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

We consider the initial value problem for a non-linear heat equation of the form {equ} {∂tu = Δu + B(u, Du) + P(u)   on [0,T]×Tdu(0,⋅) = u0(⋅) ∈C∞(Td,E) , where $\mathbb{T}^{d}=\mathbb{R}^{d}/2\pi\mathbb{Z}^{d}$ is the $d$-dimensional torus, $E$ is a vector space (over $\mathbb{R}$) of dimension $2\leq\mathop{\mathrm{dim}}\nolimits(E)<\infty$, $B\colon E\times E^{d}\to E$ is a bilinear map, $Du=(\partial_{1}u,\ldots,\partial_{d}u)$, and $P\colon E\to E$ 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 $y=(y_{1},\ldots,y_{d})\in E^{d}$ as {equ} B(⋅,y) = B_1(⋅,y_1)+…+ B_d(⋅,y_d) , where $B_{i}\colon E\times E\to E$ is bilinear, then we assume that $B_{i}$ is not symmetric for some $1\leq i\leq d$. We note that $B_{i}$ is symmetric if and only if there exists a bilinear map $\tilde{B}_{i}\colon E\times E\to E$ such that $\partial_{i}\tilde{B}_{i}(u,u)=B_{i}(u,\partial_{i}u)$ for all smooth $u$.

It is easy to show that, for $\eta>-\frac{1}{2}$, a solution $u$ to (1) exists for $T>0$ sufficiently small depending (polynomially) on $|u_{0}|_{{\mathcal{C}}^{\eta}}$, where ${\mathcal{C}}^{\eta}=B^{\eta}_{\infty,\infty}$ is the Hölder–Besov space of regularity $\eta$ (see Section 2.1 for a definition of $B^{\alpha}_{p,q}$). Furthermore, the map ${\mathcal{C}}^{\eta}\ni u_{0}\mapsto u$ is locally Lipschitz once the target space is equipped with a suitable norm and $T$ is taken sufficiently small on each ball in ${\mathcal{C}}^{\eta}$. We give a proof of these facts in Appendix LABEL:app:large_eta. (If each $B_{i}$ were symmetric, these facts would hold for $\eta>-1$.)

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.

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.

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 [Bourgain94, Burq_Tzvetkov_08, Oh_Pocovnicu_16, Pocovnicu17] and the review article [BOP19]; for recent work on parabolic PDEs, see [Sourav_flow, CCHS22, Hairer_Le_Rosati_22]. Furthermore, there have been many developments in the past decade in the study of singular stochastic PDEs with both parabolic [Hairer14, GIP15, BCCH21] and dispersive [GKO18, GKOT21, Tolomeo21] dynamics, for which it is often important to understand the effect of irregularities of (random) initial condition.

In [Sourav_flow, CCHS22], it is shown that (1) is locally well-posed¹¹1Strictly speaking, only the DeTurck–Yang–Mills heat flow, which is a special form of (1), was considered in [Sourav_flow, CCHS22] but the methods therein apply to the general form of (1); furthermore, only mollifier approximations were considered in [CCHS22], but the methods apply to Fourier truncations. for the GFF $X$ on $\mathbb{T}^{3}$, in the sense that, if $u_{0}=X^{N}$, then $u$ converges in probability as $N\to\infty$ to a continuous process with values in ${\mathcal{C}}^{\eta}$ for $\eta<-1/2$, 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 $(\mathcal{I},\|\cdot\|)$ of distributions with (continuous) inclusions $C^{\infty}\subset\mathcal{I}\subset\mathcal{S}^{\prime}(\mathbb{T}^{d},\mathbb{R})$. Let {equ} (^I,∥⋅∥) ,  ^I⊂S’(T^d,E) , denote the Banach space of all $E$-valued distributions of the form $\sum_{a\in A}x^{a}T^{a}$, where each $x^{a}\in\mathcal{I}$ and $\{T^{a}\}_{a\in A}$ is some fixed basis of $E$, with norm $\|\sum_{a\in A}x^{a}T^{a}\|\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\sum_{a}\|x^{a}\|$. We say that there is norm inflation for (1) at $x\in\hat{\mathcal{I}}$ if, for every $\delta>0$, there exists a solution $u$ to (1) such that {equ} ∥x-u_0∥¡δ   and   —∫_T^d u_t(x)​dx—¿ δ^-1 . 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 $\|u_{t}\|>\delta^{-1}$ are considered in (1). Norm inflation (in the sense that $\|u_{t}\|>\delta^{-1}$) at arbitrary points $x\in\hat{\mathcal{I}}$ 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) in particular shows that the solution map $C^{\infty}\ni u_{0}\mapsto u\in C([0,T],\mathcal{S}^{\prime})$ is discontinuous at $x$ in $\hat{\mathcal{I}}$, even locally in time. See [Iwabuchi_Ogawa_15, Carles_Kappeler_17, Oh_Wang_18, Kishimoto19] and [BP08, Cheskidov_Dai_14, Molinet_Tayachi_15] where norm inflation of various types is shown for dispersive and parabolic PDEs respectively.

Consider now a family of non-negative numbers $\sigma^{2}=\{\sigma^{2}(k)\}_{k\in\mathbb{Z}^{d}}$. We say that $\mathcal{I}$ carries the GFS with variances $\sigma^{2}$ if {equ} lim_N→∞∥X-X^N∥ = 0  in probability whenever $\{X_{k}\}_{k\in\mathbb{Z}^{d}}$ is a family of (complex) Gaussian random variables such that $\mathbb{E}|X_{k}|^{2}=\sigma^{2}(k)$ and $X^{N}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\sum_{|k|\leq N}X_{k}e^{\mathbf{i}\langle k,\cdot\rangle}$ is a real GFS for every $N\geq 1$ (see Definition 2.5), and where $X$ is a real GFS.

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 $\mathcal{I}$ is that every $\sigma^{2}(k)>0$ and that the smooth functions are dense in $\mathcal{I}$ (see Remark 2.14).

Note that the Besov space ${\mathcal{C}}^{-1/2}=B^{-1/2}_{\infty,\infty}$ is an ‘endpoint’ case since (1) is well-posed on ${\mathcal{C}}^{\eta}$ for every $\eta>-\frac{1}{2}$. In Proposition LABEL:prop:Besov_regLABEL:pt:-1 below, we show that ${\mathcal{C}}^{-1/2}$ carries a GFS with variances $\sigma^{2}$ satisfying the bounds in Assumption 2.6. Corollary 1.4, together with Remark 2.14 and Proposition LABEL:prop:Besov_regLABEL:pt:-1, therefore gives a probabilistic proof of the following result, which appears to be folklore.

We remark that ${\mathcal{C}}^{-1}$ is the scaling critical space for (1), so our results show ill-posedness above scaling criticality. Norm inflation of the same type as (1) in ${\mathcal{C}}^{-2/3}$ was shown for the cubic non-linear heat equation (NLH) $\partial_{t}u=\Delta u\pm u^{3}$ by the author and Oh–Wang [COW22] using a Fourier analytic approach taking its roots in [Iwabuchi_Ogawa_15, Oh17, Kishimoto19]. The method of [COW22] can likely be adapted to yield norm inflation for (1) in ${\mathcal{C}}^{-1/2}$, and we expect that the methods of this article can similarly be adapted to show (probabilistic) norm inflation for the cubic NLH. We remark, however, that our method appears not to reach $B^{-1/2}_{\infty,q}$ for $q<\infty$ (see Proposition LABEL:prop:Besov_regLABEL:pt:<-1), while the method in [COW22] for the cubic NLH does extend to $B^{-2/3}_{\infty,q}$ 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)) was shown in the scaling critical space ${\mathcal{C}}^{-1}$ by Bourgain–Pavlović [BP08] and which is locally well-posed in ${\mathcal{C}}^{\eta}$ for $\eta>-1$ (the main difference with (1) is that the non-linearity in NSE is a total derivative). In fact, norm inflation for NSE has been shown in $B^{-1}_{\infty,q}$ for $q>2$ by Yoneda [Yoneda10] and for all $q\in[1,\infty]$ by Wang [Wang15]. 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 ${\mathcal{C}}^{-1/2}$, is scaling subcritical. See also [Choffrut_Pocovnicu_18] and [Forlano_Okamoto_20] where norm inflation is established for the fractional non-linear Schrödinger (NLS) and non-linear wave (NLW) equations respectively above the critical scaling.

In [SunTz_20_norm_inf] and [Camps_Gassot_23], norm inflation is shown for the NLW and NLS respectively for all initial data $u_{0}$ belonging to a dense $G_{\delta}$ 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 [CCHS20, CCHS22] analysed the stochastic quantisation equations of the Yang–Mills (YM) and YM–Higgs (YMH) theories on $\mathbb{T}^{2}$ and $\mathbb{T}^{3}$ respectively (see also the review article [Chevyrev22_YM]); we also mention that Bringmann-Cao [BC23] analysed the YM stochastic quantisation equations on $\mathbb{T}^{2}$ by means of para-controlled calculus (vs. regularity structures as in [CCHS20]), and that the invariance of the YM measure on $\mathbb{T}^{2}$ for this dynamic was shown in [ChevyrevShen23]. In [CCHS22], the authors in particular constructed a candidate state space for the YM(H) measure on $\mathbb{T}^{3}$. A related construction was also proposed by Cao–Chatterjee [Sourav_state, Sourav_flow]. An ingredient in the construction of [CCHS22] is a metric space $\mathcal{I}$ of distributions such that the solution map of the DeTurck–YM(H) heat flow (or of any equation of the form (1), see [CCHS22, Prop. 2.9]) extends continuously locally in time to $\mathcal{I}$ and such that suitable smooth approximations of the GFF on $\mathbb{T}^{3}$ converge in $\mathcal{I}$; essentially the same space was identified in [Sourav_flow]. The works [Sourav_flow, CCHS22] thus provide a local well-posedness result for (1) with the GFF on $\mathbb{T}^{3}$ as initial condition.

In contrast, our results provide a corresponding ill-posedness result. Indeed, the GFF on $\mathbb{T}^{3}$ satisfies Assumption 2.6 and the DeTurck–YM(H) heat flow is an example of equation (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 $\mathcal{I}$ in [CCHS22] 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 $\mathbb{T}^{3}$ 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. [Chevyrev19YM, CCHS20] 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 [TerryPath] (see also [Lyons07, Prop. 1.29] and [FrizHairer20, Prop. 1.1]); the construction of the field $Y$ in Theorem 1.1 is inspired by a similar one in [TerryPath].