Blow-up analysis for a periodic two-component μ-Hunter–Saxton system

The two-component μ-Hunter–Saxton system is considered in the spatially periodic setting. Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works.

Integrating the first equation of system (1) over the circle S = R/Z and noting the periodicity of u, we have μ(u t ) = 0. Making use of system (1), we have that S (u 2 x + ρ 2 ) dx is conserved in time. In what follows we denote and Then μ 0 and μ 1 are constants independent of time t.
which implies that the amplitude of wave remains bounded in any time. Namely, we have which results in In fact, the initial-value problem (1) can be recast in the following: where A = μ -∂ 2 x is an isomorphism between H s and H s-2 with the inverse ν = A -1 ω given explicitly by Commuting A -1 and ∂ x , we get and Note that if f ∈ L 2 (S), we have A -1 f = (μ -∂ 2 x ) -1 f = g * f , where we denote by * convolution and g is the Green's function of the operator A -1 given by and the derivative of g can be assigned The objective of the present paper is to focus mainly on a wave-breaking criterion and wave-breaking phenomena for system (1). The local well-posedness of system (1) is firstly established by using Kato's theory. Then we present two wave-breaking criteria (see Theorem 3.1 and Theorem 3.2) and a wave-breaking phenomenon (Theorem 4.1) for system (1) in the Sobolev space H s (S) × H s-1 (S) with s ≥ 2. The results obtained in this paper are new and different from those in Liu and Yin's work [2].
The rest of this paper is organized as follows. Section 2 states local well-posedness for the periodic two-component μ-Hunter-Saxton system. In Sect. 3, we employ the transport equation theory to prove a wave-breaking criterion in the Sobolev space H s (S) × H s-1 (S) with s ≥ 2. An improved wave-breaking criterion also is presented in Sect. 3. Section 4 is devoted to the study of a wave-breaking phenomenon.

Local well-posedness
In this section, we will establish the local well-posedness for system (1) by Kato's theorem. For convenience, we present here Kato's theorem. Consider the abstract quasilinear evolution equation Let X and Y be two Hilbert spaces such that Y is continuously and densely embedded in X. Let Q : Y → X be a topological isomorphism, and let · X and · Y be the norms of the Banach spaces X and Y , respectively. Let L(Y , X) denote the space of all bounded linear operators from Y to X. In particular, it is denoted by We make the following assumptions, where μ 1 , μ 2 , μ 3 , and μ 4 are constants depending only on max{ z Y , y Y }.
(III) f : Y → Y extends to a map from X into X, is bounded on bounded sets in Y , and satisfies Kato's Theorem ( [25]) Assume that conditions (I), (II), and (III) hold. Given v 0 ∈ Y , there is a maximal T > 0 depending only on v 0 Y and a unique solution v to system (13) The local well-posedness for system (1) is collected in the following. Proof Since there are some similarities with the proof of Theorem 3.1 in [14], here we omit the proof of the theorem.

The function m(t) is absolutely continuous on
Now, consider the initial value problem for the Lagrangian flow map: where u denotes the first component of the solution z = (u, ρ) to system (1). Applying classical results from ordinary differential equations, one can obtain the result.
Theorem 3.1 Let z 0 = (u 0 , ρ 0 ) ∈ H s (S) × H s-1 (S) with s ≥ 2, and z = (u, ρ) be the corresponding solution to (1). Assume that T > 0 is the maximal existence time. Then Proof Since the two equations for u and ρ in system (7) satisfy the transport structure Therefore, we can complete the proof of Theorem 3.1 by making use of conservation laws and the localization analysis in transport equation theory (see Theorems 3.1 and 3.2) in [18]. The detailed proof can be found in [18].
Proof By Theorem 2.1 and a simple density argument, we need only to prove this theorem for s ≥ 3. We may also assume u 0 = 0, otherwise it is trivial. Let T > 0 be the maximal time of existence of the corresponding solution (u, ρ) to system (1). We first prove the case in (18). Assume that T < ∞ and (18) is not true. Then there is some positive number Ω > 0 such that Therefore, Theorem 3.1 implies that the maximal existence time T = ∞, which contradicts the assumption that T < ∞. Now, we try to prove the blow-up criterion (19). Since sup x∈S (v x (t, x)) = -inf x∈S (-v x (t, x)), we define Obviously, Since q(t, ·) defined by (14) is a diffeomorphism of the circle for any t ∈ [0, T), there exists Along the trajectory of q(t, x 1 (t)), we have Differentiating the first equation of system (7) and using the equality Along the trajectory of q(t, x 1 (t)), (25) can be rewritten as the following form: where denotes the derivative with respect to t and d(t) = -2μ 0 u + 2μ 2 0 + 1 0 ( σ 2 u 2 x + 1 2 ρ 2 ) dx. Assume that (19) is not valid, then there is some positive number Ω such that then, from (24), for each x ∈ S, we get from which we obtain In order to proceed with the proof, next we need to obtain the lower bound of d(t, -ξ (t)). Let then We now claim that Assume the contrary that there is t 0 ∈ [0, T) such that Q(t 0 ) < 0. Let t 1 = max{t < t 0 ; Q(t) = 0}. Then Q(t 1 ) = 0 and Q (t 1 ) < 0, namely and Recalling (26) and using (33), we have which is a contradiction to (35). This verifies that (33) is valid. Therefore, choosing arbitrary x ∈ S, we have recalling the assumption we get σ 2 |u x | < +∞. This contradicts our assumption T < ∞, which completes the proof of Theorem 3.2.

Wave-breaking phenomenon
In this section, we give a new blow-up phenomenon. To prove the blow-up phenomenon, the following lemma is crucial. We let then a new wave-breaking result is collected in the following theorem. and then the solution of system (1) blows up in finite time.
Proof By Theorem 2.1, we need only to prove this theorem for s ≥ 3. According to Lemma 3.1, there exists ξ (t) ∈ S such that Since q(t, ·) defined by (14) is a diffeomorphism of the circle for any t ∈ [0, T), we know that there exists x 1 (t) ∈ S such that Then (38) and (40) imply that Therefore we can choose ξ (0) = x 1 and Using Lemma 3.3, we have On the other hand, due to sup x∈S (v x (t, x)) = -inf x∈S (-v x (t, x)), we similarly define There exists x 2 (t) ∈ S such that Moreover, we have ρ t, -q t, x 2 (t) = ρ t, -η(t) = 0, ∀t ∈ [0, T).