Global estimates and blow-up criteria for the generalized Hunter-Saxton system

The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(\lambda,\kappa)\in\mathbb{R}\times\mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.

From a more heuristic point of view, (1.1) may serve as a tool to better understand the role that convection and stretching play in the regularity of solutions to one-dimensional fluid evolution equations; it has been argued that the convection term can sometimes cancel some of the nonlinear effects and contribute positively to regularity of solutions ( [49], [31], [52]). More particularly, differentiating (1.1)i) in space, and setting ω = −u xx , yields (1.5) The nonlinear terms in equation (1.5)i) represent the competition in fluid convection between nonlinear steepening and amplification due to (1 − 2λ)-dimensional stretching and 2κ-dimensional coupling ( [30], [66]). More particularly, the parameter λ ∈ R is related to the ratio of stretching to convection, while κ ∈ R denotes a real dimensionless constant measuring the impact of the coupling between u and ρ. Additional fluid models belonging to the family of equations (1.1) include: For (λ = −κ = ∞), equation (1.1)ii) reduces, after the introduction of new variables, to the well-known Constantin-Lax-Majda equation ( [17]), a one-dimensional model for the three-dimensional vorticity equation for which finite-time blow-up solutions are known to exist. If λ = −κ = 1/2, the inviscid von Karman-Batchelor flow ( [8], [31]), derived from the 3D incompressible Euler equations, is known to have periodic global strong solutions. Lastly, if ρ never vanishes in [0, 1], then which represents a slight variation of (1.1), can be obtained from the 2D inviscid, incompressible Boussinesq equations by considering velocities u and scalar temperatures θ (or densities) of the form on an infinitely long two-dimensional channel (x, y) ∈ [0, 1] × R. The Boussinesq equations model large-scale atmospheric and oceanic fluids (see for instance [25], [54], [43]). Above, θ denotes either the scalar temperature or density, and → e 2 is the standard unit vector in the vertical direction. Before giving an outline of the paper, we discuss some previous results.

Previous Results.
In this section, we review a few results on the regularity of solutions to system (1.1) ( [65], [66], [63], [41]). For additional blow-up or global-in-time criteria the reader may refer to [46] and [47].
First of all, the local well-posedness of (1.1) has already been established, see for instance [65], [66], or [63]. Then the following is known: Then u x will diverge.

Outline of the Paper
The outline for the remainder of the paper is as follows. In §3, we derive representation formulae for general solutions to (1.1). This is done using the method of characteristics to reformulate the system as a nonlinear second-order ODE, which we are then able to solve using the prescribed boundary conditions. In §4.1 we establish terminology and introduce useful preliminary results. Then, we begin our study of regularity in §4.2, which examines the case of parameter values λκ < 0, whereas, those satisfying λκ > 0 are deferred to §4.3. More particularly, Theorems 4.24 and 4.29 in §4.2 establish criteria for the finite-time convergence of solutions to steady states and their finite-time blow-up, respectively. Then Theorems 4.49 and 4.60 in §4.3 examine finite-time blow-up and global existence in time for (λ, κ) ∈ R − × R − and respectively (λ, κ) ∈ R + × R + , with the latter case leading to the "most singular" solutions. Lastly, specifics examples are provided in §5, while trivial or simpler cases are deferred to Appendix A or the Corollaries C.5 and C.6 in Appendix C.

General Solution Along Characteristics
In this section we derive new solution formulae for u x and ρ, along characteristics, for arbitrary (λ, κ) ∈ R × R.
Remark 3.45. The above formulae is also valid for u satisfying Dirichlet boundary conditions which implies that, for as long as u is defined, those characteristics that originate at the boundary stay at the boundary, As a result, the jacobian still has mean one in [0, 1] and we may proceed as in the periodic case.
Remark 3.46. Integrating the jacobian (3.38) in α yields the trajectories γ(α, t) = γ(0, t) + 1 P 0 (t) α 0 P 0 (y, t) dy, (3.47) where γ(0, t) = u(γ(0, t), t). For Dirichlet boundary conditions, γ(0, t) ≡ 0, and so u • γ is obtained fromγ = u • γ, namely However, for periodic solutions γ(0, t) ≡ 0 is generally not true, and so an extra condition is needed to determine γ(0, t). Consequently, to have a completely determined description of the problem, we will assume for the remainder of the paper that the first component solution, u, has mean zero in [0, 1], It is known in the case of the giPJ equation (see for instance [57] or [15]), that (3.48) arises naturally for particular classes of data due to certain symmetries that are preserved by the PDE. Moreover, in the context of the Euler equations, (3.48) holds as long as the pressure is periodic in one of its coordinate variables ( [60]).

Blow-up, Global Estimates, and Convergence to Steady States
In this section we study the evolution in time of (3.42) and (3.43) for parameters (λ, κ) ∈ R\{0} × R\{0}, particularly, their finite-time blow-up, persistence for all time, and convergence to steady states in finite or infinite time. Most of the regularity results established here, as well as in the Appendix, apply to initial data (u 0 , ρ 0 ) that is either smooth or belongs to a particular class of smooth functions; however, from the solution formulae derived in §3, these results can actually be extended to larger classes of nonsmooth functions, specifically, bounded functions u 0 (x) and ρ 0 (x) which are, at least, C 0 [0, 1] a.e.
First, in §4.1 we introduce some new terminology and establish useful preliminary results. Then §4.2 examines the case of parameter values λκ < 0, whereas, those satisfying λκ > 0 are deferred to §4.3. More particularly, Theorems 4.24 and 4.29 in §4.2 establish criteria for the convergence, in finite time, of solutions to steady states as well as their finite-time blow-up, respectively. Then Theorems 4.49 and 4.60 in §4.3 examine finite-time blow-up and global existence in time for (λ, κ) ∈ R − × R − and respectively (λ, κ) ∈ R + × R + , with the latter case leading to the "most singular" solutions. Lastly, we note that in §4.3 we treat the case where (3.32) vanishes earliest at some η−value of single multiplicity; the case of a double root is deferred to Corollaries C.5 and C.6 in Appendix C.

Notation and Preliminary Results.
For C(α) as in (3.28), let and so that [0, 1] = Ω ∪ Σ. Although we allow for Ω = ∅, we will assume that C(α) is not identically zero, namely, Ω [0, 1]. Moreover, and mostly for simplicity, we suppose that for Ω ∅, C(α) vanishes at a finite number of locations 2 . Lastly, we limit our analysis to the case ρ 0 (x) 0, which by (3.43) implies ρ(x, t) 0. From the formulae (3.42) and (3.43), and the time-dependent integrals in them, we see that the issue to consider while studying the evolution in time of these quantities is not just a possible vanishing of the quadratic Q in (3.32), but also what are the locations in [0, 1] that yield the least, positive η−value for which vanishing occurs. With this in mind, note that whenever α ∈ Ω, Q reduces to a linear function of η. Instead, if α ∈ Σ, then C(α) 0 and we may factor the quadratic into terms whose zeroes have either single or double multiplicity. More particularly, since the discriminant of (3.32) is given by Q admits three representations, the first being The reader may refer to Appendix A for the cases Ω = [0, 1], as well as λ = 0 and/or κ = 0. For now, we note that these special cases lead to regularity results that have already been established, in [57] or [58] for instance, or to solution formulas which simplify greatly, in some cases leading to trivial solutions. and valid whenever α ∈ Σ is such that ρ 0 (α) 0. The second is given by for α ∈ Σ and ρ 0 (α) = 0, while, if α ∈ Ω, Q reduces to Now let a finite number of points in the unit interval for which lim the earliest, and where lim for 0 < t * ≤ +∞ as in (3.44). Below we remark on the behaviour of Q relative to a possible, earliest root η * and the location(s) α where η * may be attained.
Case λκ > 0 Next suppose λκ > 0. Then the discriminant (4.3) satisfies D(α) ≥ 0 and Q now admits roots η * of either single or double multiplicity. Relative to our choice of α and initial data, the above representations (4.4)-(4.7) for Q will play an important role in our study of regularity because the λ−values for which the integral terms in (3.42) and (3.43) either converge or diverge as η approaches η * may, in turn, depend on the multiplicity of η * . Therefore, and as we will see in later estimates, it will only be necessary to consider representations (4.4) (or (4.7)) and (4.6), the single and respectively double multiplicity cases. However, the case of η * a double root, as it turns out, has already been studied (see for instance [57], [58] and the references therein). Indeed, suppose the data is such that (4.8) corresponds to some α satisfying (4.6), i.e. η * = 1 λu 0 (α) . First of all, note that the simplest instance where this occurs is when ρ 0 (α) ≡ 0, so that (3.32) reduces to (4.6) for all α ∈ [0, 1]. As discussed in §1, for ρ 0 ≡ 0, (1.1) becomes the giPJ equation, studied extensively in [57], [58]. Furthermore, even though for ρ 0 0 a double root η * may also occur if α ∈ Σ is such that ρ 0 (α) = 0, 3 then, as we show below, solutions still retain their giPJ equation behaviour.
The latter assumptions imply that the space-dependent term in (3.42) will diverge earliest, as η ↑ η * , when α = α, Similarly we find that the integral terms, for both r > 0 and η * − η > 0 small, satisfȳ Consequently, if Q has as its earliest zero, η * , a root of double multiplicity, then for η * − η > 0 small the above estimates imply that the time-evolution of (3.42) can be examined, alternatively, via the simpler estimate The right-hand-side of (4.15) was studied in [57] and [58] in connection to the giPJ equation; therefore, for the double root case, estimates on the behaviour of the integrals (4.13) and (4.14) as η ↑ η * are readily available in these works, and we direct the reader to Corollaries C.5 and C.6 in Appendix C for the corresponding regularity results. However, and for the sake of completeness, we will give a brief outline on how to obtain these estimates in the proof of Theorem 4.29.
In light of the above discussion, in §4.3 we will only be concerned with the representations (4.4) and (4.7), the case where η * is a single root of Q. Accordingly, define and Notice that, while M always exists, N may not due to the vanishing of C(α) at finitely many points, which implies that Σ is an open set. Also, note that there is no need to consider an eventual vanishing of the linear term in (4.4) involving g 2 . Indeed, if the initial data is such that g 2 (α) ≤ 0, then due to the strictly increasing nature of η(t) and η(0) = 0, such term will never vanish. Moreover, if g 2 is somewhere positive, it is easy to see that, over those α ∈ Σ where ρ 0 (α) 0, we have g 1 (α) > g 2 (α). As a result, for parameters λκ > 0, we conclude that there are two cases of interest concerning the least value η * > 0 at which Q vanishes. If N does not exist, or if it does but M > N, we set whereas, for N > M, we let See below for two simple examples involving single roots.

4.2.
Regularity Results for λκ < 0. For parameters λκ < 0, Theorem 4.24 below establishes conditions on the initial data for which both component solutions, (3.42) and (3.43), converge in finite time to steady states U ∞ and respectively P ∞ , given by where C(α) > 0 is as in (3.28) and the real numbers M > 0 and N are defined as  .3) for parameters λκ < 0 and initial data u 0 (x) and ρ 0 (x) both bounded and, at least, C 0 [0, 1] a.e. If ρ 0 (α) never vanishes, there exists a finite t ∞ > 0 such that (3.42) and (3.43) converge to the steady states in (4.22) as Proof. Suppose λ and κ are such that λκ < 0 and ρ 0 (α) is never zero on [0, 1]. The latter implies that, particularly, ρ 0 (α) 0 on Σ. This and λκ < 0 imply that (4.3) is negative and, thus, a contradiction. Therefore, 0 < Q < +∞ for all α ∈ [0, 1] and 0 ≤ η(t) < +∞, but Q → +∞ as η → +∞. Next, define real numbers M > 0 and N as in (4.23) and note that both are well-defined because λκ < 0 and ρ 0 0 imply that C(α) > 0. Then (3.32) yields, for large enough η > 0, the simple asymptotic estimates Then, if P ∞ (α) denotes the limit as η → +∞ of the right-hand-side above, we get (4.22)ii). In a similar fashion, using (4.26) on (3.42) yields (4.22)i). Finally, since (4.26)i) implies that that is, as η → +∞, t ceases to be an increasing function of η and converges to a finite value, which we denote by t ∞ . This establishes the first part of the Theorem. For the last part, denote by α i ∈ [0, 1], 1 ≤ i ≤ n, the points where ρ 0 vanishes. Moreover, assume there are finitely many of these points and suppose λu 0 ( which, once again, implies that 0 < Q < +∞ since η ≥ 0. Similarly for u 0 (α i ) = 0, in which case Q(α i , t) ≡ 1. At this point, we may now follow the argument used to prove the first part of the Theorem. This establishes our result.
blows up to negative infinity.
In this section, we are concerned with regularity properties of (3.42) and (3.43) for Ω in (4.1) non-empty and parameters λκ > 0. 7 Below we will see how, of the two cases λκ < 0 or λκ > 0, the latter represents the "most singular" in the sense that, relative to a class of smooth initial data, spontaneous singularities may now form in ρ. This should not come as a surprise if we note that for λκ > 0, as opposed to λκ < 0, the discriminant (4.3) now satisfies D(α) ≥ 0, and so a root (4.8) of single multiplicity corresponding to α ∈ [0, 1] with ρ 0 (α) 0, may now occur. Furthermore, we remind the reader that only the case where (4.8) is a single root of Q is considered in this section. Although such case arises the most for ρ 0 (α) 0, in Appendix C regularity results for the instance of a double multiplicity root are presented and examples of nonsmooth initial data for which it occurs are given. is attained. Then there exist smooth initial data and a finite t * > 0 such that , t), t) → −∞ as t ↑ t * , whereas, if α α, it remains finite for (λ, κ) ∈ (−1, 0) × R − and 0 ≤ t ≤ t * , but diverges to plus infinity, as t ↑ t * , when (λ, κ) ∈ (−∞, −1] × R − . (2) For ρ 0 (α) > 0 or ρ 0 (α) < 0, ρ(γ(α, t), t) diverges, as t ↑ t * , to plus or respectively minus infinity, but remains finite otherwise. 7 The case Ω = ∅ follows similarly.
Proof. We consider the case where η * > 0, the earliest zero of Q, has multiplicity one. Refer to Corollary C.5 in Appendix C for the double multiplicity case (N > M with ρ 0 (α) = 0), and see Appendix A for Ω = ∅. For (λ, κ) ∈ R − × R − , let α ∈ [0, 1] denote the finite number of points where the largest between M and, if it exists, N, both as defined in (4.50), is attained 8 . Without loss of generality, we will assume that N exists and N > M; otherwise, you may use an almost identical argument to the one presented below. Set Then the space-dependent term in (3.42) will vanish, first, when α = α as η ↑ η * . However, we still need to consider the behaviour of the integral terms in (4.33). As in the proof of Theorem 4.29, we begin with the simple case where (λ, κ) ∈ [−1/2, 0) × R − . For such values of λ, the integral terms in (3.42) and (3.43) remain finite for smooth enough initial data and, thus, the space-dependent term in (3.42) leads to blow-up, i.e. for α = α, In contrast, if α α, then the space-dependent term, and consequently u x (γ(α, t), t), remain finite for all 0 ≤ η ≤ η * and (λ, κ) ∈ [−1/2, 0) × R − . Moreover, (4.50)ii) implies that ρ 0 (α) 0, consequently (3.43) and boundedness of (4.33)i) yields, as η ↑ η * , but remains finite otherwise. The existence of a finite blow-up time t * > 0 in this case follows from (3.44) in the limit as η ↑ η * . Actually, because only the integral term (4.33)i) appears in (3.43), we have in fact established part (2) of the Theorem.
Remark 4.57. Notice that, if N does not exist, g 1 cannot attain positive values greater than M at infinitely many points in Σ. Indeed, suppose N does not exist and recall that Ω is assumed to be a discrete finite set. Then there isα ∈ Ω such that g 1 (α) = λu 0 (α) + λκρ 0 (α) 2 < λu 0 (α) + λκρ 0 (α) 2 (4.58) for all α ∈ Σ. Butα ∈ Ω and (3.28) imply λκρ 0 (α) 2 = λu 0 (α) 2 , which we substitute into the right-hand-side of (4.58) to obtain Clearly, if (4.18) holds, 2λu 0 (α) above becomes M forα = α. The above argument justifies (4.18) as the earliest η−value causing blow-up of the space-dependent terms in the case where N does not exist. A similar argument to the one above may be used for the case where g 2 in (4.5)ii) has a positive maximum.
In the above we used η(t) = t, a consequence of (3.41) and (5.1)i), and which yields Letting α = 1/2 in (5.2), we see that    Figure B shows a bounded ρ • γ for several choices of α and 0 ≤ t ≤ t * .
We note that we are also able to obtain the inverse jacobian function in closed-form, which is then used to plot the solution in Eulerian coordinates. Lastly, the steady states, u ∞ x and ρ ∞ , are given by Additionally, and for the sake of comparison, below we plot (3.42) and (3.43) for the same initial data but (λ, κ) = (1/2, −1). The steady states in that case are which are reached at, approximately, t ∞ ∼ 2.22. It is simple to check that both steady states coincide with the formulae in Theorem 4.24. See Figures 5 and 6 below.  In this section, we consider some special and trivial cases not studied in the paper.
A.4. Case λ = 0. For smooth enough initial data, this case may be treated by following the argument in Appendix A of [57], and leads to global-in-time solutions.

Appendix B. Further Integral Estimates
In this section, we give a brief outline of the method used to obtain several of the integral estimates used throughout the paper for (4.33)i), particularly those leading to a convergent integral as η ↑ η * for certain values of λ. First we need some auxiliary results. Recall the Gauss hypergeometric series ( [1], [20], [24]) . Then consider the following results ( [20], [24]): Proposition B.2. Suppose |arg (−z)| < π and a, b, c, a − b Z. Then, the analytic continuation for |z| > 1 of the series (B.1) is given by where Γ(·) denotes the standard gamma function.
Below we give a simple example on how to use the above results to estimate the behaviour, as η ↑ η * , of the integral 1 0 dα (1 − λη(t)u 0 (α)) 1 λ for λ ∈ (1, +∞)\{2} and where M 0 > 0 denotes the largest value attained by u 0 (α) at finitely many points α ∈ [0, 1]. For > 0 small, we start with the approximation which originates from a Taylor expansion, with non-vanishing quadratic coefficient, of u 0 about α. Now, suppose λ ∈ (1, +∞)\{2} and set b = 1 λ in Lemma B.4 to obtain where the above series is defined by (B.1) as long as ≥ −C 1 ≥ −s 2 C 1 > 0, namely −1 ≤ s 2 C 1 < 0. However, ultimately, we are interested in the behaviour of (B.8) for > 0 arbitrarily small, so that, eventually, s 2 C 1 < −1. To achieve the transition of the series' argument across −1 in a well-defined, continuous fashion, we use proposition B.2 which provides us with the analytic continuation of (B.8) from argument values inside the unit circle, particularly on the interval −1 ≤ s 2 C 1 < 0, to those found outside and thus for s 2 C 1 < −1. Consequently, for small enough, so that −s 2 C 1 > > 0, proposition B.2 implies for ψ( ) = o(1) as → 0 and C ∈ R + which may depend on λ and can be obtained explicitly from (B.3). Then, substituting = 1 λη − M 0 into (B.9) and using (B.7) along with (B.8), yields for η * − η > 0 small. We remark that the above blow-up rate for λ ∈ (1, 2) could also have been obtained for the whole interval λ ∈ (0, 2) via the simpler method used in the proofs of most Theorems in this article.

Appendix C. Roots with double multiplicity
In this section, we present blow-up and global existence results for solutions to (1.1) in the case where the quadratic (3.32) vanishes earliest at η * having double multiplicity. As shown in §4.1, in such case the asymptotic analysis of the space-dependent terms in (3.42) and (3.43), as well as that of the integrals (4.33), follows that of [57] and [58]. Consequently, we direct the reader to those works for specific details on the corresponding estimates.
Suppose the initial data is such that Q vanishes earliest at is attained at finitely many points α ∈ [0, 1]. In other words, we are assuming that the least, positive zero of the quadratic has multiplicity two (see (4.6)). In turn, this means that the discriminant of Q satisfies D(α) = 0, namely ρ 0 (α) = 0. By comparing N 1 with the maximum value over Ω of 2λu 0 or, if it exists, the maximum over Σ of g 1 in (4.5)i), both of which lead to single roots, we see that for most choices of smooth, non-trivial initial data, η * will be a zero of multiplicity one; however, double roots may occur in some trivial cases as well as for (non-)smooth data with particular growth conditions near locations where ρ vanishes. For instance, as mentioned before, for the trivial case ρ 0 (α) ≡ 0 (the giPJ equation case), η * is always a double root of Q. For the non-trivial case, consider for instance piecewise continuous data Therefore, if α ∈ [1/4, 3/4] we have that Q(α, t) = (1 − η(t)) 2 , which vanishes as η ↑ η * = 1, a root of double multiplicity. For α ∈ [0, 1]\[1/4, 3/4], Q has single roots that are either negative, or larger than η * . Notice that for the above non-smooth initial data formulas (3.42) and (3.43) are still defined. Although we are not concerned with non-smooth data in this work, the above may serve as a prototype on how double roots, with ρ 0 0, can occur. We claim that for smooth data a single root is most common because, if we are to compare the greatest values of g 1 = λu 0 + √ λκ |ρ 0 | and λu 0 over Σ for λκ > 0, then for the former to be less that than the latter, assumptions are needed on how steep u 0 and |ρ 0 | decrease and respectively increase near zeroes of ρ 0 in Σ, and that is assuming u 0 attains its greatest value there. If not, then the maximum of g 1 would be greater. Clearly, in the other case the maximum of 2λu 0 is greater than that of λu 0 .
As opposed to Theorem 4.49, which deals with an earliest root η * of single multiplicity, Corollary C.5 below considers a double multiplicity root for parameters (λ, κ) ∈ R − × R − . We find that regularity results in both cases are rather similar, with the only difference being a scaling between the λ values in both results. In contrast, Corollary C.6 represents the case where, for (λ, κ) ∈ R + × R + , Q has an earliest root of multiplicity two. As opposed to the single multiplicity case in Theorem 4.60, we find that double multiplicity in η * now allows for global solutions in certain ranges of the parameter λ.
Proof. Recall that for λκ < 0, the only instance leading to finite-time blow-up involved Q having a double root η * . Based on this assumption, we derived estimate (4.39) in the proof of Theorem 4.29. Comparing such derivation to the one leading to estimate (4.53) in Theorem 4.49, we note that the two correspond simply if we replace λ in the latter by λ 2 , you may check that both (4.36) and (4.52) coincide under the suggested substitution. In this way, for instance, the regularity result for u x on the interval −1 < λ < 0 in Theorem 4.49, will apply to our u x for −1 < λ 2 < 0, namely, −2 < λ < 0, and similarly for the remaining values. In contrast, greater care is needed when studying ρ. Since η * = 1 λm 0 is the earliest root of Q and has multiplicity two, we have, from D(α) = 0, that ρ 0 (α) = 0, whose opposite was precisely the requirement we needed on ρ 0 in Theorem 4.49 to obtain blow-up. Because this is not possible in the present double root case, we obtain part (2) of the theorem.