastic PDEs with both parabolic [Hai14, GIP15,BCCH21] and dispersive [GKO18,GKOT21,Tol21] dynamics, for which it is often important to understand the effect of irregularities of (random) initial condition.
In [CC23,CCHS22b], it is shown that (1.1) is locally well-posed 1 for the GFF X on T 3 , in the sense that, if u 0 = X N , then u converges in probability as N → ∞ to a continuous process with values in C η for η < −1/2, at least over short random time intervals; we discuss these works in more detail at the end of this section.Theorem 1.1 therefore shows the importance of taking precisely the GFF X in these works, rather than another Gaussian field which has the same regularity as X when measured in any normed space.We note that Y in Theorem 1.1 can therefore clearly not be taken independent of X.
We now state a corollary of Theorem 1.1 and of the construction of Y which elaborates on this last point and is of analytic interest.Consider a Banach space (I, denote the Banach space of all E-valued distributions of the form a∈A x a T a , where each x a ∈ I and {T a } a∈A is some fixed basis of E, with norm a∈A x a T a def = a x a .We say that there is norm inflation for (1.1) at x ∈ Î if, for every δ > 0, there exists a solution u to (1.1) such that x − u 0 < δ and This notion of norm inflation is slightly stronger than the usual one introduced by Christ-Colliander-Tao [CCT03] in their study of the wave and Schrödinger equations, for which only x = 0 and u t > δ −1 are considered in (1.2).Norm inflation (in the sense that u t > δ −1 ) at arbitrary points x ∈ Î was shown by Xia [Xia21] for the non-linear wave equation below critical scaling.Oh [Oh17] showed that there is norm inflation for the cubic non-linear Schrödinger equation at every point in negative Sobolev spaces at and below the critical scaling.Norm inflation captures a strong form of ill-posedness of (1.1): (1.2) in particular shows that the solution map ) is discontinuous at x in Î, even locally in time.See [IO15, CK17, OW18, Kis19] and [BP08,CD14,MT15] where norm inflation of various types is shown for dispersive and parabolic PDEs respectively.
Consider now a family of non-negative numbers σ 2 = {σ 2 (k)} k∈Z d .We say that I carries the GFS with variances • is a real GFS for every N ≥ 1 (see Definition 2.5), and where X is a real GFS.
Corollary 1.4.Suppose I carries the GFS with variances σ 2 = {σ 2 (k)} k∈Z d satisfying the bounds in Assumption 2.6.Suppose that the law of X in (1.3) has full support in I. Then there is norm inflation for (1.1) at every point in Î.
The proof of Corollary 1.4 is given in Section 2.4.We note that a simple criterion for the law of X to have full support in I is that every σ 2 (k) > 0 and that the smooth functions are dense in I (see Remark 2.14).
Note that the Besov space ∞,∞ is an 'endpoint' case since (1.1) is well-posed on C η for every η > − 1 2 .In Proposition A.1(i) below, we show that C −1/2 carries a GFS with variances σ 2 satisfying the bounds in Assumption 2.6.Corollary 1.4, together with Remark 2.14 and Proposition A.1(i), therefore gives a probabilistic proof of the following result, which appears to be folklore.remark, however, that our method appears not to reach B −1/2 ∞,q for q < ∞ (see Proposition A.1(ii)), while the method in [COW22] for the cubic NLH does extend to B −2/3 ∞,q for q > 3. Corollary 1.5 should be contrasted with the 3D Navier-Stokes equations (NSE) for which norm inflation (in a slightly weaker sense than (1.2)) was shown in the scaling critical space C −1 by Bourgain-Pavlović [BP08] and which is locally well-posed in C η for η > −1 (the main difference with (1.1) is that the non-linearity in NSE is a total derivative).In fact, norm inflation for NSE has been shown in B −1 ∞,q for q > 2 by Yoneda [Yon10] and for all q ∈ [1, ∞] by Wang [Wan15].We remark that our analysis (and generality of results) relies on the fact that any space carrying X as in Theorem 1.1, in particular C −1/2 , is scaling subcritical.See also [CP18] and [FO20] where norm inflation is established for the fractional non-linear Schrödinger (NLS) and non-linear wave (NLW) equations respectively above the critical scaling.
In [ST20] and [CG23], norm inflation is shown for the NLW and NLS respectively for all initial data u 0 belonging to a dense G δ subset of the scaling supercritical Sobolev space and where the approximation x is taken as a mollification of u 0 .These works in particular imply a statement similar to Corollary 1.5 for the NLW and NLS but with a more precise description of the approximating sequence that exhibits norm inflation (cf.[Oh17,Xia21]).
We finish the introduction by describing one of the motivations for this study.The author and Chandra-Hairer-Shen in [CCHS22a,CCHS22b] analysed the stochastic quantisation equations of the Yang-Mills (YM) and YM-Higgs (YMH) theories on T 2 and T 3 respectively (see also the review article [Che22]); we also mention that Bringmann-Cao [BC23] analysed the YM stochastic quantisation equations on T 2 by means of para-controlled calculus (vs.regularity structures as in [CCHS22a]), and that the invariance of the YM measure on T 2 for this dynamic was shown in [CS23].In [CCHS22b], the authors in particular constructed a candidate state space for the YM(H) measure on T 3 .A related construction was also proposed by Cao-Chatterjee [CC21,CC23].An ingredient in the construction of [CCHS22b] is a metric space I of distributions such that the solution map of the DeTurck-YM(H) heat flow (or of any equation of the form (1.1), see [CCHS22b, Prop.2.9]) extends continuously locally in time to I and such that suitable smooth approximations of the GFF on T 3 converge in I; essentially the same space was identified in [CC23].The works [CC23,CCHS22b] thus provide a local well-posedness result for (1.1) with the GFF on T 3 as initial condition.
In contrast, our results provide a corresponding ill-posedness result.Indeed, the GFF on T 3 satisfies Assumption 2.6 and the DeTurck-YM(H) heat flow is an example of equation (1.1) covered by our results (see Examples 2.2 and 2.3).Theorem 1.1 and Corollary 1.4 therefore imply that the metric space I in [CCHS22b] is not a vector space, and, more importantly, that this situation is unavoidable.That is, there exists no Banach space of distributions which carries the GFF on T 3 and admits a continuous extension of the DeTurck-YM(H) heat flow.Since the 3D YM measure (at least in a regular gauge) is expected to be a perturbation of the standard 3D GFF, this suggests that every sensible state space for the 3D YM(H) measure is necessarily non-linear (cf.[Che19,CCHS22a] in which natural linear state spaces for the 2D YM measure were constructed).
A non-existence result in the same spirit was shown for iterated integrals of Brownian paths by Lyons [Lyo91] (see also [LCL07,Prop. 1.29] and [FH20, Prop.1.1]); the construction of the field Y in Theorem 1.1 is inspired by a similar one in [Lyo91].

Preliminaries and main result 2.1 Fourier series and Besov spaces
We recall the definition of Besov spaces that we use later.Thorough references on this topic include [BCD11,Tri83]; see also [GIP15, Appendix A] and [MW17, Appendix A] for concise summaries.Let ) denote the orthonormal Fourier basis; here and below, we equip For s ∈ R and 1 ≤ p, q ≤ ∞, we define the Besov norm of We let B α p,q be the space obtained by completing C ∞ (T d ) with this norm, which one can identify with a subspace of distributions S ′ (T d ).We use the shorthand for t > 0 denote the heat flow (with P 0 = id as usual).In particular, P t e k = e −|k| 2 t e k for all k ∈ Z d .We denote by Π N :

Assumptions on the equation
Consider E, B, P as in Section 1.Without loss of generality, we will assume B 1 is not symmetric and henceforth make the following assumption.
The next two examples show that our results cover the DeTurck-YM(H) heat flow.
Example 2.2 (Yang-Mills heat flow with DeTurck term).Let E = g d , where g is a non-Abelian finite-dimensional Lie algebra with Lie bracket [•, •].We write elements of E as X = d j=1 X j dx j , X j ∈ g.The DeTurck-YM heat flow is To bring this into the form (1.1), we write elements of Then B i is not symmetric for every 1

Gaussian Fourier series
Definition 2.5.A complex Gaussian Fourier series (GFS) is an S ′ (T d , C)-valued random variable X such that {X k } k∈Z d is a family of complex Gaussians with EX2 k = 0 for all k = 0 and X k and X n are independent for all k, n ∈ Z d such that k / ∈ {−n, n}.Here, as before, we denote The sequence {E|X k | 2 } k∈Z d uniquely determines the law of a real GFS X. Conversely, a real GFS can be constructed from any sequence {σ 2 (k)} k∈Z d with polynomial growth by taking a set , defining {X k } k∈K as a family of independent complex Gaussians with E|X k | 2 = σ 2 (k) and EX 2 k = 0, and setting X −k = X k for all k ∈ K, and X 0 as a real Gaussian with E|X 0 | 2 = σ 2 (0).Assumption 2.6.Suppose {X N } N ≥1 is a family of E-valued GFSs such that 2 for all k 0 ≤ |k| ≤ N , and σ 2 N (k) ≤ C for 0 ≤ |k| < k 0 , where k 0 , C > 0 are independent of N , and σ 2 (k) = 0 for |k| > N .
Example 2.7.Let X be an E-valued GFS with for some sufficiently small ε > 0 without changing the statement of Theorem 2.13 below.We restrict to γ = −d + 1 only for simplicity.
Lemma 2.9.Let η < − 1 2 and suppose that the upper bound on Proof.This follows from a standard Kolmogorov-type argument (similar to and simpler than the proof of Lemma 3.2).It also follows directly from the sharper result of Proposition A.1(ii) combined with the obvious embedding B α p,q ֒→ B α p,∞ .
Remark 2.10.The logarithmic factors in Assumption 2.6 are considered to allow for endpoint cases; we will see in Proposition A.1 that they allow for X N def = Π N X to converge to a GFS X in C −1/2 , which would not be the case without these factors.This allows us to show norm inflation for (1.1) in C −1/2 (Corollary 1.5).If these logarithmic factors are dropped, then |Π 0 u T | > c log log log N in (2.1) can be replaced by inf t∈[N −2+ε ,T ] |Π 0 u t | > c log N for any ε > 0.

Main result
Definition 2.11.For a distribution ξ ∈ S ′ (T d , C), we define Rξ ∈ S ′ (T d , C) as the unique distribution such that, for all k ∈ Z d , (Rξ) If ξ satisfies the reality condition ξ −n = ξ n , then so does Rξ.Furthermore, if X is a real GFS, then so is RX and RX law = X.Recall now Assumption 2.1.Definition 2.12.For an E-valued GFS X, we define another E-valued GFS Y = c∈A Y c T c by setting, for each c = b, Y c law = X c and independent of X, and setting Note that Y can be defined on a larger probability space than that of X and that Y law = X by construction.The following is the main result of this article.
Theorem 2.13.Suppose B satisfies Assumption 2.1 and {X N } N ≥1 satisfies Assumption 2.6.Define Y N as in Definition 2.12, u 0 def = X N + Y N , and let u be the solution to (1.1).Then there exist M, c > 0 such that, for every ε > 0, if we set T = (log N ) −M for N ≥ 2, lim (2.1) The proof of Theorem 2.13 is given in Section 6.Before continuing, we note that Theorem 2.13 clearly proves Theorem 1.1.We can now also give the proof of Corollary 1.4.
Proof of Corollary 1.4.Let X be an E-valued GFS with E|X a k | 2 = σ 2 (k) for every a ∈ A, and let Y be defined as in Definition 2.12.It follows from the assumption that I carries the GFS with variances σ 2 (k) that the law of Z def = X + Y is a Gaussian measure on the Banach space Î and that lim By the assumption that the law of X a has full support in I, it follows that Z has full support in Î. (One only needs to be careful about the component T b for which Z b = X b + RX a since this is not independent of Z a = X a + Y a .However, since X b is independent of {X c } c =b and of Y , we indeed obtain that the law of Z has full support in Î.) Now, for every x ∈ Î and δ > 0, one has Furthermore, it follows from (2.2) that, for all N sufficiently large, The conclusion follows from Theorem 2.13 (or, more simply, Theorem 1.1).
Remark 2.14.In the context of Corollary 1.4, a sufficient condition for the law of the real GFS X in (1.3) to have full support in I is that σ 2 (k) > 0 for all k ∈ Z d and that the smooth functions are dense in I. Indeed, the condition σ 2 (k) > 0 implies that the Cameron-Martin space of X contains all trigonometric functions, which are dense in the smooth functions.
We briefly outline the proof of Theorem 2.13 and the structure of the rest of the article.Dropping the reference to N , in Section 3 we show that all the quadratic terms in B(P t (X + Y ), DP t (X + Y )) are controlled uniformly in N ≥ 1 and t ∈ (0, 1) except for the spatial mean In Section 4 we show that J − EJ is controlled uniformly in N, t, together with an upper bound EJ ≤ CN 2 ∧ t −1 uniformly in N, t and a lower bound as t → 0 and (2.3) Finally, in Section 5, we show that the solution u t to (1.1) tracks closely the first Picard iterate P t u 0 + t 0 B(P s u 0 , DP s u 0 ) ds at all times t ≤ (log N ) −M .Since this first Picard iterate blows up like log log log N at time t = (log N ) −M due to (2.3), this allows us to conclude the proof of Theorem 2.13 in Section 6.In Appendix A, we give conditions under which the Besov space B α ∞,q for α ∈ R and q ∈ [1, ∞] carries the GFS with a specified covariance σ 2 .

Decorrelated terms
In the remainder of the article, we will use x y to denote that x ≤ Cy for some fixed proportionality constant C > 0, and likewise let x ≍ y to denote C −1 x ≤ y ≤ Cx.We let x ≫ 1 and x ≪ 1 denote x > C and 0 < x < C −1 respectively for some sufficiently large fixed C > 0.
In this section, we let X and Y be real GFSs such that EX k Y n = 0 for all k = −n and such that there exists C > 0 such that, for all k ∈ Z d We will apply the results of this section to the three cases X = Y , RX = Y , or X and Y are independent.We record the following lemma, the proof of which is straightforward and we omit.
Then every moment of sup t∈(0,1) t δ |π 0 (P t X∂ i P t Y )| C β is finite and depends only on C, δ, β.Furthermore, if X and Y are independent, then the same holds for sup t∈(0,1) Proof.Let us denote Z t = t δ P t X∂ i P t Y for t ∈ (0, 1] and Z 0 = 0. We will show that there exists κ > 0 such that for all p ≥ 1 E|π uniformly over 0 ≤ t < u ≤ 1, from which the first statement follows by Kolmogorov's continuity theorem [FV10, Thm.A.10].The second statement will be shown in a similar way.It suffices to consider t < u ≤ 2t.
For every p ≥ 1 and ℓ ≥ 0, By equivalence of Gaussian moments, the final integral is bounded above by a constant C(p) > 0 times where Again by Lemma 3.1, for k = 0, Since 0 < δ ≤ 1, observe that, for κ > 0 small, uniformly in x ≥ 1 and 0 < t < u ≤ 2t < 1, where we used that e −x x p 1 for any p > 0 in the final bound.We split the sum (3.4) into two parts with |m| ≤ 2|k| and |m| > 2|k|.The first part is bounded above by a multiple of where we used (3.5) and The second part is bounded by a multiple of where we again used (3.5) and the fact that −4δ + 4κ − 2d + 4 < −d for κ > 0 small since δ > 1 − d 4 by assumption.In conclusion, E|F k t,u | 2 |t − u| κ k −4δ+4κ+4−d , and thus Consequently, by (3.2), (3.3), and (3.6), Since 2β + 4 − 4δ < 0, we can take κ > 0 sufficiently small such that the series is summable in ℓ.
For the second statement, since and thus, for κ > 0 small, where the final bound follows from (3.5) and the fact that −4δ − 2d + 4 < −d.

Correlated terms
We now take {X N } N ≥1 satisfying Assumption 2.6.We will drop N from our notation, writing simply X for X N .Recall that Assumption 2.1 is in place.We define Z : [0, 1] → R by (If i = 1, the 1st and 2nd sums cancels and the 3rd is 0, and if i = 1, the 3rd sum is still 0 but the 1st and 2nd sums are equal.)Similarly Hence, for X, Y related as in Definition 2.12, by Lemma 3.2, the quantity which we expect to be well-behaved uniformly in N is The following lemma shows that Z t is well approximated by its expectation.
Proof.Define W t = t δ (Z t − EZ t ).We will show that there exists κ > 0 such that, for all p ≥ 1, uniformly in 0 ≤ t < u ≤ 1 and N ≥ 1, from which the conclusion follows by Kolmogorov's continuity theorem [FV10,Thm. A.10].It suffices to consider t < u ≤ 2t and, by equivalence of Gaussian moments, p = 2.
Since X a n and X a m are independent if n = m with n 1 , m 1 > 0, and Recall from (3.5) that, for κ > 0 small, uniformly in 0 < t < u ≤ 2t < 1 and x ≥ 1, We can take κ > 0 sufficiently small such that −2d + 4 − 4δ + 4κ < −d, and therefore We now show that EZ t explodes as N → ∞ and t → 0 in a controlled manner.
Lemma 4.2.It holds that uniformly in t ∈ (0, 1) and in N ≥ 1.Furthermore, Proof.Using the upper bound in Assumption 2.6, EZ t is bounded above by a multiple of On the other hand, using the lower bound in Assumption 2.6, EZ t is bounded below by cK where c > 0 is uniform in t ≪ 1 and N > t −1/2 , and Define the deterministic processes H, I : [0, 1] → E by Observe that, by (4.1), uniformly in t ∈ (0, 1) and N ≥ 1, while by (4.2), uniformly in 0 < t ≪ 1 and

Deterministic bounds
In this section, we let H, I : [0, 1] → E be defined as in (4.3) from some {X N } N ≥1 satisfying Assumption 2.6.We consider N ≫ 1 and suppress it from our notation.We further fix δ ∈ ( 1 2 , 1) and β ∈ (−1, 0) such that Lemma 5.1.Suppose u 0 ∈ C ∞ and define the functions N t = B(P t u 0 , DP t u 0 ) and Q t = N t − H t .Then there exist κ > 0 small and C > 0 large, not depending on N , such that for all T ∈ (0, 1) with it holds that the solution u to (1.1) exists on [0, T ] and ) .
Proof.We decompose u t = P t u 0 + R t + I t where R solves the fixed point where we used that H t and I t are constant in space in the final equality.Define the Banach space We will show that, for T satisfying (5.1), M is a contraction on the ball of radius Recalling the heat flow estimates for α < 0 and γ ≥ 0 (see, e.g.[GIP15, Lem.A.7] and the proof of [BCD11, Thm.2.34]) (5.3) we obtain where we used β + η > −1, and where we used β > − 1 2 .Moreover, by Lemma 4.3 with p = 1, In this appendix, we prove well-posedness of (1.1) in the regime u 0 ∈ C η for η > − 1 2 (we do not require here that B is not a total derivative).We follow the notation of Sections 1 and 2.1 (and start of Section 3).By the elementary bound t 0 s a (t − s) b ds ≍ t a+b+1 for a, b > −1, we have where we used the heat flow estimates (5.3) in the second line and that η > − 1 2 in the third line to estimate the integral t 0 s η− 1 2 |v| 2 C η ds ≍ t η+ 1 2 |v| 2 C η .Using again (5.3), the same bound holds with t −η− 1 2 |M v t (R)| L ∞ replaced by t −η |M v t (R)| C 1 .It follows that there exists ε > 0 such that, for all M > 0 sufficiently large, K > 1, and v ∈ B η K , if T η+1 ≤ εM −1 K −2 , then M v stabilises the ball in X T centred at 0 of radius M K 2 .Likewise, for another v ∈ C η and R ∈ X T , a bilinear map, Du = (∂ 1 u, . . ., ∂ d u), and P : E → E is a polynomial of degree at most 3.In what follows, we assume B is not a total derivative, i.e., if we write B for y = (y 1 , . . ., y d ) ∈ E d as B(•, y) = B 1 (•, y 1 ) + . . .+ B d (•, y d ) ,

(B. 1 )
Again the same bound holds witht −η− 1 2 | • • • | L ∞ replaced by t −η | • • • | C 1 .Applying this with v = v ∈ B η K shows that M v is a contraction on the ball in X T centred at 0 of radius M K 2 provided T η+1 ≤ εM −1 K −2 .This proves the existence of a unique fixed point R = M v (R) in this ball.The uniqueness of R in all of X T follows by restarting the equation over sufficiently short time intervals.Furthermore, applying (B.1) to two fixed pointsR = M v (R) and R = M v ( R) with v, v ∈ B η K shows that |R − R| XT ε 1/2 M 1/2 K|v − v| C η provided T η+1 ≤ εM −1 K −2 ,which proves the claimed local Lipschitz property.
is anti-symmetric and non-zero for some X, Y ∈ E since g is non-Abelian.Equation (DYM) therefore satisfies Assumption 2.1.Example 2.3.The same example as above shows that the DeTurck-YM-Higgs heat flow (with or without the cube of the Higgs field, see [CCHS22b, Eq. (1.9) resp.(2.2)]) satisfies Assumption 2.1.
a |I s | p s b ds t 0 (t − s) a N 2p s b+p ds ≍ t a+b+p+1 N 2p t a+b+1 .On the other hand, if t > 2N −2 then (t − s) a ≍ t a for s ≤ N −2 and therefore, by (4.4), t 0