ON THE CAUCHY PROBLEM OF THE MODIFIED HUNTER-SAXTON EQUATION

. This paper is concerned with the Cauchy problem of the modiﬁed Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces H s with s > 3 / 2, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker H r -topology is used then it is shown that the solution map becomes H¨older continuous in H s .

1. Introduction. In this paper we study the periodic Cauchy problem of the modified Hunter-Saxton equation where k ≥ 1 is a positive integer, T = R/2πZ is the torus, H s (T) is the Sobolev space on the torus T with exponent s. For s > 3/2 we prove that the Cauchy problem (1)-(2) is locally well posed in T and that the data-to-solution map is continuous but not uniformly continuous. Furthermore, we show that the solution map is Hölder continuous in T if it is equipped with an H r -norm, 0 ≤ r < s. The equation (1) was proposed by J. Hunter and R. Saxton [48]. In [63], the author established local well-posedness in the Sobolev spaces in Sobolev space H s (T) for s > 3 2 and in C 1 (T) also studied the analytic regularity (both in space and time variables) of this problem. In [54], Kohlmann considered the initial boundary value problem of the modified Hunter-Saxton equation.

YONGSHENG MI, CHUNLAI MU AND PAN ZHENG
For k = 1, Eq. (1) reduces to the Hunter-Saxton equation [48] ∂ t u + u∂ x u = 1 2 which describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. Here, u(t, x) describes the director field of a nematic liquid crystal, x is a space variable in a reference frame moving with the linearized wave velocity, and t is a slow time variable. More precisely, the orientation of the molecules is described by the field of unit vectors (cos(u(t, x)), sin(u(t, x))) [48,49]. The Hunter-Saxton equation also has a bi-Hamiltonian structure [48,59] and is completely integrable [2,49]. and it is also the Euler equation for the geodesic flow on the quotient space of the infinite-dimensional group D(S) of orientationpreserving diffeomorphisms of the unit circle S = R/Z modulo the subgroup of rotations Rot(S) equipped with the H 1 right-invariant metric [55,56,64]. The initial value problem for the Hunter-Saxton equation on the line (nonperiodic case) was studied by Hunter and Saxton in [48]. Using the method of characteristics, they showed that smooth solutions exist locally and break down in finite time. The occurrence of blow-up can be interpreted physically as the phenomenon by which waves that propagate away from the perturbation "knock" the director field out of its unperturbed state [48]. The initial value problem for the periodic Hunter-Saxton equation was studied in [65]. Using the Kato theorem, the author obtained that this equation has solutions for initial data u 0 ∈ H s (S), s > 3 2 and showed that all the nonconstant solutions blow up in finite time [65]. Recently, global dissipative and conservative weak solutions for the Hunter-Saxton equation on the line were investigated extensively, cf. [4,24,50,51,66,67,68].
The Hunter-Saxton equationn can be regarded as a non-local perturbation of the Burgers equation. Also, it is formally obtained in the short-wave limit (t, x) → (εt, εx) for ε → 0 of the famous Camassa-Holm equation modelling the unidirectional propagation of shallow water waves over a flat bottom, u(t, x) stands for the fluid velocity at time t in the spatial direction x. It is a wellknown integrable equation describing the velocity dynamics of shallow water waves. This equation spontaneously exhibits emergence of singular solutions from smooth initial conditions. It has a bi-Hamilton structure [30] and is completely integrable [7,13]. In particular, it possesses an infinity of conservation laws and is solvable by its corresponding inverse scattering transform. After the birth of the Camassa-Holm equation, many works have been carried out to probe its dynamic properties. Such as, Eq. (5) has travelling wave solutions of the form ce −|x−ct| , called peakons, which describes an essential feature of the travelling waves of largest amplitude (see [14,15,22,17]). It is shown in [21,12,18] that the inverse spectral or scattering approach is a powerful tool to handle the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention that Eq. (5) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group [19,58], and this geometric illustration leads to a proof that the Least Action Principle holds. It is shown in [16] that the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. Moreover, the Camassa-Holm equation has global conservative solutions [5,45] and dissipative solutions [6,46]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [3,23,20,11,32,33,31,27,25,26] and the references therein. Non-uniform dependence of the Camassa-Holm type equation has been studied by several authors. In [44], Himonas and Misio lek provides the first nonuniform dependence result for CH on the circle for s ≥ 2 using explicitly constructed travelling wave solutions. This result was sharpened to s > 3/2 in [39] on the line and [40] on the circle. Both of these more recent works utilize the method of approximate solutions in conjunction with delicate commutator and multiplier estimates. For the DP equation, the first result for nonuniform dependence can be found in Christov and Hakkaev [10]. In the periodic case with s ≥ 2, nonuniform dependence of the datato-solution map for DP is established following the method of travelling wave solutions developed in [44] for the CH equation. This result has been recently sharpened in [37], where nonuniform dependence on the initial data for DP is proven on both the circle and line for s > 3/2 using the method of approximate solutions in tandem with a twisted L 2 -norm that is conserved by the DP equation. This method of approximate solutions has also been adapted to a homogeneous setting in Holliman [47] to prove nonuniform dependence of the flow map for the Hunter-Saxton equation on the circle with s > 3/2, and to the higher dimensional setting in [43] to prove nonuniform dependence for the Euler equations.
Motivated by the references cited above, we stablish the local well-posedness for the strong solutions to the Cauchy problem (1)- (2) in H s , s > 3 2 and that the datato-solution map is continuous but not uniformly continuous. Furthermore, we show that the solution map is Hölder continuous in T if it is equipped with an H r -norm, 0 ≤ r < s.
Our main results in this paper are stated as follows. , which depends continuously on the initial data u 0 . Furthermore, we have the estimate where c s > 0 is a constant depending on s.
The well-posedness of the solutions for the i.v.p. (1)-(2) is obtained by Kato's semigroup approach [63] and by classical Friedrichs's regularization method [57], respectively. However, we have not been able to find the estimates (6) in the literatures. Here, we shall give a proof of local well-posedness of the solutions for the i.v.p. (1)-(2) in the sense of Hadamard, including estimates (6) by the different which are key ingredients in our work. Our proof of Theorem 1.1 is based on a Galerkin-type approximation method, which for quasi-linear symmetric hyperbolic systems can be found in Taylor [61].
Using this result of Theorem 1.1 and the method of approximate solutions we prove the following nonuniform dependence result.  The proof of Theorem 1.2 is based on the method of approximate solutions and well-posedness estimates for the solution and its lifespan, which is motivated by the works of the [37,38,39,40,42,28,34,41,47,43]. We will choose approximate solutions to the the generalized Camassa-Holm equation such that the size of the difference between approximate and actual solutions with identical initial data is negligible. Hence, to understand the degree of dependence, it will suffice to focus on the behavior of the approximate solutions (which will be simple in form), rather than on the behavior of the actual solutions. In order for the method to go through, we will need well-posedness estimates for the size of the actual solutions to the generalized Camassa-Holm equation, as well a lower bound for their lifespan. This will permit us to obtain an upper bound for the size of the difference of approximate and actual solutions. Theorem 1.1 and 1.2 show that the generalized Camassa-Holm equation is wellposed in Sobolev spaces H s on both the line and the circle for s > 3/2 and its datato-solution map is continuous but not uniformly continuous. Here, we show that the solution map for the generalized Camassa-Holm equation is Hölder continuous in H r -topology for all 0 ≤ r < s. More precisely, we prove the following result. Theorem 1.3. If s > 3/2 and 0 ≤ r < s, then the data-to-solution map for the Cauchy problem (1)-(2), on both the line and the circle, is Hölder continuous on the space Hs equipped with the H r norm. More precisely, for initial data where the exponent α is given by and the regions A 1 , A 2 and A 3 in the sr−plane are defined by The lifespan T and the constant c depend on s, r and ρ. The proof of Theorem 1.3 follows the work of [35,9,43]. The rest of this paper is organized as follows. In Section 2, using the method of approximate solutions, we show that the solution map is not uniformly continuous. Section 3 is devoted to the study of the local wellposedness result for s > 3/2 with an accompanying solution size estimate. Finally, in Section 4, we prove Theorem 1.3.

2.
Non-uniform dependence. Before giving the proof of the non-uniform dependence for modified Hunter-Saxton equation, we introduce some definition and known results for later proof.
We define the relation and x ≈ y ⇐⇒ x y and y y.
The Fourier transform f of the function f is taken to be Therefore, the inverse relation is given by the Fourier series For any s ∈ R we take the operator D s to be defined by The Sobolev space of exponent s is defined by The homogeneous Sobolev space of exponent s to be the subspaceḢ s ⊂ H s defined byḢ Remark. As we will be working exclusively in the periodic case, we will often use the notation H s = H s (T),Ḣ s =Ḣ s (T). Keeping in mind that when we define the inverse of ∂ x we will need the resulting function to be periodic, we define the operator as follows For f ∈ H s we set It defines a continuous linear map Also, it is both a left and right inverse of the operator ∂ x : H s+1 →Ḣ s . Moreover, it satisfies the relations Since for any r ≥ 0 and g ∈Ḣ r we have g H r ≤ 2 r/2 g Ḣr−1 , we also obtain the following inequality

YONGSHENG MI, CHUNLAI MU AND PAN ZHENG
Next, we will prove Theorem 1.2 for Sobolev exponents s > 3/2. The basis of our proof rests upon finding two sequences of solutions, {u n }, {v n } in C([0, T ] : H s (T)), to (1)-(2) that share a common lifespan and satisfy

Approximate solutions
For any n, a positive integer, we define the approximate solution u ω,n = u ω,n (x, t) as where Now we compute the error of the approximate solutions (19). Note that and Thus, the error E of the approximate solutions (19) is We will now proceed to estimate the size of the error E in H σ . For the remainder of this proof, we will be taking σ ∈ ( 1 2 , 1) with the additional condition σ + 1 < s.
Proof. Applying the triangle inequality, we have Using the formulas the Algebra Property, ω k = ω and |ω| ≤ 1, we have Using inequality (16), we have

Estimating the difference between approximate and actual solutions
Now that we have these initial estimates for the approximate solutions, we will proceed to construct our family of actual solutions. Let u ω,n (x, t) be the actual u ω,n (x, 0) = ωn − 1 k + n −s cos nx.
Notice that (26) implies that the initial data u ω,n (x, 0) ∈ H s for all s ≥ 0, since for n sufficiently large. Hence, by Theorem 1.1, there is a T > 0 such that for n > 1, the Cauchy problem (31) To estimate the difference between the actual solutions and the approximate solutions, we define v = u ω,n − u ω,n which satisfies the following i.v.p.
We will now show that the H s norm of the difference v decays to zero as n goes to infinity.
Proof. Apply the operator D σ to both sides of (34), multiply by D σ v, and integrate over the torus to get We now estimate the three integrals in the right-hand side of (36).

ON THE CAUCHY PROBLEM OF THE HUNTER-SAXTON EQUATION 2055
Estimate of the first integral. Applying the Cauchy-Schwarz inequality, we have Estimate of the second integral. we begin by rewriting this term by commuting v with D σ ∂ x to arrive at The first integral can be handled by the following Calderon-Coifman-Meyer type commutator estimate that can be found in [40].
provided that ρ > 3 2 and σ + 1 ≤ ρ. Applying Lemma 2.3 with σ + 1 ≥ 0, ρ = s > 3 2 and σ + 1 ≤ s tells us Integrating by parts and using the Sobolev Theorem, we have We obtain Estimate of the third integral. For the third term, we observe that after applying the Cauchy-Schwarz inequality , the quantity w 2 v + w 3 ∂ x v will be in the H σ−1 space, which precludes use of the algebra property. To overcome this obstacle, we will apply the following multiplier estimate also found in [40].

YONGSHENG MI, CHUNLAI MU AND PAN ZHENG
We shall show that w 1 H s 1, w 2 H σ−1 1 and w 3 H σ 1. From (26), we have The actual solutions u ω,n (x, t) are bounded by the solution size estimate in Theorem 1.1 Thus, we have Similarly, we have We can refine (45) by using (47)-(49) to write Solving (49) and using the error estimate (35), we arrive at the desired estimate (35) The examination of the H s+1 norm of the difference v is summarized in the next lemma.
Proof. From the definition of v and the triangle inequality, we have v(t) H r = u ω,n − u ω,n H r ≤ u ω,n H r + u ω,n H r .
The bound on the H r -norm of the approximate solution is achieved by a straightforward calculation. We have From our hypothesis, we have r > 3 2 . As our initial data is smooth and therefore in H r , we may appeal to estimate (6) to bound the H r norm of the actual solution. Now using the fact that u ω,n(0) = u ω,n (n), we have u ω,n H r n r−s .
Combining (53) and (54) yields the desired bound. Next, using interpolation between σ and s + 1 we show that the H s -norm of v(t) decays. In fact, using (35) and (51) where have the exponent ε > 0 given by Proof of non-uniform dependence. Here we will prove Theorem 1.2 for Sobolev exponents s > 3 2 . The basis of our proof rests upon finding two sequences of

ON THE CAUCHY PROBLEM OF THE HUNTER-SAXTON EQUATION 2057
solutions to the MHS i.v.p. (1)-(2) that share a common lifespan and satisfy three conditions. For k-odd, we take the sequence of solutions with ω = ±1 and for k-even the sequence of solutions with ω = 0, 1. The three conditions they satisfy are as follows (1) Property (1), for k-even or odd follows from the solution size estimate in Theorem 1.1. We have Property (2), for k-odd, follows from the definition of our approximate solutions (19). We have Similarly, for k-even we have For property (3), we consider k-odd first. Using the reverse triangle inequality we get from which we obtain that Since, by the trigonometric identity cos(α − β) − cos(α + β) = 2 sin α sin β, we have inequality (58) gives which completes the proof of property (3) in the case that k is odd.
We now consider k-even. Similarly, using the reverse triangle inequality, the definition of approximate solutions, and the fact that the difference between solutions and approximate solutions decays, we get where now Therefore (60) gives where for each ∈ (0, 1], the operator J is the Friedrichs mollifier. We fix a Schwartz function j ∈ S(R) that satisfies 0 ≤ j(ξ) ≤ 1 for every ξ ∈ R and j(ξ) for ξ ∈ [−1.1] This allows us to define the periodic functions j as Then J is defined by This construction of j results in two lemmas that will prove repeatedly useful throughout the paper.
Lemma 3.1. [47] Let s > 0 and J be defined as in (65). Then for any f ∈ H s , we have J f → f in H s .

Lemma 3.2. [47]
Let r < s ,the map I − J : H s → H r and J be defined as in (65). Then for any f ∈ H s , we have J f → f in H s . satisfies the operator norm estimate Hence, for each ∈ (0, 1], (62)-(63) has a unique solution u with lifespan T > 0. Our strategy is now to demonstrate that the Cauchy problem (62)-(63) satisfies the hypotheses of the Fundamental ODE Theorem. We will therefore obtain a unique solution u (·, t) ∈ H s , |t| < T , for some T > 0. This idea is summarized in the following lemma. Proof. The map F : H s → H s is well-defined. The only remaining hypothesis of the Fundamental ODE Theorem, we need to show is satisfied is that F is a continuously differentiable map. In this case, derivative of F can be explicitly calculated at each u 0 ∈ H s as Hence, for each ∈ (0, 1], (62)-(63) has a unique solution u with lifespan T > 0.
3.1. Energy estimate and lifespan of solution u For each , there is a solution u to the mollified MHS (62)- (63). The lifespan of each of these solutions has a lower bound T . In this subsection, we shall demonstrate that there is actually a lower bound T > 0 that does not depend upon . This estimate is crucial in our proofs.
To show the existence of T , we shall derive an energy estimate for the u . Applying the operator D s to both sides of (62), multiplying by D s u , and integrating over the torus yields the H s -energy of u To bound the energy , we will need the following Kato-Ponce [53] commutator estimate.
We now rewrite the first term of (67) by first commuting the exterior J and then commuting the operator D s with (J u ) k arriving at Setting v = J u ,we can bound the first term of (69) by first using the Cauchy-Schwarz inequality and then applying lemma 3.4 to arrive at For the second term of (69), we have The second term on the right hand side of (67) is bounded by first applying the Cauchy-Schwarz inequality and then using of the operator norm of ∂ −1 x and the 2060 YONGSHENG MI, CHUNLAI MU AND PAN ZHENG algebra property of H s . Here we have Using the fact that and combining these results we conclude that u satisfies the differential inequality This together with Sobolev's lemma gives Set y = u (t) 2 H s , the differential inequality (75) Integrating (76) from 0 to t gives Solving for y(t) and substituting back in y = u (t) 2 H s gives us the inequality Setting T = 1 2csk u0 k H s , we see from (78) that the solution u exists for 0 ≤ t ≤ T and satisfies a solution size bound Therefore we see that T is a lower bound for the lifespan of u independent of .

Existence
We may thus apply Alaoglu's theorem deduce that {u } is be precompact in B (0, 2 u 0 H s ) ⊂ L ∞ ([0, T ]; H s ) with respect to the weak * topology. Therefore we may extract a subsequence {u υ } that converges to an element u ∈ B(0, 2 u 0 H s ) weakly * . Given this construction, u will satisfy our solution size estimate (6).
Strong convergence in C([0, T ]; H s−1 ). We will prove that the family {u } ∈(0,1] satisfies the hypotheses of Ascoli's Theorem. We begin with the equicontinuity condition. For t 1 , t 2 ∈ [0, T ], we have Using the fact that u satisfies (62)- (63), we see that the H s−1 norm of ∂ t u can be bounded independently of as follows Next, we observe that for each t ∈ [0, T ] the set U (t) = {u } ∈(0,1] is bounded in H s . Since T is a compact manifold, the inclusion mapping i : H s → H s−1 is a compact operator, and therefore we may deduce that U (t) is a precompact set in H s−1 . As the two hypotheses of Ascoli's Theorem have been satisfied, we have a subsequence {u υ } that converges in ([0, T ]; H s−1 ). By uniqueness of limits, this subsequence must converge to u.
Strong convergence in C([0, T ]; H s−σ ) for σ ∈ (0, 1]. As in the previous case, we will prove that the family {u } satisfies the hypotheses of Ascoli's Theorem. To establish the equicontinuity condition, we see that for t 1 , t 2 ∈ [0, T ] that we have Our objective therefore is to bound u C σ ([0,T ];H s−σ ) independently of . We begin with the definition of this norm as The first term on the right hand side of (85) is bounded by application of (79), giving us For the second term, we begin with two elementary inequalties. First, as σ ∈ (0, 1) we have 1

YONGSHENG MI, CHUNLAI MU AND PAN ZHENG
Using this inequality, we may further deduce that We therefore can therefore bound this term by Combining (86) and (89), we have the independent bound The precompactness condition is established in exactly the same fashion as the previous case as the inclusion mapping of H s into H s−σ is a compact operator. As the two hypotheses of Ascoli have been satisfied, we may extract a subsequence that converges to u in C([0, T ]; H s−σ ).
Strong convergence in C([0, T ]; C 1 (T)). We will now σ ∈ (0, 1) so that s − σ > 3 2 . The Sobolev lemma then tells us that H s−σ embeds continuously into C 1 (T), which therefore implies that u → u in C([0, T ]; C 1 (T)). We will next prove that Strong convergence of ∂ t u in C([0, T ]; C 1 (T)). From (62) we have As we have already established that u → u in C([0, T ]; C 1 (T)) it follows that ∂ x u → ∂ x u. Using the fact that this space is an algebra, then continuity of ∂ −1 x implies the convergence of the nonlocal term Next, we observe that J u → u in C([0, T ]; C(T)) as For the first term of this sum, choose r with 1 2 < r < s. Then applying Lemma 3.2, we see that for t ∈ [0, T ] Estimating the second term immediately follows from the fact that we have established u →u in C([0, T ]; C(T)) We then examine ∂ x as above, and similarly conclude Proof. Let u 0 ∈ H s and let u and w be two solutions to the Cauchy problem(1.1)-(1.2) with u(x, 0) = u 0 (x) = w(x, 0). Then the difference v = u − w satisfies the following Cauchy problem where w 1 = (u) k + (u) k−1 w + · · · + u (w) k−1 + (w) k , Let 1 2 < σ < min{s − 1, 1}. The H σ -energy estimate is then given by 1 2 To bound (111), we commute D σ ∂ x and v, which results in two integrals. The commutator integral is estimated by applying the Cauchy-Schwarz inequality followed by Lemma 2.3 and the solution size estimate (6). The second integral is bounded using integration by parts, the Sobolev Embedding Theorem and the solution size bound (6). The non-local term is estimated by the Cauchy-Schwarz inequality and the continuity of ∂ −1 x . The resulting energy estimate 1 2 which we solve to find the inequality We recall that v = u − w where u and w are both solutions to the MHS i.v.p.

Continuity of the data-to-solution map
Our approach is to use energy estimates. To avoid some of the difficulties of estimating the term involving u k ∂ x u, we will use the J convolution operator to smooth out the initial data. Let ∈ (0, 1]. We take u to be the solution to the MHS i.v.p. with smoothed initial data J u 0 = j * u 0 . Similarly, let u n be the solution with initial data J u 0,n . Applying the triangle inequality, we arrive at We will prove that for any η > 0, there exists an N such that for all n > N , each of these terms can be bounded by η 3 for suitable choices of and N . We note that the choice of a sufficiently small will be independent of N and will only depend on η; whereas, the choice of N will depend on both η and . However, after has been chosen, N can be chosen so as to force each of the three terms to be small. Estimation of u − u n C([0,T ];H s ) . We can bound this term directly using an H senergy estimate. Let v = u − u n . Then v satisfies the following Cauchy problem where w 1 = (u ) k + (u ) k−1 u n + · · · + u (u n ) k−1 + (u n ) k ,