The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

The CH equation was derived independently by two groups of researcheres, Fuchssteiner and Fokas [2] and Camassa and Holm [3]. Fuchssteiner and Fokas derived (4) in studying completely integrable generalizations of the KdV equation with bi-Haniltonian structures, while Camassa and Holm proposed (4) to describe the unidirectional propagation of shallow water waves over a flat bottom. Many physicists and mathematicians have paid much attention to the CH equation and a series of achievements had been made. For example, consider the following authors. Constantin [4] and Misiolek [5], investigated the Cauchy problem. Constantin, Escher, and McKean, and so forth [6– 8] studied the wave-breaking and so on. Xin and Zhang [9] proved that there are global weak solutions for any data in

proved the existence of the global conservative and dissipative solutions.The CH equation arises also as an equation of the geodesic flow for the  1 metrics on the Bott-Virasoro group [14,15].Ivanov [16] extended the CH hierarchy and obtained their conserved quantities.Constantin and Ivanov [17] studied an integrable two-component CH shallow water system.
The DP equation can be regarded as a model for nonlinear shallow water dynamics derived by Degasperis and Procesi [18] in 1999.In 2003, the DP equation was derived by Dullin et al. [19] as a shallow water approximation to the Euler equation.The Cauchy problem for the DP equation has been studied extensively.The local and global wellposedness for the strong solutions, the global existence of weak solution, the blow-up phenomena, and nonuniformly continuous dependence on the initial data can be seen in [20,21] and the references therein.
In 2003, Holm and Staley [22] studied the exchange of stability in the dynamics of solitary wave solutions for -equation.The well-posedness, blow-up phenomena, and global solutions for the -equation can be found in [23,24] and the additional references therein.
It is showed that Novikov equation possesses a bi-Hamilton structure and an infinite sequence of conserved quantities and admits exact peaked solutions (, ) = ±√e −|−| with  > 0 (see [25]), as well as the explicit formulas for multipeakon solutions (see [25,26]).By using the Littlewood-Paley decomposition and Kato's theory, the Novikov equation's well-posedness was studied in Besov spaces   , (R) and in the Sobolev space   (R) (see [27,28]).Wu and Yin [29] established some results on the existence and uniqueness of global weak solutions to the Novikov equation.Jiang and Ni [30] obtained some results about blow-up phenomena of the strong solution to the Cauchy problem for the Novikov equation.For the periodic case, Tiglay [31] investigated the Cauchy problem for the periodic Novikov equation and proved that for  > 5/2 the periodic Novikov equation is local well-posedness in   (T).The range of local well-posedness for the periodic Novikov equation was extended to the  > 3/2 in [32].When  < 3/2, Grayshan [33] proved that the properties of the solution map for the Novikov equation are not uniformly continuous in Sobolev spaces   .
The issue of continuity of the solution map has been the subject of many papers.For the Burgers equation, Kato [34] showed that the solution map  0  →  is not Hölder continuous regardless of the Hölder exponent.However, for certain general quasilinear hyperbolic systems, Kato obtained uniform continuity of the data to solution map for initial data in Sobolev spaces with integer index (measured in a weaker Sobolev norm).Tao [35] obtained Lipschitz continuity of the solution map for the Benjamin-Ono equation for  1 (R) initial data measured in  2 .Herr et al. and so forth [36] have also obtained Lipschitz continuity in a weaker topology for the Benjamin-Ono with generalized dispersion.Davidson [37] studied the continuity of the solution map for the generalized reduced Ostrovsky equation.Karapetyan [38] proved the Hölder continuity of the solution map for the hyperelastic rod equation.For the continuity of solution map for some CH type equation and incompressible Euler equations in Besov spaces, we refer to [39][40][41].These works lead to a natural question, whether a result similar to these holds for (1) when ,  satisfy some assumptions.
Motivated by the results mentioned above, this paper deals with the problem (1)- (2).The aim of this paper is to prove local well-posedness of strong solutions in Besov spaces, Hölder continuity of the solution map in   , equipped with a weak topology.
We formulate the periodic boundary value problem ( 1)-(2) in Besov space as Now we introduce some notations and make some assumptions to () and ().
Notations.In this paper we adopt the following notations.
The notation ≲ denotes the estimates that hold up to some "harmless" constant which may change from line to line but whose meaning is clear from the context.D(T) is the space of all infinitely differentiable functions on T and D  (T) is its dual space (the details on the periodic distributions can be found in, e.g., [42]).Assuming that all function spaces are over T, hence we drop T in our notations of function spaces if there is no ambiguity.
(2) When  >  0 , the solution map  0  →  defined by the problem (8)  , ), where  = ().More precisely, let  0 , V 0 ∈ (0, ) ⊂   , and , V be the solutions corresponding to the initial data  0 , V 0 , respectively.Then we have where the exponent  satisfies Remark 2. As mentioned before, we can view (1) as a generalized Burgers-type equation: with a nonlocal perturbation ().To obtain the wellposedness, we need to assume some smooth properties on () and ().Condition (A1) implies that, for We note that the issue of nonuniform dependence on initial data has been the subject of many papers (see, e.g., [32,43]).In this paper, it is to be regretted that we can not find a feasible method to study the uniform continuity of the solution map  0  →  defined by problem (1)-(2) for general () and ().But when () =  and () =  2 , the solution map  0  →  defined by problem (1)-( 2 We now conclude this introduction by outlining the rest of the paper.In Section 2, we give some preliminaries.In Section 3, we prove Theorem 1.In Section 4, we demonstrate Proposition 3.

Preliminaries
In this section, we shall recall some basic facts on the Littlewood-Paley theory, Besov spaces, and the transport equations theory that will be used in this paper.We refer to [44][45][46] for the elementary properties of them.
The following lemma summarizes some useful properties of   , .

Proof of Theorem 1
The proof of Theorem 1 includes the following several steps.

Approximate Solution.
Starting from  1 = 0 and by induction, we define a sequence of smooth functions {  },  ∈ N by solving the following transport equation iteratively: Since (), () satisfy the continuous assumption and all the data belong to  ∞ , , from Lemma 8 and by induction, we see that, for all  ≥ 1, the above equation has a global solution  +1 belonging to (R + ,  ∞ , ).

Existence of the
where  =  −   .From ( 62) and ( 63), we see that {  } converges to  in (;    , ) for all   < ; this enables us to deduce that  indeed solves (1) in the sense of distributions.

Continuity of the Solution Map. The continuity with respect to the initial data in
or in can be obtained directly by Lemma 10.We now prove that the continuity holds true up to index  >  0 ≥ 1+1/,  ̸ = 2+1/.We state the following lemma first.
It is obvious that   solves the transport equation: We first consider V 0 ∈   , and  ∈  1 (0, ;   , ).Note that  < ∞; by Lemma 7, we obtain that, for  ∈ N, V  ∈ ([0, ];   , ) and Since  ̸ = 2 + 1/, for  ∈ N, Lemma 7 tells us that Since  > 1 + 1/, using (A1) yields that Therefore, we obtain the following estimate by gathering the above inequalities: which yields the desired result of convergence.For the general case V 0 ∈  −1 , and  ∈  1 (0, ;  −1 , ), we have where V   satisfies the "cut-off " equation: For V  − V   , we see that V  − V   solves the equation: . (76) Similarly, we can obtain Inserting ( 75 ∈  1 (0, ).Using the Lebesgue dominated convergence theorem yields that the first two terms of the righthand side of the above estimate may be arbitrary small for  large enough.Take  large enough; then fix .Let  tend to infinity so that the last term of right-hand side tends to zero.We complete the proof of Lemma 11.