On Stability Estimates for the Inviscid Boussinesq Equations

We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow v=(y,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v=(y,0)$$\end{document} and a stratified temperature θ=αy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha y$$\end{document} with α>14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >\frac{1}{4}$$\end{document}. We show that for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document} there exist non-trivial explicit solutions, which are initially perturbations of size ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}, and grow to size 1 on a time scale ϵ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon ^{-2}$$\end{document}. Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than ϵ-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon ^{-4}$$\end{document}.


Introduction and Main Results
In this article we consider the stability of the incompressible, inviscid Boussinesq equations in a two-dimensional periodic channel near the stationary solution v = (y, 0), θ = αy, (2) where α > 1 4 is a constant.The Boussinesq equations are a common model of the evolution of a heat conducting fluid in terms of its velocity v and temperature θ and may additionally incorporate viscosity or thermal dissipation.In particular, questions of well-posedness and asymptotic behavior in regimes with partial dissipation [TWZZ20, MSHZ20, DWZ20, WXZ19, DWZZ18, LT16, CW13] or the inviscid problem [EW15,Wid18] have been an area with strong research activity in recent years.
The term θe 2 models buoyancy and causes hotter fluid to rise above colder fluid, where −e 2 is the direction of gravity.It is well known that, in the case without shear (v = 0), this buoyancy might lead to the so-called Rayleigh-Bénard instability, if hotter fluid is below colder fluid, α < 0. In contrast, if α > 0 is sufficiently large then the Miles-Howard criterion [How61] rules out spectral instability, which is the setting considered in this article.As we state in Lemma 1.2 and recall in Section 2 the linearized equations around the stationary solution (2) are stable in arbitrary Sobolev regularity globally in time.However, for the nonlinear equations we construct explicit solutions growing as ǫ(1 + t 2 ) 1/4 as t increases, which hence are only small on a time scale t < ǫ −2 .Moreover, even when restricting to this time scale, higher, Gevrey regularity is required in order to establish stability [BBCZD21,TW19].In the corresponding viscous problem instead Sobolev regularity is required, however with a smallness condition depending on the size of the viscosity [ZZ22].
The aims of the present article are two-fold: • For related equations such as the Euler equations [DM18, DZ19], Vlasov-Poisson equations [MV11,Bed20] or partially viscous Boussinesq equations [Zil21b] it is known that the norm inflation of the nonlinear dynamics is tied to the interaction of non-trivial low frequency solutions, which we call traveling waves, and their interaction with high frequency perturbations.We thus construct these traveling waves for the present problem and discuss for which choices of perturbations and parameters one might expect the largest possible norm inflation.• For these linearized equations we identify multiple frequency regimes depending on the initial size ǫ > 0 of the waves and the time interval under consideration.For frequencies |ξ| < ǫ −4 we establish an upper bound for perturbations concentrated at frequency ξ by exp((ǫξ) 2/3 ).In particular, if |ξ| ≤ ǫ −α with 1 < α ≤ 4 this factor is bounded by exp(ξ 2/3(1−1/α) ).This bound hence matches the control by exp( √ ξ), that is Gevrey 2 regularity, as in the nonlinear problem [BBCZD21] for ξ = ǫ −4 , but exhibits improved bounds if ξ is smaller.As a complementary result, if ξ > Cǫ −4 with a sufficiently large r constant C > 1, we instead obtain an upper bound which is uniform in ξ and ǫ on the time scale under consideration.
We remark that for technical reasons we consider a simplified model, which fixes the underlying shear flow.As we discuss in Section 4 this simplification can be removed in time intervals where the main norm inflation takes place and for large times.For small times we provide a rough bound for the non-simplified model, but expect that it can be improved to a uniform bound with substantial additional technical effort.
Before stating our main results, we recall that the linearized problem around the stationary solution (2) is stable, when working in coordinates moving with the shear and choosing suitable unknowns.The following lemma is adapted from [BBCZD21,TWZZ20].
Lemma 1.1.Let α > 1 4 .Then the linearized Boussinesq equations around the stationary solution (2) are stable in the sense that for any initial data ω, θ with ωdx = θdx = 0 the energy is bounded above and below for all times, uniformly in terms of its initial value, with a constant depending only on α.
As we discuss in Section 2.1 the choice of unknowns moving with the underlying shear flow Z(t, x, y) := √ α (∂ −2 x ∆) −1/4 ω (t, x − ty, y), Q(t, x, y) := (∂ −2 x ∆) 1/4 ∂ x θ (t, x − ty, y) (3) is natural.These unknowns have previously been used in [BBCZD21] and we use the same notation.We remark that in the (partially) viscous setting other choices of unknowns are natural [ABSPW22, ACW10, LWX + 21, TWZZ20, TW19, DWZZ18, CW13, ZZ22].We further observe that the linearized problem around the stationary solution (2) in terms of (Z, Q) reads Since the operator on the right-hand-side is a (time-dependent) constant coefficient Fourier multiplier the evolution of (Z, Q) decouples in Fourier space with respect to both x and y.Therefore all stability estimates hold frequency-wise and hence extend to arbitrary Sobolev, Besov or Gevrey spaces.However, this stability can be understood as an artifact of the fact that the stationary solution (2) is independent of x and that perturbations therefore decouple in frequency and cannot propagate along chains of resonances.For this reason, in order to capture instabilities of the nonlinear problem, instead of a stationary solution we consider nearby x-dependent explicit solutions.
Lemma 1.2 (compare Proposition 2.1 in [Zil21b] and [BBCZD21]).Let α ≥ 0 be given, then there exist non-trivial functions f (t) and g(t) such that are a solution of the nonlinear inviscid Boussinesq equations for all times.We call these solutions traveling waves.Moreover, if α > 1 4 it holds that for some constants 0 < c < C < ∞ depending on α.
We remark that in terms of the unknowns (3) these traveling waves read They are global in time, low-frequency solutions and remain uniformly bounded in any suitable Sobolev or Gevrey space for all times.
The functions f (t) and g(t) can be computed explicitly in terms of hypergeometric functions and the above results can hence be obtained by explicit computation (as was done in [Zil21b]).Furthermore, as shown in Lemma 1.1 the stability can also be obtained as a special case of an energy estimate similar to the one of [BBCZD21].
In the following we will consider a simplified version of the linearization of the Boussinesq equations in terms of (Z, Q) around these traveling waves and establish upper bounds on the possible norm inflation.The reduction of the non-simplified linearized Boussinesq equations is discussed in Section 4.
We note that for the traveling waves of Lemma 1.2 for large times the vorticity grows as Therefore if the traveling wave is initially of size ǫ > 0 it will remain a small perturbation of the stationary state (2) only on time scales where 0 < δ < 0.1 is a constant.Hence, similarly as in [BBCZD21] we restrict to studying stability and norm inflation on that time interval.
Theorem 1.3 (Stability and upper bounds on norm inflation).Let 0 < δ < 0.1 and 0 < ǫ < 0.1 be given and consider the simplified linearized Boussinesq equations around the traveling waves Then there exists C > 0 and |γ| < δ such that for any initial data (Z 0 , Q 0 ) whose Fourier transform satisfies the corresponding solution remains regular up to a loss in the constant C.That is, for all times t > 0 it holds that Here k ∈ Z denotes the frequency with respect to x and ξ ∈ R denotes the frequency with respect to y.
Moreover, there exists a constant c = c(α) such that if the Fourier transform of the initial data is supported in the region |ξ| ≥ cǫ −4 then the stability estimate improves to a uniform estimate Let us comment on these results: • Since the traveling waves are independent of y, these equations decouple with respect to the Fourier frequency ξ ∈ R corresponding to y.We may hence interpret (5) as an upper bound on the possible norm inflation factor for frequency-localized initial data by In particular, we emphasize that this multiplier strongly differs from the Euler case.As we discuss in Section 3.1 we expect that this bound is optimal in the sense that this norm inflation is attained (possibly with slightly smaller constant C) for all frequencies ǫ −1 ≤ |ξ| ≤ ǫ −4 .However, since estimates in certain time regimes are technically very involved (in particular for the non-simplifed problem) in this article we only establish upper bounds.• A corresponding nonlinear result has been established in [BBCZD21] using different methods with an upper bound on the norm inflation by exp(Cξ σ ) with σ > 1 2 .The present result recovers this bound with σ = 1 2 for ξ = ǫ −4 in the linearized problem around traveling waves.As major novelties, in this article we prove that for the present model: -The upper bound on the norm inflation factor is different and, in particular, much smaller when |ξ| is much smaller than ǫ −4 .-For large frequencies |ξ| ≥ cǫ −4 the norm inflation is bounded by a constant factor instead (see Proposition 3.7 for a more detailed statement).Compared to the estimates of [BBCZD21] we further exploit that the underlying traveling wave is much smaller for small times and that the time cut-off imposes an upper bound on the frequencies of resonances.
• In our simplified equations we omit the term from the linearized Boussinesq equations.As we discuss in Section 4 in the main time regime t > ξ 2/3 ǫ −1/3 , where the resonance mechanism takes place, this simplification can be removed.For the regime of small times, for the non-simplified problem we instead obtain a rough growth bound by exp(c √ ξ).However, we expect that this bound can be improved to a uniform bound (as for the simplified model) with more technical effort.
The remainder of the article is structured as follows: • In Section 2 we discuss the linearized problem around a ground state (2) as formulated in Lemma 1.1.In particular, we introduce the unknowns and system formulation used throughout the article.• In Section 3.1 we discuss the underlying resonance mechanism for a toy model.In particular, this allows us to clearly present the norm inflation mechanism and compare it with the Euler equations or the partially viscous problem.• Based on the insights derived from this model we show in Sections 3.2.1 and 3.2.2 that norm inflation cannot happen outside a specific time interval depending on the size of ξ and ǫ. • The main result of the paper is established in Section 3.2.3,where we establish bounds on the norm inflation achieved.• Finally, in Section 4 we show that the previously derived norm inflation estimates also extend to the non-simplified model.In particular, the omitted terms are only non-negligible perturbations for small times, where they can be absorbed by a loss of Gevrey regularity.As we discuss, we do not expect this loss to be attained.However, since the main focus of this article lies in the resonance mechanism for large times, we do not pursue this further.

Notation.
In this section we collect some notation used throughout the article for easier reference.
Our main object of interest are the (simplified) linearized Boussinesq equations around the traveling waves of Lemma 1.2 which we write in the form where are the coefficients of the traveling waves of Lemma 1.2.These equations decouple after a Fourier transform in y.Hence we view these equations as equations for (F y Z)(t, x, ξ), (F y Q)(t, x, ξ) for any fixed frequency ξ ∈ R and with slight abuse of notation write Z(t, x), Q(t, x) again.These equations may equivalently be expressed as coupled system for Fourier modes Z k , Q k as stated in Definition 3.3: with the coefficient functions stated in (12) (for k ± 1 = 0): Here c ± k is used as short-hand-notation for c + k or c − k .Similarly, we use c ∓ k±1 to refer to c − k+1 and c + k−1 .As noted after Lemma 1.1 for simplicity of notation the estimates of this article are stated for L 2 (dxdy) or ℓ 2 (Z) (with respect to k).However, since the above system includes only nearest neighbor interaction all estimates extend to the case of weighted ℓ 2 spaces, provided the weight λ(k) is such that |λ(k)/λ(k ± 1) − 1| is small enough.In particular, this allows for λ(k) = 1 + c|k| N for any N ∈ N and λ(k) = exp(c|k| s ) for any 0 < s < 1 and hence to establish stability in Sobolev or Gevrey spaces.
Throughout this article in several estimates it suffices to control quantities only in terms of upper and lower bounds within a constant factor.Hence, in order to simplify notation, we sometimes approximate values.For instance, we write to denote that for any k ∈ N, k = 0 the last two terms are comparable within a factor at most 10.

The Homogeneous Problem, Waves and Good Unknowns
As remarked following Lemma 1.1 the explicit solutions of the Boussinesq equations of the form may be found by inserting this ansatz into the Boussinesq equations, which reduce to an ODE for the coefficient functions f, g.
In the following we provide a different perspective on these solutions as low frequency waves.Thus consider the perturbations in coordinates moving with the affine flow.Then the full nonlinear Boussinesq equations are given by where we used the cancellation of ∇ ⊥ •∇.In particular, the left-hand-side has a very similar structure as the Boussinesq equations in vorticity formulation except that the equation satisfied by the stream function perturbation Φ now is time-dependent.With respect to these unknowns the traveling waves of Lemma 1.2 take the form They are explicit non-trivial solutions of the nonlinear problem, which are smooth, low frequency and initially are small perturbations of (0, 0).
In the following subsection we discuss the linearized equation around (0, 0) and introduce the associated (frequency-localized) solution operators.In particular, we show that the linearized problem around (0, 0) (which we call the homogeneous problem) is stable in arbitrary regularity.In contrast, as we sketch in Section 3.1 for a toy model the norm inflation of the corresponding non-linear problem is closely linked to the interaction of high and low frequencies by means of the nonlinearity In particular, this mechanism is not present in the linearized problem around (0, 0) but is present in the linearized problem around traveling waves.The main aim of the remainder of this article is to show that this linearized problem around such waves indeed captures this norm inflation mechanism and to identify the sharp regularity classes corresponding to this norm inflation.

Stability of the Homogeneous Problem.
A natural first step towards understanding the nonlinear behavior of initially small solutions of ( 7) is to study the linearized problem Given this form, we symmetrize the problem by introducing the good unknowns (3) (as in [BBCZD21]): We remark that the problem decouples after a Fourier transform in x and hence in our definition of Z, Q we may choose any power of ∂ x instead of |∂ x | 1/2 .This particular choice is made to simplify calculations for ∂ −2 x ∆ and to exploit slightly improved cancellation properties in Proposition 4.4.For the x-average we omit the |∂ x | 1/2 and define The following proposition states the stability result of Lemma 1.1 in terms of these unknowns.
Proposition 2.1.Let α > 1 4 and consider the linear system is approximately constant in the sense that there exist constants 0 < c < C < ∞, depending only on α, such that We remark that all operators involved are constant coefficient Fourier multipliers.The problem hence decouples in frequency and we may therefore replace the L 2 space in the definition of E(t) by any Fourier-based Hilbert space such Sobolev spaces H s , Besov spaces or Gevrey spaces and obtain the same result.We further remark that, as a decoupled ODE system in Fourier space, the solution operator could be computed explicitly.However, in view to later perturbed estimates, where upper bounds are sufficient, we instead employ an energy estimate approach as for instance used in [BBCZD21].
Proof of Proposition 2.1.We observe that the operator on the right-hand-side of the equation Since L t is a bounded operator one may already obtain a rough upper bound by employing Gronwall's lemma (we remark that at this point we do not yet require α > 1 4 ).In order to improve this estimate we further use that also the diagonal entries are symmetric.Therefore we may compute that Here the first two terms exactly cancel with the ones above and thus Similarly as in [BBCZD21, Zil20, DWZZ18] we observe that the operator norm of is strictly smaller than 2 if (and only if) α > 1 4 .Therefore, in that case E(t) is a positive definite bilinear form in (Z, Q) and it follows that The result hence follows by Gronwall's lemma and noting that the problem decouples in frequency.More precisely, instead of controlling the time integral of the operator norm of ), it suffices to control the time integral of the Fourier symbol for each fixed frequency: which is uniformly bounded.
Having established stability of the linearized problem around (Z, Q) = (0, 0) in the following we study the linearization around traveling waves.

Echo Chains and Gevrey Regularity
While the results of Section 2.1 establish linear stability of (Z, Q) = (0, 0), nonlinear stability (see [BBCZD21]) and asymptotic behavior for large times are very challenging problems.As a first step towards understanding the (optimal) long time behavior of the nonlinear Boussinesq equation (7) near a shear and hydrostatic balance, in the following we consider a toy model highlighting the role of the nonlinearity and of traveling waves.

A Toy Model and Optimal Gevrey Classes.
Resonance chains in phasemixing problems often manifest as a low frequency part of the solution interacting with the high frequency part by means of the nonlinearity (see for instance [DM18, BMM16, DZ19, Zil21a]).Based on this heuristic in this section we introduce a toy model capturing this mechanism, which allows us to identify an expected optimal regularity class.The main aim of the remainder of the paper then is to show that the linearized equations around a traveling wave indeed exhibit this growth.
We recall that the nonlinear Boussinesq equations near a traveling wave read Here we consider Z and Q to be at high frequency and thus the right-hand-side can be seen as paraproduct decomposition into low-high, high-low and high-high frequency products.
In order to derive our toy model in a first step we ignore all terms except the low-high term, which drives the resonance mechanism and arrive at .
We observe that in this model the evolution for Z decouples and, since the coefficient functions do not depend on y the equations further decouples after a Fourier transform in y (which is also true for the linearized problem around a wave, but not for the nonlinear problem).Let thus ξ > 1000 be a given frequency in y and suppose that Z is localized at frequency k in x and ξ in y.Then the multiplier is small unless t ≈ ξ k .As in [BBCZD21] for the toy model we thus consider a twodimensional system, which is supposed to approximate the evolution of F (Z)(k, ξ) and F (Z)(k − 1, ξ) on a time interval, where t ≈ ξ k .More precisely, we replace powers of ∆ t by its Fourier symbol and for simplicity of the model approximate Then the toy model reads: In [BBCZD21] the authors construct a toy model in the same way, but further estimate f (t)(1 + t 2 ) 1/4 ≤ 1 from above.Recalling from Lemma 1.2 that f (t) ≤ Cǫ √ 1 + t uniformly in time, this upper bound is achieved for t being comparable to ǫ −2 .However, for smaller times this upper bound is a (potentially large) overestimate, leading to a larger growth bound on the chain of resonances.
More precisely, we observe that by our choice of time interval t ≈ ξ k .This toy model thus suggests a growth by , which is potentially much smaller: Lemma 3.1.Let ξ ≥ 100 and k ∈ N be given and define Then on the time interval I k = (t k , t k−1 ) we consider the simplified toy model Then if Z N R (t k ) = 0 there exists a constant 1 < C < 10 such that , where we use ≈ to denote upper and lower bounds within a factor 10.
• We remark that here we neglected the evolution of Z R , which in turn effects the evolution of Z N R .As we discuss in Section 3.2.3taking this coupling into account results in a slightly modified growth bound by • By the restriction on the time scale it holds that ǫ ξ k ≈ ǫ √ t ≤ 1.Thus the amount of growth may be estimated from above by ξ k 2 , uniformly in ξ.However, this is an over estimate for most values of k and ξ. • As we discuss following the the proof of this lemma, the dependence on ǫ and k here strongly differs from the one of the Euler equations.
Proof of Lemma 3.1.The integral formula is immediate.We further observe that The result hence follows by observing that if ξ k 2 is sufficiently large the integral over I k is comparable to the integral over all of R.
Iterating this heuristic growth bound we may conjecture a total growth of by choosing k 0 ≈ (ǫξ) 2/3 and using Stirling's approximation.Thus at first sight this toy model suggests stability for Gevrey 3/2 regular initial data, uniformly in 0 < ǫ < 1 and for all times.However, since our evolution is restricted to the time interval (0, δǫ −2 ) this is an overestimate for large values of ξ.Indeed, for a resonance to happen the time t ≈ ξ k needs to have passed.Thus if ξ is very large then in the above product we may only consider those k for which while by the above consideration our cascade should start at k 0 to maximize the product and we thus arrive at an estimate of the possible norm inflation by Lemma 3.2.For 0 < ǫ < 0.1 and 0 < ξ < ǫ −4 consider the function Then it holds that Moreover, this bound is attained in the sense that for any 1 < σ < 4 we may consider ξ = ǫ −σ and there exists ǫ σ > 0 such that for 0 and that the product is empty if ξ is larger than this.In particular, for ξ ≥ ǫ −4 we may expect to obtain a uniform bound instead of norm inflation (this is shown in Proposition 3.6).
Proof of Lemma 3.2.For simplicitiy of notation let k 0 = ⌊(ǫξ) 2/3 ⌋.Then for the upper bound we may replace the starting point of the product by 1 to obtain For k 0 ≤ 100 we may control this quantity by a constant uniformly in ǫ and ξ, since (ǫξ) 2/3 ≤ k 0 + 1.It hence suffices to discuss the case when k 0 is large, where by Stirling's approximation formula it holds that We may therefore further estimate , which yields the desired upper bound by noting that the last factor is controlled by ((k 0 + 1)/k 0 ) k0 and hence uniformly bounded.
For the lower bound we argue similarly, but now have to take into account that the product starts at k 1 := ⌊ǫ 2 ξ⌋ and hence We thus need to show that the second factor is not too small and hence cannot cancel the growth.Here we again may restrict to the case when k 1 is large and use Stirling's approximation to compute It thus suffices to estimate exp 3/2k 0 − 3/2k 1 (1 + log(ξǫ 4 )) .
Here we observe that for ξ = ǫ −σ it holds that Since σ > 1 the power of ǫ in the formula for k 1 is negative and for σ < 4 it holds that Hence for 0 < ǫ < ǫ σ sufficiently small 3/2k 0 dominates.
Thus on this time interval the total growth is limited in terms of ǫ (by a frequency cut-off) and letting ǫ tend to zero the optimal Gevrey regularity class is expected to be given by 2.
Conjecture.Let 0 < ǫ ≤ 0.1 and consider the nonlinear Boussinesq equations perturbed around traveling waves of size ǫ.Then on the time interval (0, ǫ −2 ) the optimal space for stability is given by the Fourier weight exp((ǫξ We remark that stability in Gevrey 2, that is a bound by exp(C √ ξ), has been established in [BBCZD21], which coincides with the above weight for ξ = ǫ −4 .The conjecture suggests an improvement to these estimates when ξ is much smaller or much larger than ǫ −4 and that for ξ < ǫ −4 the exponent of growth (ǫξ) 2/3 < ξ 1/2 is attained.As first step towards proving this conjecture in this article we show that this statement is true for the (simplified) linearized Boussinesq equations around traveling waves.We remark that the above heuristic also suggests growth bounds for t > ǫ −2 for data in higher regularity classes (e.g.global in time for Gevrey 3 2 ).However, at that point the toy model simplification ceases to be justified and the toy model has to be replaced.The question of stability on larger time scales than (0, ǫ −2 ) for more regular data thus remains an interesting problem for future research.
For comparison we also note that for the Euler equations and a wave initially of size ǫ the growth of the vorticity ω (instead of Z) is bounded by exp( ǫξ) and thus stability in Gevrey 2 regularity holds when considering arbitrarily large times [DM18, DZ19,BM15].When also restricting to the time interval t < ǫ −2 we require that ξ is such that and thus exp( ǫξ) ≤ exp(min(ξ 1/3 , ǫ −1 ).
From this heuristic model we can thus already see that the hydrostatic balance for α > 1 4 yields a strong change of the stability and norm inflation behavior of the Boussinesq equations compared to the Euler equations: The growth of the underlying traveling wave results in larger norm inflation.

The Inhomogeneous Problem and Upper Bounds.
Building on the heuristic of the previous model in the following we consider the simplified linearized Boussinesq equations around a traveling wave where we omitted the low frequency velocity contribution As we discuss in Section 4 this term does not qualitatively change the dynamics at large times, but is technically challenging to control for small times.
As suggested by the notation we consider this problem as a (possibly large) perturbation of the inhomogeneous problem of Section 2. In particular, if we denote by S(t 2 , t 1 ) the solution operator of the homogeneous problem, then we may equivalently express the above differential equation as the integral equation We next recall that the solution operator S(•, •) is given by a Fourier multiplier and decouples in frequency.Hence, taking a Fourier transform in both x and y we arrive at a system with nearest neighbor interaction.

Definition 3.3 (Inhomogeneous system). The simplified linearized Boussinesq equations around a traveling waves for a perturbation frequency localized at
where we introduced the coefficient functions and denote by Z k , Q k the Fourier modes at frequency k ∈ Z in x and frequency ξ ∈ R in y.As the system decouples in ξ we treat it as a fixed parameter and suppress it in our notation.
The toy model of Section 3.1 here omitted all terms except the main resonance mechanism due to c + k−1 .Our main aim in the following is to show that this model indeed provides an accurate heuristic and that all other contributions can be controlled.
• The main time regime of interest is given by time intervals I k , where t ≈ ξ k , for which c + k−1 is comparatively large.This regime is studied in Section 3.2.3.Here the coupling of modes leads to a modified growth behavior as compared to the toy model of Section 3.1.
• In Sections 3.2.1 and 3.2.2we show that in the remaining time intervals resonances are too small to have a large effect on the dynamics and the evolution is at most algebraically unstable.

The Long Time Regime.
In this section we consider the regime of "large" times, where 2ξ < t < δǫ −2 .
As suggested by the heuristic model of Section 3.1 for such large times there are no resonances and hence the evolution is at most algebraically unstable.
Proposition 3.4.Let 0 < ξ < 2δǫ −2 be given and consider the time interval Then on I the solution to the system (11) grows at most algebraically in any Sobolev or suitable Gevrey space in the sense that the Fourier projections away from and onto the modes k = −1, 1 satisfy for any t ∈ I and any 1/2 < γ < 1.
This proposition implies a bound on the norm inflation on this time interval by t 3/2 ≤ ǫ −3 , which is much smaller than the exponential growth bound expected on earlier time intervals.We further remark that this time interval is empty if ξ > ǫ −2 and that this proposition is hence only concerned with "small" frequencies.
Proof of Proposition 3.4.We recall from Proposition 2.1 of Section 2.1 that the solution operators are bounded by a constant C α uniformly in k.
It thus suffices to control the corrections of (11) dt in a suitable way to invoke Gronwall's lemma, where we will distinguish between the case where |k| ≥ 2 and the cases k = −1, 0, 1.For simplicity of presentation in the following we establish estimates in the unweighted space ℓ 2 .The case of weighted spaces with a weight λ k can be reduced to this case by considering modified coefficient functions of the form More precisely, for instance for Sobolev spaces we may choose λ k = 1 + c|k| s , where c is a small constant and hence deduce that λ k±1 λ k is bounded above and below by constants close to 1. Hence all estimates below extend to this case with possibly a small loss of constants.
In the following we estimate the coefficient functions.We observe that for t > 2ξ all frequencies k = 0 are non-resonant in the sense that In particular, recalling the definition of the coefficient functions (12) as long as none of k, k − 1, k + 1 are zero, we may bound for some universal constant C, where we with slight abuse of notation estimated g(t) ≤ f (t)/t.We further recall that by our choice of time interval Hence, for these coefficient functions we obtain uniform L 1 estimate in time (more precisely, the supremum in k is still in L 1 ).It hence only remains to discuss the cases k ∈ {−1, 0, 1}.Here we observe that while Therefore, we cannot hope for better growth estimates than for the simple ODE system for t > ξ.Note that after rescaling we may without loss of generality set ξ = 1.We may then introduce 1/2 < γ < 1 and consider We observe that the diagonal entries are integrable in time by our choice of γ, while the bottom right-entry is negative.Hence, by Gronwall's lemma 3.2.2.The Small Time or High Frequency Regime.By the results of the preceding Section 3.2.1 any possible norm inflation has to happen for times 0 < t < 2ξ.
Thus similarly to the setting of the Euler or Vlasov-Poisson equations we partition this time interval into regions in which t is comparable to ξ k for some k ∈ N. Definition 3.5.Let ξ > 0 be given.Then for any k ∈ N we define and the associated time intervals We further define We recall from the toy model of Section 3.1 and from the structure of the coefficient functions c ± k , d ± k stated in (12) that on a given time interval I k the main resonance mechanism is expected to be determined by In particular, this value is bigger than 1 for k ≤ k 0 and smaller than 1 if k ≥ k 0 + 1.We further remark that, if ξ is much bigger than ǫ −4 or if t is small, then we expect resonances to only result in small perturbation of the dynamics, as we prove in the following propositions.The main resonance mechanism in the remaining time interval is then studied in Section 3.2.3.As a first result we note that if ξ > ǫ −4 is very large all permissible choices of k (that is, with t k < ǫ −2 ) are non-resonant and stability estimates can be obtained by a simple ode-type estimate.Proposition 3.6 (High frequency I).Let ξ > ǫ −4 , then there exists as constant C depending only on α such that for any choice of initial data it holds that Proof of Proposition 3.6.We recall that by Definition 3.3 we may equivalently consider a system of integral equations In particular, using the bounds on S k of Section 2.1 it follows that the solution satisfies the integral inequality where C is a constant which may depend on α.The claimed bound hence follows by an application of Gronwall's inequality provided sup Indeed, we observe that for t ∈ I k it holds that is uniformly bounded by assumption on ξ and that f (t) < 1 by our choice of time interval.The estimates on d ± l follow analogously by noting that g(t) √ 1 + t is uniformly bounded by our choice of time interval.Thus the estimate (13) holds, which concludes the proof.
We remark that this bound is very rough and not expected to be sharp for most choices of ξ.Indeed as suggested by the model of Section 3.1 if ξ is much larger than ǫ −4 we obtain no norm inflation at all.Proposition 3.7.Let C α denote the operator norm of semi-group of the homogeneous problem, that is Then for all ξ > 2C α ǫ −4 and all t ∈ (0, ǫ −2 ) it holds that The evolution is uniformly bounded.
Proof of Proposition 3.7.We claim that for this choice of ξ it holds that Recalling the integral equation we thus deduce that In particular, we may consider the supremum in t ≤ τ on both sides and consider (suitably weighted) ℓ 2 norms to obtain that Since the factor 1 2 on the right-hand-side is smaller than 1 we may subtract it from both sides and obtain that Finally we observe that sup and that for a time independent function (such as (Z, Q)(0)) equality holds.
It thus only remains to establish the estimates (14).Indeed we observe that if On the remaining interval and for all other l we may estimate f (t) ≤ δ and observe that and that for l = 0 The same method of proof can also be applied for general ξ when restricting to suitably small times.
Further suppose that δ < 1 Cα with C α as in Proposition 3.7.Then for all 0 ≤ t ≤ min(T, δǫ −2 ) it holds that Proof of Proposition 3.8.We claim that for this choice of T it holds that The result then follows by the same argument as in the proof of Proposition 3.7.Indeed, we observe that for k such that t k ≤ T (that is for all k larger than k 1 with t k1 ≈ T ) it holds that If instead k is such that t k < T (or if we integrate over (0, T ) \ I l±1 ) then integral is not (yet) resonant and hence For times larger than T resonances are possibly very large and thus the preceding argument does not work anymore, since estimates of the form do not control the norm.In the following Section 3.2.3we thus instead establish growth bounds on each interval I k which mimic the growth of the toy model of Section 3.1 with slight changes to the exponent.

Main Echo Chains.
In this section we consider the main norm inflation mechanism of the (simplified) linearized Boussinesq equations as compared to the toy model of Section 3.1.Here, similarly to the Euler setting [DZ19], it turns out for large frequencies the back-coupling between resonant and non-resonant modes results in correction of the growth bounds, which has to be taken into account.As we discuss in Section 4 the following results remain valid for the non-simplified linearized Boussinesq equations as well.
As a preliminary step we consider a more accurate toy model and establish more accurate bounds Lemma 3.9.Let ξ k 2 ≥ 100 be given, let 0 ≤ f (t) < δ and consider the differential inequalities Then there exists a constant 0 < γ < 2δ such that That is, we obtain an upper bound on the norm inflation by f L ∞ ( ξ k 2 ) 1/2+γ .We remark that if f (t) is replaced by a constant and if we consider a differential equation instead of an inequality, then the ODE system can be solved explicitly by noting that u solves This is the defining equation of Legendre functions and the above estimates hence follow from the known asymptotics of these functions.
The main aim of this lemma is thus to provide a more robust energy-based proof, which also extends to differential inequalities with time-dependent coefficients.
Proof of Lemma 3.9.We consider the following energy: We note that E(t) is continuous and that both (1 + t 2 ) 1/2 and (1 + t 2 ) −1/2 are decreasing on the respective time intervals.Hence by direct computation Integrating this inequality on (− ξ k 2 , ξ k 2 ) we obtain that In particular, it holds that and by construction We have thus established upper bounds for general initial data.
We next consider the differences compared to the initial data.By the fundamental theorem of calculus for any This is almost the desired bound except that we are still missing one factor of f L ∞ .It is however already sufficient to estimate Z R as Finally we may return to the bound for Z N R and split and insert the just derived bound.
Having established this improved model we next show that also the (simplified) linearized Boussinesq equations exhibit this modified growth (as compared to the toy model of Section 3.1; the non-simplified equations are studied in Proposition 4.2).Here in addition to the above growth bounds we have to take into account the evolution by the homogeneous semigroup (see Section 2.1).We further recall that for the linearized Boussinesq system Proposition 3.10.Let 0 < ξ < ǫ −4 , α > 1 4 and 0 < ǫ < δ and consider the linearized Boussinesq equations (11) on the time interval with 1 ≤ k ≤ k 0 and t k , t k−1 as in Definition 3.5.Then there exists C = C(α) and 0 < γ < δ such that for all choices of data at time t k and all t ∈ I k it holds that Corollary 3.11.Under the same assumptions as in Proposition 3.10 for any l ≤ k 0 it holds that Thus the total possible norm inflation on (t k0 , δǫ −2 ) is bounded by the exponential factor stated in Theorem 1.3.
Proof of Corollary 3.11.The result follows by repeated application of the estimate of Proposition 3.10.In particular, choosing t l maximal we obtain the products discussed in Section 3.1 with an additional correction in the exponent.
Proof of Proposition 3.10.Based on the structure of the homogeneous problem as studied in Section 2.1 we consider

On the Model Reduction
In this section we discuss the non-simplified linearized Boussinesq equations which we may also express in an integral system as in Definition (11) by introducing the coefficients We observe that by our choice of time interval is small and integrable and that f (t) cos(x)∂ y is a transport operator which corresponds to a change of variables (x, y) → (x, y − F (t) sin(x)), which is an analytic change of variables and a small perturbation of the identity (for ǫ small).
In view of this smallness and in order to simplify the analysis of the model and the presentation of the resonance mechanism, throughout this article we have considered the simplified linearized Boussinesq equations which omitted these terms.
In the following we show that this simplification indeed does not change the results of the long-time regime of Section 3.2.1 and the echo chains of Section 3.2.3.For the small time regime of Section 3.2.2we obtain (much) rougher bounds for the full model.We expect that with (considerable) technical effort it should be possible to improve these bounds after incorporating an additional change of variables (see the discussion following Proposition 4.2).
It hence only remains to discuss the "small time" regime of Section 3.2.2.Here we observe that f (t) 1 + t 2 ξ ≤ ǫ √ t 1 + t 2 ξ ≤ cǫξ in general is not small enough to employ the contraction argument of Proposition 3.8 unless ξ is smaller than ǫ −1 .Indeed also for the change of variables (17) we cannot expect good bounds in high Sobolev or Gevrey norms for frequencies larger than ǫ −1 .In order to obtain better bounds we thus have to take these changes into account.
One option here is to consider the unknowns (Z, Q) in the coordinates (17).However, here F (t) sin(x) introduces further nearest neighbor coupling in x, which makes terms such as ∆ −1/4 t very technically challenging to study.A second option, which sidesteps this issue, is to restrict to studying stability estimates in ℓ 2 (Z) (that is, with respect to Fourier modes in x for a fixed frequency in y), following the argument of [BBCZD21].Since f (t)/(1 + t 2 ) cos(x)∂ y is an anti-symmetric operator in L 2 , this space allows us to exploit cancellation and hence to estimate As a first preliminary result we consider an adaptation of the "intermediate time" estimate of [BBCZD21, Section 6.3.2], which allows for loss of regularity in Gevrey σ with σ < 2 in the time interval where t > √ ξ.Proof.We observe that for t → ∞ or l → ∞ We may hence exploit the fact that the operator f (t) cos(x)∂ y is anti-symmetric and thus have to estimate f (t)/(1 + t 2 )ξ((1 + (ξ/l − t) 2 ) 1/4 (1 + (ξ/(l ± 1) − t) 2 ) −1/4 − 1).

Lemma 4. 3 .
Let ξ and T be as in Proposition 3.8 and suppose that √ ξ < T .Then on the time interval ( √ ξ, T ) the maximal possible norm inflation is bounded by exp(δc σ ξ σ )for any σ ≥ 1 2 .Proof.Arguing similarly as in the proof of Proposition 3.8 we consider the energy exp(λ(t)ξ σ ) (Z, Q) 2

L 2
with λ(t) decreasing in time and bounded below, still to be determined.Computing the time derivative it then suffices to show that λ(t) can be chosen such thatλ(t)ξ σ + |g ± l | + |h ± l |.