Global Existence and Finite-Time Blow-Up for a Nonlinear Nonlocal Evolution Equation

We study the Cauchy problem for a nonlinear nonlocal evolution equation arising in the study of oceanic flows in equatorial regions. We present a well-posedness result and show that while some initial data develop into solutions that exist for all times, others lead to blow-up in finite time.


Introduction
We consider the Cauchy problem for spatially periodic solutions of the evolution equation u t + uu x + (Hu)(Hu) x + β Hu − μ u x x = k(t, x) (1.1) with initial data where u = u(t, x) ∈ R for (t, x) ∈ (0, T ) × T, β ∈ R and μ > 0 are constants, and k : [0, T ) × T → R is a known smooth forcing with zero mean at every t ∈ [0, T ); here T = R/Z is the one-dimensional torus. Equation (1.1) arises in the context of equatorial ocean flows, describing the evolution of the restriction of the horizontal fluid velocity u to a fixed depth, with the forcing term k due to the corresponding horizontal pressure gradient (see Sect. 2). The study of (1.1) is not only of theoretical interest. Indeed, while submerged velocimeters and pressure transducers are commonly used devices to assess the state of ocean flows, the investigations that explore the global implications for the bulk of the fluid flow are typically pursued within the simpler setting of travelling waves (see the discussion in [4]). Equation (1.1) opens up a wider range of physically relevant possibilities.
In Sect. 2 we derive (1.1) from the governing equations for equatorial ocean flow (the Navier-Stokes equation with Coriolis effects accounted for, and the equation of mass conservation for an incompressible homogeneous fluid). In Sect. 3 we discuss the well-posedness of (1.1) and in Sect. 4 we prove global existence for small initial data. The decomposition in Fourier modes performed in Sect. 5 permits us to obtain a global existence result for suitable large initial data. In Sect. 6 we show that some large initial data develop into solutions that blow-up in finite time.

Derivation of the Model Equation
We now present a derivation of the model equation (1.1) from the governing equations for ocean flow in equatorial regions. The limited latitudinal extent of equatorial flows makes the effects of the Earth's sphericity unimportant and we can therefore use the Cartesian f -plane representation of the equations of motion (see [17]). We choose a coordinate system with the origin at a point on the Equator, with the x-axis pointing horizontally due East, the y-axis horizontally due North and the z-axis upward. We denote by u, v, w, the corresponding fluid velocity components in the direction of increasing azimuth, latitude and elevation. If is the (constant) rotational speed of the Earth round the polar axis toward the East, t stands for time, g is the (constant) gravitational acceleration at the Earth's surface, μ 1 and μ 2 are the (constant) horizontal and vertical eddy viscosity coefficients, respectively, ρ is the constant water density, and P is the pressure, the governing equations in the f -plane approximation are the Navier-Stokes equations (see [16]) coupled with the equation of mass conservation and with the constraint (u, v, w) → 0 for z → −∞, uniformly in (x, y), (2.5) that ensures that the water is practically at rest at great depths. A significant feature of equatorial air dynamics is that the change of sign of the Coriolis force across the Equator produces an effective waveguide: the Equator acts as a (fictitious) natural boundary that facilitates azimuthal flow propagation, with a negligible meridional velocity component (see [5,6]). Imposing a vanishing meridional velocity (v ≡ 0), for flows periodic in the x-variable (with normalized wavelength λ = 2π ) and independent of the y-variable the equations (2.1)-(2.4) simplify to Assuming ψ to be harmonic ensures that w and u are the real and imaginary parts of an analytic function of the complex variable x + iz. The Cauchy-Riemann equations would then permit us to formulate the restriction of the first equation in (2.6) to a submerged level z = z 0 as the evolution equation (1.1) in the single horizontal space-variable x, with given that where H is the Hilbert transform acting on square integrable functions of mean zero on [0, 2π ] as the Fourier multiplier The second relation in (2.10) follows because (2.8) and the periodicity yield and now (2.5) ensures that w has zero horizontal mean at every fixed time: We have μ > 0 since the vertical eddy viscosity coefficient is several orders of magnitude smaller than the horizontal one (see [16]). Note that reversing the flow direction (x → −x and u → −u) has the effect of changing β to −β in (1.1) and a solution to (1.1) determines the deep-water flow beneath the level z = z 0 . The harmonic hypothesis is convenient from the point of view of analytical tractability and presents also the advantage of ensuring that we investigate the propagation of largeamplitude oscillations along the submerged level. Indeed, (2.13) and the fact that the Hilbert transform is an isomorphism on the space of L 2 (T)-functions of mean zero ensure that at the level z = z 0 the horizontal and vertical kinetic energies are of the same order of magnitude. Note that the harmonicity of ψ beneath the submerged level z = z * 0 for some z * 0 > z 0 captures the physically realistic assumption of irrotational flow below z = z * 0 . Indeed, equatorial currents are confined to a near-surface layer (see the data in [6]) and the assumption of zero vorticity = 0, where is consistent with the equations (2.6)-(2.8) since using (2.8) and taking the curl of (2.6)-(2.7) yields (2.12) Zero initial data ( = 0 at t = 0) ensures = 0 at all subsequent times for the solution to the linear parabolic equation (2.12), for any bounded and continuously differentiable velocity field (u, w). Let us also point out that (1.1) captures the entire dynamics throughout the region of irrotational flow beneath z = z 0 . To see, this, it is convenient to introduce the velocity potential ϕ(t, x, z) as the harmonic conjugate of the stream function at every fixed time, uniquely determined up to an additive function of time by In terms of ϕ, we can recast the validity of the equations (2.6)-(2.7) in the region z < z * 0 as which is the viscous geophysical analogue of the Bernoulli equation for irrotational inviscid two-dimensional flow (see [3] for the relevance of the latter to water flows). Equation (1.1) is obtained by differentiating the restriction of (2.14) to z = z 0 with respect to the x-variable. This derivation forces k(t, ·) to have mean zero for all t ∈ [0, T ).

Well-Posedness
In the sequel we denote by L 2 0 (T) and H s 0 (T), s ∈ R, the closed subspaces of zero-mean functions in L 2 (T) and H s (T), respectively. The local well-posedness of (1.1)-(1.2) can be obtained by adapting the standard approaches that were developed for the heat and the (viscous) Burgers equation (see for instance [14,15,18,19] and also [13] for heat equations and [8], [2] for the Burgers equation). Here we follow mainly the point of view used by Dix in [8]. We choose to present two different results. In both cases we work with the integral equation x } t>0 denotes the heat semigroup generated on L 2 (T) by μ∂ 2 x . Our aim is to prove that the associated map is 1/2-contractive in a ball of a suitable metric function space. In our approach we will make use of the two following technical lemmas. The first one deals with classical product estimates in Sobolev spaces (see for instance [1] or [12]). Lemma 3.1. Let (t, r 1 , r 2 ) ∈ R 3 with r 1 + r 2 > t + 1/2, r 1 + r 2 ≥ 0 and r 1 , r 2 ≥ t. Then for any u ∈ H r 1 (T) and v ∈ H r 2 (T), we have uv ∈ H t (T) with
The second lemma concerns the regularizing effect of the heat kernel in Sobolev spaces.
Proof. Relation (3.3) follows directly from the exact formula Indeed, the Plancherel identity yields and this leads to 3.1. The case s ≥ 0. We start by an approach that enables us to get the unconditional local-wellposedness of (1.1)-(1.2) in H s (T) for s ≥ 0. The notion of unconditional local-wellposedness, introduced by Kato in [11], means that uniqueness holds in L ∞ (0, T ; H s (T)), which, roughly speaking, ensures the uniqueness of weak solutions. Proof. We first notice that since u 0 has zero mean-value and k ∈ L ∞ (R + ; H s 0 (T)), u 0 maps zero mean-value functions to zero mean-value functions. Now, a direct application of (3.3) with θ = 13 8 , in combination with (3.2) for r 1 = r 2 = s ≥ 0 and t = s − 5 8 , leads us to where in the fourth step we used that the Hilbert transform is a non-expansive map in H s (T) for any s ∈ R; here the constant C μ may change from line to line.
On the other hand, using the contractivity of the semigroup e μt∂ 2 x , t > 0, in Sobolev spaces and Lemma 3.2 with θ = 1, it is straightforward to check that Using these two estimates we obtain and This proves that for the map u 0 is 1/2 contractive on the ball centered at the origin of C([0, T s ]; H s 0 (T)), with radius 2 u 0 H s . Consequently, we showed the unconditional well-posedness of (1.1)-(1.2) in H s 0 (T).
To prove that the maximal time of existence only depends on u 0 L 2 (T) we proceed by induction on s. Let us denote by T * (s) > 0, s ≥ 0, the maximal time of existence of u in H s 0 (T). For s ∈]0, 5/8], we notice that Lemma 3.1 ensures Inserting this inequality in the third step of (3.5) we eventually get Translating this estimate in time and setting Assuming that T * (s) < T * (0) and taking t 0 < T * (s) close enough to T * (s) would lead us to which contradicts the definition of T * (s). It thus follows that T * (s) = T * (0) for 0 < s ≤ 5 8 . Since according to Lemma 3.1, we see that we can reach any s > 0 by repeating this argument a finite number of times. It remains to prove that u ∈ C(]0, T * (0)[; H s +1 (T)). For this we show a smoothing effect for our solution by making use of a space of weighted continuous function in time introduced by Kato and Fujita in [10]. More precisely, we define the Banach space We first notice that Lemma 3.2 ensures Then proceeding exactly as for (3.5), applying (3.3) with θ = 13 8 but this time (3.2) with r 1 = r 2 = s ≥ 0 and t = s + 1 16 Finally, applying (3.3) with θ = 1 16 and θ = 9 8 , we get . (3.11) The estimates (3.10)-(3.11) ensure that, starting with e μt∂ 2 x u 0 ∈ C([0, T ]; H s 0 (T))∩Y T , the function sequence constructed by the Picard iterative scheme associated with u 0 that, according to the contraction theorem, converges to the solution u in C([0, T s ]; H s (R)), is a Cauchy sequence in Y T s and thus u ∈ Y T s . Hence u ∈ C(]0, T s ]; H s+ 1 16 (T)) and we can re-apply the same argument with u(0+) ∈ H s+ 1 16 (T) as initial datum to get u ∈ C(]0, T s ]; H s+ 1 8 (T)). Actually, the argument can be repeated as long as s + n 16 ≤ s + 1 + 1 16 . This shows that u ∈ C(]0, T * (0)]; H s +1 (T)) since, according to the above considerations, T * (s + 1) = T * (0).

3.2.
The case −1/2 < s < 0.. Taking k = 0, a classical dilation argument (see [8], [2]) ensures that H − We then obtain in the same way as above where in the last step we used that s/4−5/8 > −1/8−5/8 > −1 and in the next-to-last step we performed the change of variables t = tτ . Moreover, it is easy to check that In view of (3.13) we easily get for 0 < T < 1 and v i ∈ X T , i = 1, 2, that and Combining these estimates with (3.12) we infer that, for s > −1/2, u 0 is 1/2contractive on X M,T with M ∼ u 0 H s (T) and This leads to the existence and uniqueness in X T s of solutions to (1.1)-(1.2) with u 0 ∈ H s 0 (T). Now to prove that the solution u belongs to C([0, T s ]; H s 0 (T)) we start by noticing that (3.4) ensures e μt∂ 2 x u 0 ∈ C(R + ; H s 0 (T)).

Global Existence for Small Data
To fully justify the following calculations we may assume that u ∈ C([0, T ]; H 4 0 (T)) and then extend the obtained L 2 (T)-bound to u ∈ C([0, T ]; L 2 0 (T)) by invoking the continuity with respect to initial data, established in Proposition 3.3.
Taking the L 2 (T)-scalar product of (1.1) with u and integrating by parts we get Since u has zero mean-value, we have using the inequality ab ≤ a 2 + b 2 /4 with a = μ 4 u x L 2 (T) and b = 4 μ u 3 L 2 (T) in the last step of (4.3). Combining (4.1), (4.2) and (4.3), we obtain Assuming that yields therefore u(t) 2 L 2 (T) ≤ μ 2 2 4 for all positive existence times since − 5μ 8 In view of Proposition 3.3, we thus obtain the following global existence result for small data: (4.5). Then the solution u to (1.1)-(1.2) belongs to C(R + ; H s 0 (T)) and is bounded in L 2 (T) uniformly in time.

Global Existence for Large Data
In this section we prove a global existence result for large data in the case k ≡ 0, corresponding to an isobaric level set z = z 0 .
For spatially 2π -periodic square-integrable real functions of mean zero we have the Fourier modes decomposition and, since u is real-valued, we also have The evolution equation (1.1) decomposes into the infinite system of ordinary differential equationṡ or, equivalently, the case of η m with m < 0 being covered by (5.1). Consequently we havė This ensures that an initial data u 0 supported by a finite number of Fourier modes (with η j (0) = 0 for | j| ≥ N ) gives rise to a solution supported by a finite of Fourier modes with vanishing Fourier modes of index j with | j| > N . In particular, if for some integer N ≥ 2, where {ζ } and {ζ } stand for the real and imaginary part of the complex number ζ , respectively, then (5.2) particularizes for N ≥ 3 to the linear system ⎧ ⎨ and for N = 2 to the nonlinear system ⎧ ⎨ ⎩η 1 + 4i η 2 η 1 + μ − iβ η 1 = 0, η 2 + 4μ − iβ η 2 = 0.
develops into a solution of (1.1) that is defined for all t ≥ 0.

A Blow-Up Result
We show that for k ≡ 0 and β > 0 the solution of (1.1) that emerges from a suitable initial data u 0 with mean-value zero and only supported by the first three Fourier modes blows-up in finite time: Proposition 6.1. For k ≡ 0 and β > 0 an initial data of the form develops into a solution to (1.1) with a finite maximal existence time 0 < t 0 < π 12β such that Proof. The equations (5.2) show that the solution emanating from u 0 will also have mean-value zero and will be supported by the three first Fourier modes for as long as it is defined. In particular, (5.2) simplifies to the following differential system ⎧ ⎨ The third equation can be solved explicitly, giving so that, by setting we transform (6.1) to the equivalent system Denoting now with real coefficients a 1 , b 1 , a 2 , b 2 , A, (a 1 , b 1 , a 2 We now prove that if the initial data satisfies then the unique solution to the system (6.6)-(6.9) belongs to A. By continuity this solution verifies (6.11) on some interval [0, t) with t > 0 small enough. If the claim does not hold, there exists some minimal value t * ∈ 0, min{t 0 , π However, from (6.10) and (6.8) we geṫ while from (6.10) and (6.9) we obtaiṅ and therefore (6.12) ensures Similarly, (6.10), (6.7) and (6.11) yielḋ so that (6.12) ensures b 1 (t * ) < −3A < 0. (6.14) On the other hand, from (6.6) and (6.7) we geṫ which, since (6.12) and the fact that b 1 is negative and strictly decreasing on (0, t * ) ensure leads tȯ
In particular, since 3a 2 (0) + b 2 (0) > 0 due to (6.12), we deduce that These considerations show that the only option left is that a 1 (t * ) = 0. But then, taking into account (6.13), (6.14), (6.9) and (6.15), we would obtain from (6.6) and (6.10) thaṫ which is a contradiction with the assumption a 1 (t) > 0 = a 1 (t * ) for t ∈ (0, t * ). This proves the claim about the solution developing from a data subject to the constraints (6.12) remaining in the set A for all t ∈ 0, min{t 0 , π 12β } . We now claim that for a suitable choice of initial data subject to (6.12), the resulting solution will blow-up in the finite time t 0 ∈ 0, π 12β , with Indeed, if the negative function b 1 (t) stays bounded from below on [0, t 0 ), then the relation 0 < a 1 (t) < −b 1 (t) yields that a 1 (t) stays bounded on [0, t 0 ) and the linear equations (6.8)-(6.9) now ensure this also for the remaining two Fourier modes a 2 (t) and b 2 (t).
Acknowledgement The authors declare no conflict of interest. They are grateful for helpful comments from the referees.
Funding Open access funding provided by University of Vienna.

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