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J. Evol. Equ. 18 (2018), 1147–1171 © 2018 The Author(s). This article is an open access publication Journal of Evolution 1424-3199/18/031147-25, published online March 2, 2018 Equations https://doi.org/10.1007/s00028-018-0435-5 Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces Mats Ehrnström and Long Pei Abstract. For both localized and periodic initial data, we prove local existence in classical energy space s 3 H , s > , for a class of dispersive equations u +(n(u)) +Lu = 0 with nonlinearities of mild regularity. t x x Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and s+2 n(·) in the local Sobolev space H (R). In particular, the results include non-smooth and exponentially loc growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on R to the periodic setting by using the difference–derivative characterization of Besov spaces. 1. Introduction We consider nonlinear dispersive equations of the form u + (n(u)) + Lu = 0, (1.1) t x x where n denotes the nonlinearity and the linear (dispersive) operator L is defined as a Fourier multiplier operator by F (Lf )(ξ ) = m(ξ )F ( f )(ξ ), (1.2) for some real and measurable function m. A large class of equations, including the Korteweg–de Vries (KdV) [6] and Benjamin–Ono (BO) equations [14], are covered by (1.1). But our main inspiration comes from [10], in which the existence of soli- tary waves was established for a class of nonlocal equations of Whitham type (1.1), and energetic stability of solutions was obtained based on an a priori well-posedness assumption. The Whitham equation itself corresponds to the nonlinearity u and non- local dispersive operator L with Fourier symbol tanh(ξ )/ξ, and is one of several equations of the form (1.1) arising in the theory of water waves [22]. Equations of the form (1.1) are in general not completely integrable, and one has to apply contraction or energy methods to obtain estimates for both the linear and Mathematics Subject Classification: 47J35 (primary); 35Q53, 45J05, 76B15 Keywords: Local well-posedness, Dispersive equations, Composition theorems, Besov spaces. Both authors acknowledge the support by Grant No. 231668 by the Research Council of Norway. M.E. additionally acknowledges the support by Grant No. 250070 by the same source. 1148 M. Ehrnström and L. Pei J. Evol. Equ. nonlinear terms to prove existence of solutions. Our linearity will be skew-adjoint (the symbol m(ξ ) will be real and even), but the only additional assumption is that m is of moderate growth, that is, |m(ξ )| (1 +|ξ |) , for some l ∈ R and all ξ ∈ R, which is to guarantee that the domain of L is dense in L . Note that this class of symbols is very large, covering both homogeneous and inhomogeneous symbols. For the same reason scaling argument cannot be applied, and the difficulties increase in determining the critical energy spaces for (1.1) and in acquiring the decay of solutions over time needed for the global existence of solutions. Moreover, if l above is sufficiently negative then the dispersion of L is very weak, and global well-posedness of (1.1) fails in classical energy spaces. In fact, the Whitham equation as a typical representative of (1.1) is locally well-posed in Sobolev spaces H , s > 3/2, with localized or periodic initial data [9,20], but exhibits finite-time blow-up (wave-breaking) for some sufficiently smooth initial data [7,13,23] so that global well-posedness in H (R), s > 3/2, is not possible. This kind of break-up phenomena cannot be observed in equations with strong dispersion like KdV and BO, which are globally well-posed in H (R) for all s ≥ 1 (see [6] and [27], respectively). Our main concern, however, is the nonlinearity n(·). The most well-studied non- linearities are pure power type nonlinearities u , p ∈ Z .For afixed p, such non- linearities have moderate growth rate far away from the origin, and it is possible to adjust p so that solutions will exist globally provided that dispersion is not too weak [5]. Otherwise, if we fix the dispersion but allow the nonlinearity to grow fast enough, the solutions may blow up within finite time [2]. Another important feature of pure power nonlinearities is that they define smooth, regularity-preserving maps on Sobolev spaces H , s > 1/2, in one dimension so that it is easy to obtain a contraction mapping from the solution operator based on either Duhamel’s principle or classical energy estimates. These features, however, fail to hold for general nonlinearities. The s+2 nonlinearity in our consideration shall belong to the local Sobolev space H (R), loc s > 3/2. For such nonlinearities and dispersive operators L as mentioned above, we establish local well-posedness for data in H , s > 3/2. Due to the locality of the space H (R), our result shows that dispersive equations of type (1.1) are locally well-posed loc in high-regularity spaces even for nonlinearities n of arbitrarily fast growth, and for dispersive operators L with arbitrarily weak or strong dispersion (as long as the dis- persion is not extreme in the sense that L ∈ S (R), the space of distributions on s+2 Schwartz space). It is a point in our investigation that the nonlinearity n ∈ H (R) is loc p p−1 not necessarily smooth and covers standard nonlinearities of the form u and |u| u as well as others. Naturally, the regularity of the nonlinearity will affect the energy space, as shown in Theorem 2.1. s+2 The local space H (R) first comes into play in obtaining the required commutator loc estimate for an operator-involving nonlinearity, needed in the semigroup theory. The key for such a commutator estimate in our case are the mapping properties of the Vol. 18 (2018) Classical well-posedness in dispersive equations 1149 nonlinearity n over the energy spaces in consideration, which is guaranteed by a composition theorem [3] when the initial data are in Sobolev spaces on the line. Namely, a composition operator T is said to act on a function space X if T (g) = f ◦ g ⊂ X, for all g ∈ X. The composition theorem in [3] holds that a function f acts on Besov s s spaces B (R),1 < p < ∞,0 < q ≤∞, s > 1+1/p,if, and only if, f ∈ B (R) pq pq,loc and f (0) = 0. That composition theorem, when applied to a non-smooth nonlinearity n, guarantees a contraction mapping in the desired solution spaces. We mention here that the Cauchy problem (1.1) with a non-smooth nonlinearity is also considered in [5], in which the author establishes global well-posedness in H (R), σ = max{α, 3/2+} with |α|≥ 1, > 0, by nonlinear semigroup theory when the dispersion is not too weak (m(ξ ) =|ξ | ). The nonlinearity in their consideration belongs to the Hölder α+1 s space C (R). Similar well-posedness results in H , s > 1/2, for generalized KdV equations can be found in [26]. In the periodic setting, a composition theorem like the one in [3] is not available in the literature. However, we prove in Theorem 5.3 that f acts on the periodic Besov s 1 s space B (T),1 < p < ∞,1 < q ≤∞ and s > 1+ ,if, and only if, f ∈ B (R), pq p pq,loc where T denotes the one-dimensional torus (the requirement that f (0) = 0 appearing in [3] is not necessary due to the periodicity, else the results are comparable). Our proof relies on the composition theorem for non-periodic Besov spaces and what can be called a localizing property of periodic Besov spaces (see Lemma 5.2): Smooth but compactly supported extensions of periodic functions from a n-dimensional torus T to the whole space R will be controlled by the latter periodic function norms. A feature of our proof is that we work directly with the difference–derivative characterization of Besov spaces, not the Littlewood–Paley decompositions (used for example in [3]). In this case, the main difficulty arises at integer regularity s ∈ N, where the estimates are particularly cumbersome. It could be mentioned that the composition theorem on R mentioned earlier has only been proved in one dimension, even though it is conjectured to hold in higher dimensions too (cf. [3]). Finally, we mention that a related paper for the periodic setting is [11], in which KdV-type equations are considered, and which is based on the work [4] of Bourgain. The outline is as follows. In Sect. 2 we state our assumptions and the main result. −tL∂ In Sect. 3 we reformulate (1.1) in terms of the semigroup e generated by L. Following [8] and [17], we then introduce Kato’s method from [16]. Thesamesection concludes with the composition theorem for operators on Besov spaces on the line. −tL∂ In Sect. 4, we give a detailed analysis of the generator of semigroup e and the proof of our local well-posedness result on the line. Attention needs to be paid in the case of a general dispersive symbol and a nonlinearity of mild regularity. In particular, one finds a proper domain and an equivalent definition for the dispersive operator L∂ on which it is closed and generates a continuous semigroup. Finally, in Sect. 5, we prove the above-mentioned composition theorem for Besov spaces on the torus T 1150 M. Ehrnström and L. Pei J. Evol. Equ. (see Theorem 5.3), this being the key for the well-posedness for periodic initial data. The proof of Theorem 5.3 is based on the localizing property detailed in Lemma 5.2. With all this in place, local well-posedness on H (T) is straightforward, as it can be acquired analogously to the non-periodic case, and we only state the idea and point out some key points. 2. Assumptions and main results In (1.1), the Fourier transform F ( f ) of a function f is defined by the formula −i ξ x F ( f )(ξ ) = √ e f (x ) dx , 2π R extended by duality from the Schwartz space of rapidly decaying smooth functions S(R) to the space of tempered distributions S (R). The notation f will be used inter- changeably with F ( f ). For all s ∈ R, we denote by H (R) the Sobolev space of tempered distributions whose Fourier transform satisfies 2 s 2 (1 +|ξ | ) |F ( f )(ξ )| dξ< ∞, and equip it with the induced inner product (the exact normalization of the norm will not be important to us). Sobolev spaces are naturally extended to the Besov spaces s n B (R ), s > 0, 0 < p, q ≤∞; however, we will postpone the exact definition pq for Besov spaces and introduce them in the periodic setting, since the well-known s n s n identity H (R ) = B (R ), s > 0, is the only property of Besov spaces that we will use for the well-posedness for localized initial data. Finally, we use C to denote the C (smooth) functions with compact support. Note that although the solutions of interest in this paper are all real valued, and although the operator L defined in (1.1) maps real data to real data, the Fourier transform is naturally defined in complex- valued function spaces. Hence, the function spaces used in this investigation should in general be understood as consisting of complex-valued functions. For convenience, we shall sometimes omit the domain in the notation for function spaces, and we use the notation A B if there exists a positive constant c such that A ≤ cB. Our assumptions for (1.1) are stated in Assumption A. Note that we have no regu- larity assumptions for m except for it being measurable, but the below conditions will guarantee that m ∈ L . The growth condition on m is to guarantee the density ∞,loc of the domain of the linear operator in L (R). As what concerns the nonlinearity, the assumption on it is completely local. In particular, n(x ) may grow arbitrarily fast as |x|→∞. ASSUMPTION A. (A1) The operator L is a symmetric Fourier multiplier, that is, F (Lf )(ξ ) = m(ξ ) f (ξ ) for some real and even measurable function m : R → R. Vol. 18 (2018) Classical well-posedness in dispersive equations 1151 (A2) The symbol m is temperate, that is, |m(ξ )| (1 +|ξ |) , for some constant l ∈ R and for all ξ ∈ R. 3 s+2 (A3) There exists s > such that the nonlinearity n belongs to H (R). 2 loc Denote by T the one-dimensional torus of circumference 2π. We now state our main theorem, yielding local existence for a fairly large class of equations. Note in the s−l−1 following that when l > s − 1, the Sobolev space H becomes strictly larger than L , although our interest mainly arises from the case l < 0. THEOREM 2.1. Let F ∈{R, T}. Under Assumption A and given u ∈ H (F) for 3 s s > , there is a maximal T > 0 and a unique solution u to (1.1) in C ([0, T ); H (F))∩ 1 s−max{1,l+1} C ([0, T ); H (F)). The map u → u(·, u ) is continuous between the 0 0 above function spaces. k,α REMARK 2.2. Note that the Hölder spaces C (R) are continuously embedded into the local Sobolev spaces H (R) for k = s when α = 0 and for k + α> s loc when α ∈ (0, 1). Hence, Hölder-continuous functions are concrete examples of the nonlinearities considered in this paper. As described in the introduction, our proof of Theorem 2.1 relies on a combination of Kato’s classical energy method, and recent composition theorems for Besov spaces (see Theorems 3.3 and 5.3). 3. General preliminaries Consider the transformation −tL∂ u(t ) = e v(t ), (3.1) −tL∂ where a detailed analysis of the semigroup e , t ≥ 0, is given in Sect. 4.1. Substitution of (3.1)into(1.1) yields the quasi-linear equation dv + A(t,v)v =0(3.2) dt for the new unknown v, where tL∂ −tL∂ −tL∂ x x x A(t, y) = e [n (e y)∂ ]e , (3.3) −tL∂ + and n (e y) acts by pointwise multiplication. Given a function y ∈ H the −tL∂ operator u → n (e y)u is a bounded linear operator L → L . Therefore, A(t, y) 2 2 is a well-defined bounded linear operator H → L . For the purpose of applying Theorem 3.1 below, we fix an arbitrary value of s > and consider spaces X = L (R) and Y = H (R), (3.4) 2 1152 M. Ehrnström and L. Pei J. Evol. Equ. s 2 between which = (1 − ∂ ) defines a topological isomorphism (isometry, under the appropriate norms). Let furthermore B be the open ball of radius R in H , for an arbitrary but fixed radius R > 0. To state Theorem 3.1,let T be an operator on a Hilbert space H. We denote the space of all bounded linear operators on H by B(H ). Following [18], we call an operator T on a Hilbert space H accretive if Re T v, v ≥ 0 holds for all v ∈ dom(T ), and quasi-accretive if T + α is accretive for some α ∈ R. −1 If (T + λ) ∈ B(H ) with −1 −1 (T + λ) ≤ (Re λ) B(H ) for Re λ> 0, then T will be called m-accretive, and if T + α is m-accretive for some scalar α ∈ R it will be called quasi-m-accretive. THEOREM 3.1. [16] Let X, Y, B and be as above. Consider the quasi-linear Cauchy problem du + A(u)u = 0, t ≥ 0, u(0) = u , (3.5) dt and assume that (i) A(y) ∈ B(Y, X ) for y ∈ B , with (A(y) − A(z))w y − z w , y, z,w ∈ B , X X Y R and A(y) uniformly quasi-m-accretive on B . s −s (ii) A(y) = A(y) + B(y), where B(y) ∈ B(X ) is uniformly bounded on B , and (B(y) − B(z))w y − z w , y, z ∈ B ,w ∈ X. X Y X R Then, for any given u ∈ Y , there is a maximal T > 0 depending only on u and 0 0 Y a unique solution u to (3.5) such that u = u(·, u ) ∈ C ([0, T ); Y ) ∩ C ([0, T ); X ), where the map u → u(·, u ) is continuous Y → C ([0, T ); Y ) ∩ C ([0, T ); X ). 0 0 The continuity of the operators A and B in Theorem 3.1 can be reduced to a commutator estimate. This is where our composition theorem will play a role in order to control the nonlinearity n. To this aim, we introduce the concept of action of a composition operator. DEFINITION 3.2. (Action property) For a function f ,let T denote the composi- tion operator g → f ◦ g. The operator T is said to act on a function space W if for any g ∈ W one has T (g) ∈ W , that is, T W ⊂ W. f Vol. 18 (2018) Classical well-posedness in dispersive equations 1153 For some Besov spaces over R, the set of acting functions has been completely characterized in terms of a local space, as described in the following result. THEOREM 3.3. [3] Let 1 < p < ∞, 0 < q ≤∞ and s > 1 + (1/p).For a Borel measurable function f : R → R, the composition operator T acts on B (R) pq exactly when f (0) = 0 and f ∈ B (R). In that case, T is bounded. pq,loc It is obvious that for spaces of functions with decay the condition f (0) = 0 is nec- essary. It is clear, too, that f necessarily must have the same (local) regularity as prescribed by the space B , since otherwise a smoothened cutoff of the function pq x → x would be mapped out of the space by f . What is less obvious is that these two properties are actually equivalent to the action property, and in the one-dimensional case in fact to boundedness of f on B (R),see [3]. We record here also the following pq definition and consequence of Theorem 3.3. REMARK 3.4. Let φ ∈ C (R) be a smooth cutoff function such that 0 ≤ φ ≤ 1, φ(x ) =1for |x|≤ 1, (3.6) and supp(φ) ⊂[−2, 2].For a > 0, we denote by ϕ (x ) = φ −1 the a -dilation of φ. Then the following inequality is a consequence of the proof of Theorem 3.3 in [3]. For a =g , one has s−1− s s s f (g) ( f ϕ ) s−1 g (1 +g ) . (3.7) B (R) a B (R) B (R) pq B (R) pq pq pq We are now ready to move on to the existence result on R. 4. Localized initial data 4.1. The operator L∂ Most of the material in this subsection is standard, and we present it in condensed form. Often, though, in the literature details are only given in the case when the assumptions on L are much more restrictive and n is a pure power nonlinearity. For a classical paper on well-posedness of nonlinear dispersive equations, see, e.g., [1]. Here, we follow the route of [8] and start by defining the domain of ∂ L = L∂ in x x L (R) (from now on only L )by 2 2 dom(L∂ ) ={ f ∈ L : L∂ f ∈ L }. (4.1) x 2 x 2 LEMMA 4.1. Let S be the set of all f ∈ L for which there exists g ∈ L with 2 2 ·, g =−L∂ ·, f (4.2) L x L 2 2 Then dom(L∂ ) = S and L∂ =[ f → g]. x x 1154 M. Ehrnström and L. Pei J. Evol. Equ. Proof. Since L is symmetric, for any f ∈ dom(L∂ ) and any φ ∈ C (R),wehave φ, L∂ f =−L∂ φ, f . x L x L 2 2 Thus L∂ f ∈ L yields dom(L∂ ) ⊂ S. To see that L∂ f = g, note that x 2 x x φ, g =−L∂ φ, f =L∂ f,φ , L x L x L 2 2 2 for any f ∈ dom(L∂ ), bythe skew-symmetryof L∂ . Thus, if we knew that x x S ⊂ dom(L∂ ), we could conclude L∂ =[ f → g].For f in S, L∂ f, · = x x x − f (L∂ ·) dx is clearly a well-defined distribution. In view of (4.2), we have L∂ f − g,φ= 0, and therefore L∂ f = g in D (R). (4.3) Since g ∈ L we deduce that L∂ f ∈ L , which in turn implies f ∈ dom(L∂ ).This 2 x 2 x concludes the proof. We record the following properties of L∂ . LEMMA 4.2. L∂ is densely defined on L , closed, and skew-adjoint on S. x 2 Proof. The denseness of dom(L∂ ) in L follows by (A2) (see Assumption A). Simi- x 2 larly, closedness in L follows from that L is a symmetric Fourier multiplier operator, ∗ ∗ cf. (A1). Now, let (L∂ ) be the L -adjoint of L∂ . Then, for any g ∈ dom((L∂ ) ) x 2 x x and any φ∈C ⊂ dom(L∂ ),wehave φ, (L∂ ) g =L∂ φ, g , x L x L 2 2 which implies that g ∈ dom(−L∂ ) and (L∂ ) ⊂−L∂ . To prove the inverse x x x relation, we use that for any f ∈ dom(L∂ ) there is a sequence { f } of smooth x n functions such that L L 2 2 f → f and (L∂ ) f → (L∂ ) f, (4.4) n x n x as n →∞.Here f = f ∗ , where (x ) = n (nx ) and ∈ C is a mollifier n n n ∞ ∞ satisfying (x ) ≥ 0 and dx = 1. Since ∈ C ,wehave f ∈ C ∩ L and n n 2 clearly f → f in L as n →∞. Furthermore, n 2 F (L∂ f ) = i ξm(ξ )F ( f )F ( ) = F ( f )F (L∂ ) = F ( f ∗ (L∂ )), x n k x n x n (4.5) and because L∂ is a Fourier multiplier, we also have that f ∗ (L∂ ) = (L∂ f ) ∗ . (4.6) x n x n By (4.5) and (4.6), one has L∂ f − L∂ f =(L∂ f ) ∗ − L∂ f →0as n →∞, x n x L x n x L 2 2 Vol. 18 (2018) Classical well-posedness in dispersive equations 1155 which establishes the density of C ∩ L in the graph norm of L∂ on dom(L∂ ). 2 x x For each g ∈ dom(L∂ ), we can find {g }⊂ C ∩ L . Then, for any f ∈ dom(L∂ ) x n 2 x we have L∂ f, g = lim L∂ f, g =− lim f, L∂ g = f, −L∂ g . x L x n L x n L x L 2 2 2 2 n→∞ n→∞ Therefore, −L∂ ⊂ (L∂ ) , and the operator L∂ is skew-adjoint on S. x x x From Lemma 4.2 and Stone’s theorem, one then obtains the following standard result. −tL∂ s LEMMA 4.3. L∂ generates a unitary group {e } on H ,s ≥ 0, where −tL∂ it ξm(ξ ) F (e f ) = e F ( f ), f ∈ L . 4.2. Properties of the operators A and B We now study the operator A(t, y) for a fixed y ∈ B ⊂ H . All estimates to come are uniform with respect to such y. To prove that the operator A(t, y) is quasi- m-accretive, we establish that both A(t, y) and its adjoint are quasi-accretive. It then follows by [24, Corollary 4.4] that A(t, y) is quasi-m-accretive, as proved in the following lemma. LEMMA 4.4. For any fixed y ∈ B , the operator −A(t, y) is the generator of a C -semigroup on X, and A(t, y) is uniformly quasi-m-accretive on B . In particular, 0 R for all y ∈ B , one has the uniform estimate (A(t, y)w, w) −w , (4.7) for w ∈ C . Proof. With −tL∂ −tL∂ 1 x x dom(A(t, y)) ={u ∈ L : n (e y)e u ∈ H }, (4.8) A(t, y) is densely defined in L , and closed: Take {u }⊂ dom(A(t, y)) with u → 2 n n u ∈ L and A(t, y)u → v ∈ L . Abbreviate A = A(t, y). Then, for any φ ∈ C , 2 n 2 v, φ = lim Au ,φ L n L 2 2 n→∞ = lim u , A φ n L n→∞ =u, A φ ∗ ∗ =u,(A + A ∂ )φ x L 1 2 2 =A u,φ +A u,φ , 1 L 2 L 2 2 with tL∂ −tL∂ −tL∂ −tL∂ x x x x A =−e n (e y)(e y )e , 1 x 1156 M. Ehrnström and L. Pei J. Evol. Equ. tL∂ −tL∂ −tL∂ x x x A =−e n (e y)e , both self-adjoint and bounded on L . Therefore, (A u) = v − A u ∈ L , 2 1 2 u ∈ dom(A(t, y)), and A(t, y) is closed. In order to prove that A(t, y) is quasi-m- accretive, we define tL∂ −tL∂ −tL∂ tL∂ −tL∂ −tL∂ −tL∂ x x x x x x x Gu = (e n (e y)e u) − e n (e y)e y e u, x x tL∂ −tL∂ −tL∂ x x x G u =−(e n (e y)e u) , 0 x with dense domain tL∂ −tL∂ −tL∂ 1 x x x {u ∈ L : e n (e y)e u ∈ H }. ∞ ±t ∂ L The density of C in L then implies that A(t, y) = G. Recall that e is unitary r r 1 on H , for all r ∈ R, and that H → BC for r > . Hence, for any fixed y ∈ B , −tL∂ −tL∂ −tL∂ x x x ∂ (n (e y)) =n (e y)e y x L x L ∞ ∞ −tL∂ −tL∂ −tL∂ x x x ≤(n (e y) − n (0))e y +n (0)e y x L x L ∞ ∞ (4.9) −tL∂ (n (e y) − n (0)) y +y s−1 s−1 s−1 x x H H H s− 2 R + (1 + R) R , where we have applied Theorem 3.3 to n (·) − n (0). With (4.9) and using the skew- adjointness of ∂ , quasi-accretiveness of both G and G can be proved using integration x 0 by parts. (This is structurally equivalent to proving quasi-accretiveness of uφu for a well-behaved function φ.) We now show that G and G are closed and adjoints of each other. For closedness, this is analogous to the above proof of that A is closed. To see that G is the adjoint of G in L , consider v ∈ dom(G ) and φ ∈ C ⊂ dom(G). One then has 2 0 φG v dx = Gφv dx R R tL∂ −tL∂ −tL∂ x x x = (e n (e y)e φ) v dx tL∂ −tL∂ −tL∂ −tL∂ x x x x − e n (e y)e y e φv dx tL∂ −tL∂ −tL∂ x x x = (e n (e y)e φ )v dx tL∂ −tL∂ −tL∂ x x x = (e n (e y)e v)φ dx , ∞ ∗ which implies that v ∈ dom(G ). The density of C in L directly yields that G ⊂ 0 2 G . To obtain the opposite inclusion, note that just as in the proof of Lemma 4.2,for 0 Vol. 18 (2018) Classical well-posedness in dispersive equations 1157 any u ∈ dom(G ) there is a sequence of smooth functions u converging to u in L , 0 k 2 such that tL∂ −tL∂ −tL∂ tL∂ −tL∂ −tL∂ x x x x x x (e n (e y)e u ) → (e n (e y)e u) , (4.10) k x x as k →∞. As above, write f for the convolution f ∗ , where is a standard k k k mollifier. It is then clear (cf. the proof of Lemma 4.2) that Q(∂ )( f ∗ ) = (Q(∂ ) f ) ∗ → Q(∂ ) f, x k x k x for any Fourier multiplier operator Q(∂ ) for which Q(∂ ) f ∈ L .If n is a bounded x x 2 function, one therefore immediately obtains the required convergence nQ(∂ )( f ∗ ) → nQ(∂ ) f . In the case of an operator ∂ [nQ(∂ )] as in (4.10), where n is a k x x x bounded function such that ∂ n ∈ L , one notes that x ∞ ∂ [nQ(∂ )]= (∂ n)Q(∂ ) + n∂ Q(∂ ), x x x x x x and both terms which are of the form n ˜ Q(∂ ). This argument is valid for any fixed y ∈ B ⊂ H . Thus, for v ∈ dom(G ), there exists a sequence {v } satisfying (4.10) R 0 k such that (Gu)v tL∂ −tL∂ −tL∂ tL∂ −tL∂ −tL∂ −tL∂ x x x x x x x = e n (e y)e u) − e n (e y)e y e u v dx x x k tL∂ −tL∂ −tL∂ x x x =− (e n (e y)e u)(v ) k x tL∂ −tL∂ −tL∂ −tL∂ x x x x + (e n (e y)e y e u)v dx x k tL∂ −tL∂ −tL∂ x x x =− u e n (e y)e (v ) k x tL∂ −tL∂ −tL∂ −tL∂ x x x x + (e n (e y)e y e v ) dx x k tL∂ −tL∂ −tL∂ x x x =− u e n (e y)e v dx . (4.11) Taking the limit with respect to k in (4.11), we deduce for all u ∈ dom(G) that (Gu)v = u(G v), R R ∗ ∗ which means that G ⊂ G . Therefore, G = G .By[24, Corollary 4.4], G and 0 0 hence A(t, y) are quasi-m-accretive. Denote by [T , T ]= T T − T T 1 2 1 2 2 1 1158 M. Ehrnström and L. Pei J. Evol. Equ. the commutator of two general operators T and T . Since ∂ and L are both multiplier 1 2 x operators, clearly [∂ , L]= 0 in a Sobolev setting. Let s −s s −s B(t, y) = (A(t, y)) − A(t, y) =[ , A(t, y)] , (4.12) s 2 with = (1−∂ ) , and A(t, y) defined as in (3.3). We prove the Lipschitz continuity of the operators A(t, ·) and B(t, ·). LEMMA 4.5. The operator B(t, y) from (4.12) is bounded in B(X ), with (B(t, y) − B(t, z))w y − z w , (4.13) X Y X uniformly for all y, z ∈ B ⊂ Y and all w ∈ X. The estimate (4.13) holds if we replace A(t, ·) by B(t, ·) and interchange the norms in X and Y spaces on the right-hand side. Proof. Because Fourier multipliers commute, one has s tL∂ s −tL∂ −tL∂ x x x [ , A(y)]= e [ , n (e y)]e ∂ , and with classical commutator estimates (cf. [15,16]) that s −tL∂ 1−s −tL∂ x x [ , n (e y)] ∂ (n (e y)) , s−1 for all y ∈ B . With Theorem 3.3, one further has that −tL∂ ∂ (n (e y)) s−1 −tL∂ −tL∂ −tL∂ x x x (n (e y) − n (0))e y s−1 +n (0)e y s−1 x x H H (4.14) −tL∂ −tL∂ x x n (e y) − n (0) s−1 e y s−1 +|n (0)|y x H H H s− 2 (1 + R) R + R, where all estimates depend upon the radius of the ball B in which y lies, and the final estimate also on the nonlinearity n. Thus, for any z ∈ L ,wehave s −tL∂ 1−s s−1 −s B(t, y)z =[ , n (e y)] ∂ z L x L 2 2 s −tL∂ 1−s s−1 −s ≤[ , n (e y)] ∂ z x L s− 2 (1 + R) R + R z , whence B(t, y) is bounded on L , uniformly for y ∈ B . 2 R To prove the Lipschitz continuity in y, notice that for any y, z ∈ B and w ∈ X, one has the uniform estimate B(y)w − B(z)w s −tL∂ −tL∂ 1−s s−1 −s x x [ , n (e y) − n (e z)] ∂ w x L −tL∂ −tL∂ x x ∂ (n (e y) − n (e z)) s−1 w x L 2 Vol. 18 (2018) Classical well-posedness in dispersive equations 1159 −tL∂ −tL∂ x x n (e y) − n (e z) w . H L Appealing to Theorem 3.3, one furthermore estimates −tL∂ −tL∂ x x n (e y) − n (e z) −tL∂ −tL∂ x x = n (e (z + t (y − z)))e (y − z) dt s− R(1 + R) + 1 y − z s , where we have used the same splitting of n as in (4.14). Thus, B(y) satisfies condition (ii) in Theorem 3.1 for y ∈ B ⊂ H , s > . The proof of (4.13) with A(t, y) substituted for B(t, y) is structurally similar, but easier, than the above proof, and we omit the details. Since the operator A(t, y) relies on t, besides the assumptions in Theorem 3.1 one also needs to verify the continuity of the map t → A(t, y) ∈ B(Y, X ) for each y ∈ B ⊂ Y . As remarked in [16], it, however, suffices to prove that t → A(t, y) is strongly continuous. LEMMA 4.6. The map t → A(t, y) ∈ B(Y, X ) is strongly continuous. −tL∂ Proof. Since ∂ is bounded from Y to X and e is a strongly continuous uni- tary group on both X and Y , it is enough to prove that the multiplication operator −tL∂ n (e y) − n (0) ∈ B(Y, X ) is strongly continuous in t. In view of Theorem 3.3, −tL∂ s that map is continuous even in norm, since t → e y is continuous R → H for s > . We are now ready to prove the main theorem for initial data u ∈ H (R). Proof of Theorem 2.1. BasedonLemmata 4.4, 4.5 and 4.6, we may apply Theo- rem 3.1 to find a solution v to equation (3.2) in the solution class C ([0, T ); H (R)) ∩ 1 s−1 3 C ([0, T ); L (R)). Because H (R), s > , is an algebra, and tL∂ −tL∂ −tL∂ x x x v → e n (e v)e ∂ v s s−1 maps H (R) continuously into H (R) one, however, sees that tL∂ −tL∂ −tL∂ s−1 x x x v =−A(t,v)v =−e n (e v)e ∂ v ∈ H (R). t x −tL∂ s Recall that {e } forms a unitary group on H , for any s ≥ 0. Hence, 1 s−1 s v ∈ C ([0, T ); H (R)). Also, since [v → v] is continuous H (R) → s s s−1 C ([0, T ), H (R)), and ∂ maps H (R) continuously into H (R), the same argu- ment can be used to conclude that s 1 s−1 [v → v]∈ C (H (R), C ([0, T ), H (R))). 0 1160 M. Ehrnström and L. Pei J. Evol. Equ. The transformation (3.1) gives a solution u of (1.1) in the class of C ([0, T ); H ) and s s [u → u]∈ C (H (R), C ([0, T ), H (R)).Now, −tL∂ −tL∂ x x u (t, x ) =−e L∂ v(t, x ) + e v (t, x ), (4.15) t x t so u in general inherits its smoothness from the minimal regularity of v and L∂ v. t t x By assumption A, L is a lth-order operator, whence we deduce that s−max{1,l+1} u (t, x ) ∈ C ([0, T ); H ). A technique similar to the one above gives the continuous dependence upon initial data u ∈ H . 5. Well-posedness for periodic initial data This section focuses on the Cauchy problem (1.1) in the periodic setting H (T).We first introduce some notation. For a function f defined on R or the n-dimensional torus T , we define the mth-order difference about f by 1 n f (x ) = f (x + h) − f (x ), h, x ∈ F , m−1 m 1 n f (x ) = ( f )(x ), m = 2, 3,..., h, x ∈ F , h h h where F ∈{R, T}. It is easy to verify that 2( f )(x ) = ( f )(x ) − ( f )(x ). (5.1) h 2h Denote by N and N the positive and nonnegative integers, respectively. For a multi- |α| α ∂ index α = (α ,...,α ), α ∈ N ,weuse ∂ = to denote multi-index 1 n j 0 α α 1 n ∂ ···∂ x x derivatives with the index α, where |α|= α +···+ α .For s > 0, we use the 1 n − + − + decomposition s =[s] +{s} where [s] is an integer and {s} ∈ (0, 1]. There is a general characterization for Besov spaces based on spectral decom- position and distribution theory (cf. [28]) but the derivative–difference character- s n ization is enough here since only the classical, normed, Besov spaces B (T ), pq s > 0, p ∈ (1, ∞), q ∈[1, ∞], appear in our work. With the derivative–difference s n characterization, the Besov spaces B (T ), q ∈[1, ∞), have norms (cf. [25]) pq + dh −{s} q 2 α s n − f = f + |h| ∂ f . B (T ) [s] n pq n L (T ) p n W (T ) p n |h| |α|=[s] (5.2) 1 s n s n Note that the Besov spaces B (R ) or B (T ) are only quasi-normed spaces in general; however, they pq pq are normed spaces for the indices s, p, q in the current setting. The case p = 1 is excluded since in this case the composition theorem is not expected to hold for spaces over a torus, as indicated in Theorem 3.3 for spaces over the whole space. Vol. 18 (2018) Classical well-posedness in dispersive equations 1161 m n where W (T ), m ∈ N , are the standard Sobolev spaces with norms f m n = ∂ f n , W (T ) L (T ) |α|≤m and L (T ) is defined in (5.4). In the case when q =∞, the above norm must be modified to −{s} 2 α f n = f − + sup |h| ∂ f n . (5.3) B (T ) [s] L (T ) n h p p∞ W (T ) p n 0=h∈T |α|=[s] n n A function f defined on T is naturally identified with a 2π-periodic function on R , and we denote this other function still by f . With that identification, one can define the Lebesgue measure on T and set (cf. [19]) π π f = ... | f | dx . (5.4) L (T ) −π −π Our composition theorem for periodic Besov spaces relies on Theorem 3.3 for spaces n s n over R . The Besov spaces B (R ), s > 0, p ∈ (1, ∞), q ∈[1, ∞], can be defined p,q n n just as in (5.2) and (5.3) by replacing T with R [28]. The following remark follows from a direct calculation. n n REMARK 5.1. Let 0 <δ 1. Replacing the integral domain T or R by the s n n domain |h| <δ in the norms (5.2) and (5.3) of the Besov spaces B over T and R , pq respectively, yields an equivalent norm on the same Besov spaces. We proceed by establishing a relation between periodic and non-periodic Besov spaces. This relation and the techniques used in its proof are key ingradients for the proof of our composition theorem for Besov spaces on a torus. LEMMA 5.2. (A localizing property) Let s > 0,p ∈ (1, ∞) and q ∈[1, ∞].For ∞ n any ∈ C (R ), one has s n s n f f , B (R ) B (T ) pq pq s n uniformly for all f ∈ B (T ). pq ∞ n s n Proof. Let ∈ C (R ) be as in the lemma. Since f ∈ B (T ) can be identified with c pq a2π-periodic function on R (still denoted by f ) as mentioned above, the finiteness s n of f (as a function on R ) under the norm · follows from the smoothness B (R ) pq and compact support of . In the following, we give the full details for the case 1 ≤ q < ∞; the proof for q =∞ then follows with minor changes using the same procedure. We also consider separately the cases when s is, or is not, an integer. The case when s is not an integer. By the Leibniz rule for differentiation, it suffices − + to consider the case when s ∈ (0, 1), where [s] = 0 and {s} = s. By Remark 5.1, we then have dh −sq 2 s n n f f + |h| ( f ) , B (R ) L (R ) n p h L (R ) pq p n |h| |h|<δ 1162 M. Ehrnström and L. Pei J. Evol. Equ. 1 n for a fixed δ ∈ (0, ).Given there exists N such that supp( ) =[a, b] ⊂ [−N π, N π ] . Then n n n n f ≤ f ≤ N f , (5.5) L (R ) L ([−N π,N π ] ) L (R ) L (T ) p p ∞ p and 2 2 n n ( f ) = ( f ) , (5.6) L (R ) L ([a−2δ,b+2δ] ) h p h p where we used the identification between functions on T and 2π-periodic functions on R . In view of (this can be seen by rewriting) 2 2 2 ( f )(x ) = ( f )(x ) (x + h) + f (x + 2h)( )(x ) + ( f )(x )( )(x ), 2h h h h h (5.7) we derive from (5.5) and (5.6) that f s n N n f n B (R ) L (R ) L (T ) ∞ p pq dh −sq 2 + |h| ( f ) (·+ h) h L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh −sq 2 (5.8) + |h| f (·+ 2h) L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh −sq + |h| f 2h h n L ([a−2δ,b+2δ] ) p n |h| |h|<δ n n =: N f + T + T + T . L (R ) L (T ) 1 2 3 ∞ p By the periodicity of f , k ≥ 1, we estimate T as follows dh −sq 2 T ≤ n |h| f 1 L (R ) n ∞ h L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh n −sq 2 ≤ (N + 2) n |h| f , L (R ) n ∞ h L (T ) n n |h| where in the last inequality we used the fact that [a −2δ, b+2δ] ⊂[−(N +2)π, (N + n n 2)π ] and the identification between functions on T and the 2π-periodic functions on R . Before estimating the term T , we observe from mean-value theorem for scalar [12] and vector-valued [21] multi-variable functions that | |= |[ (x + 2h) − (x + h)]−[ (x + h) − (x )]| =|∇ (ξ ) · h −∇ (ξ ) · h| 1 2 (5.9) =| ∇ (∇ )(t ξ + (1 − t )ξ ) dt · (ξ − ξ )||h| 1 2 1 2 ≤ 2sup |Hess( )(t ξ + (1 − t )ξ )||h| , 1 2 t ∈[0,1] Vol. 18 (2018) Classical well-posedness in dispersive equations 1163 where ∇ and Hess denote the gradient and the Hessian Matrix, respectively; ξ lies on the segment with endpoints x + 2h and x + h, ξ lies on the segment with endpoints x + h and x. Then, T can be estimated as dh (2−s)q T ≤ 2Hess( ) n |h| f (·+ 2h) 2 L (R ) n L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh (2−s)q ≤ 2Hess( ) n f n |h| L (R ) L ([a−4δ,b+4δ] ) ∞ p |h| |h|<δ (N + 4) Hess( ) n f n , L (R ) L (T ) ∞ p (2−s)q dh wherewehaveusedthat |h| is finite and absolutely bounded for s = |h|<δ |h| + 1 {s} ∈ (0, 1) and δ ∈ (0, ). In the first inequality above we applied (5.9), in the second the variable substitution x + 2h → x and in the last that [a − 4δ, b + 4δ] ⊂ [−(N + 4)π, (N + 4)π ] .The term T can be similarly estimated: dh (1−s)q T ≤∇ n |h| f (·+ 2h) 3 L (R ) n L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh (1−s)q +∇ n |h| f L (R ) n L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh (1−s)q ≤ 2∇ n f n |h| L (R ) L ([a−4δ,b+4δ] ) ∞ p |h| |h|<δ (N + 4) ∇ n f n , L (R ) L (T ) ∞ p 1 + where again the estimate is absolute because δ ∈ (0, ) and s ={s} ∈ (0, 1).Inthe first inequality, we relied on the mean-value theorem for , in the second inequality on the variable substitution x + 2h → x, and in the third on that [a − 4δ, b + 4δ] ⊂ [−(N + 4)π, (N + 4)π ] . By inserting above estimates for T , T and T into (5.8), 1 2 3 we have dh −sq 2 s n n f f + |h| f s,p,q,n, n B (R ) L (R ) p h L (T ) pq p n n |h| which concludes the proof for the case when s is a non-integer. The case when s is an integer. As above, we still pick δ ∈ (0, ). For an integer s,we − + have [s] = s − 1 and {s} = 1. As mentioned above, by Leibniz’s rule it suffices to − + consider the case [s] = 0 and {s} = 1. Similar to (5.8) we deduce from Remark 5.1 that 1164 M. Ehrnström and L. Pei J. Evol. Equ. f s n N n f n B (R ) L (R ) L (T ) ∞ p pq dh −q 2 + |h| ( f ) (·+ h) L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh −q 2 (5.10) + |h| f (·+ 2h) L ([a−2δ,b+2δ] ) p n |h| |h|<δ dh −q + |h| f 2h h L ([a−2δ,b+2δ] ) n |h| |h|<δ =: N n f n + S + S + S . L (R ) L (T ) 1 2 3 ∞ p Analogous to the estimates for T and T , we get control of S and S . 1 2 1 2 dh n −q 2 S ≤ (N + 2) |h| f , 1 L (R ) n ∞ h L (T ) p n n |h| n n S (N + 4) Hess( ) f , 2 L (R ) L (T ) ∞ p q dh where in the second estimate we have used that |h| is absolutely bounded |h|<δ |h| for δ ∈ (0, ), q ∈[1, ∞). However, the argument for estimating T is not suitable for S , since the integration near the origin would lead to a logarithmic blow-up and therefore fails to give a finite control for S . Instead, we use the difference iteration formula (5.1) and carefully analyze the integrals over each sub-interval. The estimate for S starts as follows. dh S ≤∇ f 3 L (R ) 2h ∞ L ([a−2δ,b+2δ] ) |h| |h|<δ 1 dh ≤ ∇ f L (R ) ∞ 2h L ([a−2δ,b+2δ] ) p n 2 |h| |h|<δ (5.11) 1 dh + ∇ f L (R ) 4h n ∞ L ([a−2δ,b+2δ] ) p n 2 |h| |h|<δ =: ∇ (S + S ) , L (R ) 31 32 where the first inequality used the mean-value theorem for scalar functions with several variables; the second inequality used the interation formula (5.7). We first make the variable substitution 2h → η in S and divide the interval of integration as dη (S ) = f 32 2η L ([a−2δ,b+2δ] ) |η| |η|<2δ dη = + f . 2η n L ([a−2δ,b+2δ] ) p n |η| |η|<δ δ<|η|<2δ Vol. 18 (2018) Classical well-posedness in dispersive equations 1165 Then (5.11) and the above equality for S imply that dh 2h L ([a−2δ,b+2δ] ) n |h| |h|<δ (5.12) dh ≤ S + f . 31 2h L ([a−2δ,b+2δ] ) n |h| δ<|h|<2δ Inserting (5.12)into(5.11), we get dh S ≤∇ n S + f . (5.13) 3 L (R ) 31 2h L ([a−2δ,b+2δ] ) n |h| δ<|h|<2δ To estimate S , since q > 1 we again use the substitution 2h → η in S to get 31 31 dη q 2 (S ) = f L ([a−2δ,b+2δ] ) n |η| |η|<2δ (5.14) dη nq −q 2 ≤ (N + 2) |η| f , L (T ) n n |η| −q where the factor |η| may be trivially introduced since |η|≤ 1 and q ≥ 1. For the remaining term in (5.13), we have dh 2h L ([a−2δ,b+2δ]) |h| δ<|h|<2δ dh q q f (·+ 2h) + f L ([a−2δ,b+2δ]) L ([a−2δ,b+2δ]) n p p |h| δ<|h|<2δ (5.15) dh L ([a−4δ,b+4δ]) n |h| δ<|h|<2δ dh nq (N + 4) f . L (T ) n |h| δ<|h|<2δ Then (5.13), (5.14) and (5.15) give the desired control of S : dη n −q 2 S ∇ n (N + 2) |η| f 3 L (R ) ∞ η L (T ) n n |η| dh +(N + 4) ∇ n f (x ) n . L (R ) L (T ) ∞ p |h| δ<|h|<2δ Inserting the above estimates for S , S and S into (5.11), we have 1 2 3 dh −q 2 f s n f n + |h| f , B (R ) L (T ) p h pq L (T ) n n |h| where the estimate depends only on , n and δ in the way detailed in the above inequalities. This concludes the proof for the case when s is an integer. 1166 M. Ehrnström and L. Pei J. Evol. Equ. Using Lemma 5.2, we shall now prove the following result. THEOREM 5.3. (A composition theorem for Besov spaces on T) Let 1 < p < ∞, 1 < q ≤∞ and s > 1 + . For a Borel measurable function f : R → R,the s s composition operator T acts on B (T) if and only if f ∈ B (R). Moreover, T f f pq pq,loc is bounded if T acts on B (T).If f (0) = 0 holds additionally, we have pq s−1− f ◦ g s ( f ϕ ) s−1 g s (1 +g s ) , (5.16) B (T) a B (T) B (T) pq B (R) pq pq pq −1 where a =g and ϕ is the a -dilation of φ defined in Remark 3.4. L a Proof of Theorem 5.3. As before, we will only give full detail for the case 1 ≤ q < ∞, since the proof for q =∞ can be treated using the same procedure with only minor changes, and is in fact easier. In addition, since f (x ) = ( f (x ) − f (0)) + f (0) and s 1 B → L for s > , it suffices to prove the theorem for f with f (0) = 0. pq s s We first prove the sufficiency part of the result, namely that T : B (T) → B (T) pq pq s s ∞ if f ∈ B (R).Let g ∈ B (T), and choose ∈ C (R) such that = 1on pq,loc pq c (−4π, 4π) with supp( ) ⊂[−6π, 6π ]. Denoting by (g ) the periodic extension per of g from [−π, π ] to R, we notice that the 2π-periodic function (g ) on R can per be identified with g on T.By (5.2) and (5.4), we then have f ◦ g = f ◦ (g ) − per B (T) [s] pq W ([−π,π ]) dh −{s}q 2 α + |h| ∂ [ f ◦ (g ) ] per h L ([−π,π ]) |h| [−π,π ] |α|=[s] dh −{s}q 2 α ≤ f ◦ (g ) − + |h| ∂ f (g ) . [s] h L (R) W (R) |h| |α|=[s] Then Theorem 3.3, Remark 3.4 and Lemma 5.2 imply that s s f ◦ g ≤ f ◦ (g ) B (T) B (R) pq pq s−1− s s ( f ϕ ) s−1 g (1 +g ) g B (R) B (R) ∞ B (R) pq pq pq s−1− s s ( f ϕ ) s−1 g (1 +g ) , g B (T) B (T) L∞ B (R) pq pq pq with ϕ as in Remark 3.4. (·) We now prove the necessity part of the result, showing that f ∈ B (R) if T acts p,q,loc s ∞ on B (T). For any ∈ C (R) with compact support supp( ) =[a, b],dividethe p,q c interval I =[a − 1, b + 1] into a finite number of subintervals I =[a − 1 + ( j − 1)η, a − 1 + j η], j = 1, 2,...,(b − a + 2)/η, of fixed but small length η 1 such that (b−a+2)/η is an integer. Then define smooth and compactly supported functions g , j = 1,...,(b −a +2)/η, with | supp(g )|≤ 1 j j and g (x ) = x , x ∈ I . j j Vol. 18 (2018) Classical well-posedness in dispersive equations 1167 Because the support of g has length less than 2π, we may extend g 2π-periodically j j to R; denote this periodic extension by g . We furthermore identify g as a j,per j,per function on the torus T, with support of at most unit length. By definition of the Besov spaces and by Remark 5.1,wehave f s ≤ − f − B (R) [s] [s] pq C (R) W ([a,b]) + dh −{s} q 2 α + |h| ∂ ( f ) , L ([a−2δ,b+2δ]) |h| |h|≤δ |α|≤[s] (5.17) where we consider δ 1 fixed but small enough for 4δ ≤ 1 (for larger δ one needs to adjust the intervals I above). The estimate for f − is straightforward j [s] W ([a,b]) since f − ≤ f − [s] [s] W ([a,b]) W (I ) p p ≤ f ◦ g − j [s] W (I ) p j (5.18) ≤ f ◦ (g ) . j,per [s] W (T) Note that the last term in (5.18) is finite since, by assumption, T acts on B (T).The pq estimate of the second term in (5.17), however, needs a more careful analysis, similar to the one in the proof of Lemma 5.2, but it suffices to consider the case when α = 0 since the case when α ∈[1, [s] ] may be reduced to the former case by the Leibniz rule. We are then left to control the term + dh −{s} q 2 S( f, ) = |h| ( f ) (5.19) L ([a−2δ,b+2δ]) |h| |h|≤δ from (5.17). The argument for estimating S( f, ) varies according to whether s is an integer or not. Therefore, we consider the two cases separately. The case when s is not an integer. In this case {s} ∈ (0, 1), and by (5.7)wehave + dh −{s} q 2 S( f, ) ≤ |h| ( f ) (·+ h) L ([a−2δ,b+2δ]) |h| |h|≤δ + dh −{s} q 2 + |h| f (·+ 2h) h L ([a−2δ,b+2δ]) |h| |h|≤δ + dh −{s} q + |h| ( f ) 2h h L ([a−2δ,b+2δ]) |h| |h|≤δ =: T + T + T . 1 2 3 1168 M. Ehrnström and L. Pei J. Evol. Equ. We first estimate T : + dh 1 −{s} q 2 T = |h| ( f ◦ g ) 1 L (R) j ∞ h 2 L (I ∩[a−2δ,b+2δ]) p j |h| |h|≤δ + dh −{s} q 2 ≤ |h| ( f ◦ g ) L (R) j h L (R) |h| |h|≤δ + dh −{s} q 2 ≤ |h| ( f ◦ g ) . L (R) j,per ∞ h L (T) |h| By the mean-value theorem and the variable substitution x + 2h → y, we may then estimate T similarly as follows. + dh 1 −{s} q 2q T = |h| |h| f 2 L (R) 2 ∞ L ([a−2δ+2h,b+2δ+2h]) |h| |h|≤δ + dh (2−{s} )q ≤ |h| f ◦ g L (R) j L (I ) p j |h| |h|≤δ + dh (2−{s} )q ≤ |h| f ◦ g L (R) j L (R) ∞ p |h| |h|≤δ + dh (2−{s} )q ≤ |h| f ◦ g . L (R) j,per L (T) ∞ p |h| |h|≤δ The term T can be estimated similarly as T . 3 2 dh (1−{s} )q T |h| f 3 L (R) 2h L ([a−2δ,b+2δ]) |h| |h|≤δ + dh (1−{s} )q |h| f L (R) L (I ) |h| |h|≤δ + dh (1−{s} )q |h| f ◦ g . L (R) j,per L (T) ∞ p |h| |h|≤δ By the assumption that f acts on B (T) it is clear from the above estimates that pq S( f, ), and thus also f , is bounded for each ∈ C . B (R) pq The case when s is an integer. In this case {s} = 1, and we first notice that dh −q 2 S( f, ) ≤ |h| ( f ) (·+ h) h L ([a−2δ,b+2δ]) |h| |h|≤δ dh −q 2 + |h| f (·+ 2h) h L ([a−2δ,b+2δ]) |h| |h|≤δ Vol. 18 (2018) Classical well-posedness in dispersive equations 1169 dh −q + |h| ( f ) 2h h L ([a−2δ,b+2δ]) |h| |h|≤δ =: S + S + S . 1 2 3 We estimate S and S analogously to T and T ,by 1 2 1 2 dh −q 2 S |h| ( f ◦ g ) , 1 L (R) j,per ∞ h L (T) |h| dh S |h| f ◦ g . 2 L (R) j,per L (T) ∞ p |h| |h|≤δ Theestimatefor S , on the other hand, starts in a way identical to (5.11)–(5.13). Just as in (5.13) that yields dh S ≤∇ϕ S + f , (5.20) 3 L (R) 31 2h L ([a−2δ,b+2δ]) |h| δ<|h|<2δ with dh S = f 2h L ([a−2δ,b+2δ]) |h| |h|<δ dη −q 2 ≤ |η| ( f ◦ g ) L (I ) p j |η| |η|<2δ (5.21) dη −q 2 ≤ |η| ( f ◦ g ) η L (R) |η| |η|<2δ dη −q 2 ≤ |η| ( f ◦ g )] , j,per L (T) |η| by the construction of the functions {g } . For the remaining term in (5.20), it may j j similarly be estimated as dh dh f ≤ 2 f ◦ g . 2h j,per L (T) L ([a−2δ,b+2δ]) |h| |h| δ<|h|<2δ δ<|h|<2δ (5.22) Then (5.20), (5.21) and (5.22) together give control of S : dh −q 2 S ≤ |h| ( f ◦ g ) 3 L (R) j,per ∞ h L (T) |h| (5.23) dh + 2 f ◦ g j,per L (T) |h| δ<|h|<2δ 1170 M. Ehrnström and L. Pei J. Evol. Equ. Since we are still proving the necessity part of the theorem, f ◦ g ∈ B (T) j,per pq by assumption. It is then clear from the above estimates that S( f, ), and thus f , is bounded. B (R) pq,loc For s > 0 and p = 2, it is well known that fractional Sobolev spaces are equivalent s s n n with the Bessel–Potential spaces, W (X ) = H (X ) with X ∈{R , T } in our case. A consequence of this is that the proof of Theorem 2.1 can be followed in detail with the standard Sobolev spaces replaced by their periodic counterparts and the Fourier transform replaced accordingly, as described above. 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Triebel, Topics in Fourier Analysis and Function Spaces,AWiley- Interscience Publication, A Wiley-Interscience Publication, John Wiley & Sons Ltd., Chichester, [26] G. Staffilani, On the generalized Korteweg-de Vries-type equations, Differential and Integral Equations, 10 (1997), pp. 777–796. [27] T. Tao, Global well-posedness of the Benjamin–Ono equation in H (R), J. Hyperbolic Differ. Equ., 1 (2004), pp. 27–49. [28] H. Triebel, Theory of Function Spaces, vol. 38 of Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], Akademische Verlags- gesellschaft Geest & Portig K.-G., Leipzig, 1983. Mats Ehrnström and Long Pei Department of Mathematical Sciences NTNU Norwegian University of Science and Technology 7491 Trondheim Norway e-mail: longp@kth.se Mats Ehrnström e-mail: mats.ehrnstrom@math.ntnu.no Long Pei Department of Mathematics KTH Royal Institute of Technology 10044 Stockholm Sweden
Journal of Evolution Equations – Springer Journals
Published: Mar 2, 2018
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