The Periodic Boundary Value Problem for the Weakly Dissipative μ-Hunter-Saxton Equation

The Hunter-Saxton (HS) equation is an asymptotic equation for rotators in liquid crystals and modeling the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal [2]. The HS equation has a bi-Hamiltonian structure [3, 4] and is completely integrable [5]. The initial value problem of the HS equation also has been studied extensively; see [2, 6, 7]. We refer to [8–12] for the weak solution to the HS equation. When μ(u) = 0, the global existence and blow-up phenomena for the dissipative HS equation can be found in [13]. When λ = 0, (1) becomes the μ-HS equation

When  = 0, (1) becomes the -HS equation which is derived and studied in [14].Khesin et al. [14] showed that the -HS equation describes the geodesic flow on   (S) with the right-invariant metric given at the identity by the inner product Moreover, if interactions of rotators and an external magnetic field are allowed, then the -HS equation can be viewed as a natural generalization of the rotator equation.The periodic peaked solution and the multipeakons solution to -HS equation were showed in [14,15].Another important model which possesses some similar structures to the -HS equation is the Camassa-Holm (CH) equation: The Camassa-Holm (CH) equation was derived independently by Fokas and Fuchssteiner in [16] and by Camassa and Holm in [17].Fokas and Fuchssteiner derived (6) in studying completely integrable generalizations of the KdV equation with bi-Hamiltonian structures, while Camassa and Holm proposed (6) to describe the unidirectional propagation of shallow water waves over a flat bottom.In [18], Constantin and Lannes proved that the CH equation, as a model for the propagation of shallow water waves, is valid approximations to the governing equations for water waves.The relation 2

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between the CH and HS equation is that the HS equation can be viewed as the high-frequency limit of the CH equation [5].
In [17], Camassa and Holm discovered the bi-Hamiltonian structure of CH, which ensures the existence of an infinite number of conservation laws.The integrability of CH (as an infinite-dimensional Hamiltonian system) was studied in [19,20].Moreover, the CH equation is such an equation that exhibits both phenomena of soliton interaction (peaked soliton solutions) and wave breaking (the solution remains bounded while its slope becomes unbounded in finite time [21]).The global existence of strong and weak solution and blow-up phenomena for the CH equation can be found in [21][22][23][24][25].
In general, it is almost impossible to avoid energy dissipation in the real world.Therefore, it is reasonable to study the model with energy dissipation.For example, the weakly dissipative CH equation and weakly dissipative DP equation have been studied in [26] and [27][28][29], respectively.
Notations.We use ≲, ≳, and ≈ to denote estimates that hold up to some universal constant. is also a generic constant that may assume different values in different lines.D(S) is the space of all infinitely differentiable functions on S and D  (S) is its dual space.All function spaces are over S and we drop S in our notations of function spaces if there is no ambiguity.
The main results of this paper are as follows.
The remainder of this paper is organized as follows.In the next section, we state some preliminaries.We prove Theorem 1 in Section 3.

Preliminaries
Now we state some preliminaries on the Littlewood-Paley theory.We refer to [34][35][36] for the elementary properties of them.
Let ,  be two smooth radial functions satisfying We decompose  ∈ D  (S) into Fourier series; that is,  = (1/2) ∑ ∈Z F  ()  , where F  () = ∫ S  − ()d is the Fourier transform and the inverse transform is given by F −1  () = (1/2) ∑ ∈Z ()  .Now we define the periodic dyadic blocks as Then we define the low frequency cut-off   as    = ∑ −1 =−1 Δ  .Direct computation implies that, for any 1 ≤  ≤ ∞, we have the quasi-orthogonality properties: Furthermore, for all  ∈ where The following lemma summarizes some useful properties of   , .

and
and (5) If {  } ∈N is bounded in   , and   converges to  in D  (S), then  ∈   , and For  = ( −  2  ), we see that , which means that  −1 is bounded from   , into  +2 , .Now we recall some results of the transport equation (see [34] for the details).
We need the following lemmas.

Proof of Theorem 1
In this section, we prove Theorem 1 by the following several steps.

Approximate Solutions and Their Uniform Bounds.
Starting from  1 = 0 and by induction, we define a sequence of smooth functions {  },  ∈ N by solving the following transport equation iteratively: where () = ∑ 1≤≤3   () and   () given in (9).Since all the data belong to  ∞ 2,1 , from Lemma 5 and by induction, we can show that, for all  ≥ 1, the above equation has a global solution  +1 belonging to (R + ,  ∞ 2,1 ).

Hölder Continuity of the Solution Map.
The continuity of the solution map can be obtained by following the steps in [31] and using Lemma 13 and we omit the details in this paper.Now we consider the Hölder continuity of the solution map.For the initial data  0 ∈ (0, ) ⊂ where T does not depend on .Therefore, we can find T > 0 such that, for all  0 ∈ (0, ) ⊂  3/2 2,1 , the corresponding solution  ∈ ([0, T];  3/2 2,1 ).Directly from (13) and Lemma 12,(14) is proved.
We complete the proof of Theorem 1.
) d d d.