GLOBAL REGULARITY, AND WAVE BREAKING PHENOMENA IN A CLASS OF NONLOCAL DISPERSIVE EQUATIONS

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1 GLOBAL EGULAITY, AND WAVE BEAKING PHENOMENA IN A CLASS OF NONLOCAL DISPESIVE EQUATIONS HAILIANG LIU AND ZHAOYANG YIN Abstract. This paper is concerne with a class of nonlocal ispersive moels the θ-equation propose by H. Liu [ On iscreteness of the Hopf equation, Acta Math. Appl. Sin. Engl. Ser. 4(3)(008)43 440]: ( u (1 x)u t + (1 θ x) ) ( u ) = (1 4θ) x, x x incluing integrable equations such as the Camassa-Holm equation, θ = 1/3, an the Degasperis-Procesi equation, θ = 1/4, as special moels. We investigate both global regularity of solutions an wave breaking phenomena for θ. It is shown that as θ increases regularity of solutions improves: (i) 0 < θ < 1/4, the solution will blow up when the momentum of initial ata satisfies certain sign conitions; (ii) 1/4 θ < 1/, the solution will blow up when the slope of initial ata is negative at one point; (iii) 1 θ 1 an θ = n n 1, n N, global existence of strong solutions is ensure. Moreover, if the momentum of initial ata has a efinite sign, then for any θ global smoothness of the corresponing solution is prove. Proofs are either base on the use of some global invariants or base on exploration of favorable sign conitions of quantities involving solution erivatives. Existence an uniqueness results of global weak solutions for any θ are also presente. For some restricte range of parameters results here are equivalent to those known for the b equations [e.g. J. Escher an Z. Yin, Well-poseness, blow-up phenomena, an global solutions for the b-equation, J. reine angew. Math., 64 (008)51 80.] Contents 1. Introuction. Preliminaries 4.1. The θ equation an its variants 4.. Local well-poseness an a priori estimates 5 3. Global regularity Key invariants an favorable sign conitions Global existence: proof of Theorem Blow up phenomena: proof of Theorem A etaile account of the case θ = Global weak solutions 15 eferences 19 Date: November 17, 009. Key wors an phrases. Dispersive equations, global regularity, wave breaking, global weak solutions. 1

2 HAILIANG LIU AND ZHAOYANG YIN 1. Introuction In recent years nonlocal ispersive moels have been investigate intensively at ifferent levels of treatments: moeling, analysis as well as numerical simulation. The moel erives in several ways, for instance, (i) the asymptotic moeling of shallow water waves [43, 0, 1]; (ii) renormalization of ispersive operators [43, 37]; an (iii) moel equations of some ispersive schemes [38]. The peculiar feature of nonlocal ispersive moels is their ability to capture both global smoothness of solutions an the wave breaking phenomena. In this work we focus on a class of nonlocal ispersive moels the θ-equation of the form ( ) ( ) u (1 x)u t + (1 θ x) u = (1 4θ) x, (1.1) x x subject to the initial conition The equation can be formally rewritten as u(0, x) = u 0 (x), x. (1.) u t u txx + uu x = θuu xxx + (1 θ)u x u xx, (1.3) which when 0 < θ < 1 involves a convex combination of nonlinear terms uu xxx an u x u xx. This class was ientifie by H. Liu [38] in his stuy of moel equations for some ispersive schemes to approximate the Hopf equation u t + uu x = 0. The moel (1.1) uner a transformation links to the so calle b moel, u t α u txx + c 0 u x + (b + 1)uu x + Γu xxx = α (bu x u xx + uu xxx ). which has been extensively stuie in recent years [17, 18, 4, 5, 8, 9]. Both classes of equations are containe in the more general class erive in [37] using renormalization of ispersive operators an number of conservation laws. In (1.1), two equations are worth of special attention: θ = 1 3 an θ = 1 4. The θ-equation when θ = 1 3 reuces to the Camassa-Holm equation, moeling the uniirectional propagation of shallow water waves over a flat bottom, in which u(t, x) enotes the flui velocity at time t in the spatial x irection [3, 19, 30]. The Camassa-Holm equation is also a moel for the propagation of axially symmetric waves in hyperelastic ros [13, 15]. It has a bi-hamiltonian structure [6, 33] an is completely integrable [3, 7]. Its solitary waves are smooth if c 0 > 0 an peake in the limiting case c 0 = 0, cf. [4]. The orbital stability of the peake solitons is prove in [1], an that of the smooth solitons in [14]. The explicit interaction of the peake solitons is given in [1]. The Cauchy problem for the Camassa-Holm equation has been stuie extensively. It has been shown that this problem is locally well-pose [8, 41] for initial ata u 0 H 3/+ (). Moreover, it has global strong solutions [6, 8] an also amits finite time blow-up solutions [6, 8, 9]. On the other han, it has global weak solutions in H 1 () [, 10, 11, 44]. The avantage of the Camassa-Holm equation in comparison with the KV equation, u t + uu x + Γu xxx = 0, lies in the fact that the Camassa-Holm equation has peake solitons an moels the peculiar wave breaking phenomena [4, 9]. Taking θ = 1 4 in (1.1) we fin the Degasperis-Procesi equation [18]. The Degasperis- Procesi equation can be regare as a moel for nonlinear shallow water ynamics an its asymptotic accuracy is the same as that for the Camassa-Holm shallow water

3 ON NONLOCAL DISPESIVE EQUATIONS 3 equation [0, 1]. An inverse scattering approach for computing n-peakon solutions to the Degasperis-Procesi equation was presente in [36]. Its traveling wave solutions were investigate in [3, 4]. The formal integrability of the Degasperis-Procesi equation was obtaine in [16] by constructing a Lax pair. It has a bi-hamiltonian structure with an infinite sequence of conserve quantities an amits exact peakon solutions which are analogous to the Camassa-Holm peakons [16]. The stuy of the Cauchy problem for the Degasperis-Procesi equation is more recent. Local well-poseness of this equation is establishe in [46] for initial ata u 0 H 3/+ (). Global strong solutions are prove in [, 34, 47] an finite time blow-up solutions in [, 34, 46, 47]. On the other han, it has global weak solutions in H 1 (), see e.g. [, 47] an global entropy weak solutions belonging to the class L 1 () BV () an to the class L () L 4 (), cf. [5]. Though both the Dgasperis-Procesi an the Camassa-Holm equation share some nice properties, they iffer in that the DP equation has not only peakon solutions [16] an perioic peakon solutions [48], but also shock peakons [35] an the perioic shock waves [3]. The main quest of this paper is to see how regularity of solutions changes in terms of the parameter θ. With this in min we present a relative complete picture of solutions of problem (1.1)-(1.) for ifferent choices of θ. Theorem 1.1. [Global regularity] Let u 0 H 3/+ () an m 0 := (1 x)u 0. i) For any θ 0, if in aition u 0 L 1 (), an m 0 has a efinite sign (m 0 0 or m 0 0 for all x ), then the solution remains smooth for all time. Moreover, for all (t, x) +, we have (1) m(t, x)u(t, x) 0 an m 0 L 1 () = m(t, ) L 1 () = u(t, ) L 1 () = u 0 L 1 (). () u x (t, ) L () u 0 L 1 () an u(t, ) L () u(t, ) 1 e 1 θ 3 t u 0 L 1 () u0 1. ii) For 1 θ 1, if in aition u 0 W, 1 θ (), then the solution remains smooth for all time. iii) For θ = n n 1 (1, ), n N, if in aition u 0 W 3, θ θ 1 (), then the solution remains smooth for all time. emark 1.1. The result state in i) recovers the global existence result of strong solutions to the Camassa-Holm equation in [8] an the Degasperis-Procesi equation in [47]. Theorem 1.. [Blow up criterion] Let u 0 H 3/+ () an m 0 := (1 x)u 0. i) For 0 < θ 1 4 an a fixe x, if u 0 (x + x) = u 0 (x x) an (x x )m 0 (x) 0 for any x, then the solution must blow up in finite time strictly before T = 1 u x (0,x ) provie u x(0, x ) < 0. ii) For 1 4 θ < 1, if u 0(x +x) = u 0 (x x) for any x an u x (0, x ) < 0, then the solution must blow up in finite time strictly before T = θ θ (θ 1)u x (0,x ). emark 1.. The result state in Theorem 1. shows that strong solutions to the θ-equation (1.1)-(1.) for 0 < θ < 1 may blow up in finite time, while Theorem 1.1 shows that in the case 1 θ 1 every strong solution to the θ-equation (1.1) exists globally in time. This presents a clear picture for global regularity an blow-up phenomena of solutions to the θ-equation for all 0 < θ 1.

4 4 HAILIANG LIU AND ZHAOYANG YIN We shall present main ieas for proofs of the above results, a further refine analysis coul be one following those presente in [4] for the b equation. Note that the case θ = 0 is a borerline case an not covere by the above results, we shall present a etaile account for this case. For completeness, we also present global existence results for weak solutions to characterize peakon solutions to (1.1) for any θ. The rest of this paper is organize as follows. In, we present some preliminaries incluing how the θ equation relates to other class of ispersive equations, the local well-poseness an some key quantities to be use in subsequent analysis. In 3, we show how global existence of smooth solutions is establishe. The ieas for eriving the precise blow-up scenario is given in 4. A etaile account for the case of θ = 0 is presente in 5. Two existence an uniqueness results on global weak solutions an one example for peakon solutions to (1.1) for any θ are given in 6.. Preliminaries.1. The θ equation an its variants. The θ-equation of the form ( ) ( ) u (1 x)u t + (1 θ x) u = (1 4θ) x, (.1) x x up to a scaling of t t θ where Q = 1 e x an for θ 0, can be rewritten into a class of B-equations u t + uu x + [Q B(u, u x )] x = 0, (.) B = ( ) ( 1 u θ ) u x θ. The B class with B being quaratic in u an u x was erive in [37] by using a renormalization technique an examining number of conservation laws. In this B class the Camasa-Holm equation correspons to B(u, u x ) = u + u x/; an the Degasperis-Procesi equation correspons to B(u, u x ) = 3u /. The local wellposeness for (.) with initial ata u 0 (x) was establishe in [37]. Theorem.1. [37] Suppose that u 0 Hx 3/+ an B(u, p) are quaratic functions in its arguments, then there exists a time T an a unique solution u of (.) in the space C([0, T ); H 3/+ ()) C 1 ([0, T ); H 1/+ ()) such that lim t 0 u(t, ) = u 0 ( ). If T < is the maximal existence time, then lim t T sup 0 τ t u x (, τ) L (Ω) =, where Ω = for initial ata ecaying at far fiels, or Ω = [0, π] for perioic ata. Wave breaking criteria are ientifie separately for several particular moels in class (.), using their special features, see [37] for further etails. For θ 0, the class of θ-equations can also be transforme into the b equation of the form u t α u txx + c 0 u x + (b + 1)uu x + Γu xxx = α (bu x u xx + uu xxx ). (.3) In fact, if we set u(t, x) = c 0 θ + ũ(τ, z), ( z = α x θ (1 + Γα ) ) t, τ = αθt,

5 ON NONLOCAL DISPESIVE EQUATIONS 5 then a straightforwar calculation leas to (1 α z)ũ τ + c 0 ũ z + 1 θ ũũ z + Γũ zzz = α (( 1 θ 1 ) ũ z ũ zz + ũũ zzz ). Setting θ = 1 b + 1 an changing variables (ũ, τ, z) back to (u, t, x) we thus obtain the so-calle b equation (.3). Note that the θ equation oes not inclue the case b = 1, which has been known un-physical. Also θ = 0 case is not in the class of B equation (.) either... Local well-poseness an a priori estimates. In orer to prove our main results for ifferent cases, we nee to establish the following local existence result. Theorem.. [Local existence] Let u 0 H 3/+ (), then exists a T = T (θ, u 0 3/+ ) > 0 an a unique solution in C([0, T ); H 3/+ ()) C 1 ([0, T ); H 1/+ ()). The solution epens continuously on the initial ata, i.e. the mapping u 0 u(, u 0 ) : H s () C([0, T ); H s ()) C 1 ([0, T ); H s 1 ()), s > 3/ is continuous. Moreover, if T < then lim t T u(t, ) s =. The proof for θ 0 follows from that for the b equation in [4] or for the B equation in [38]. Furthermore we have the following result. Theorem.3. Let u 0 H 3/+ () be given an assume that T is the maximal existence time of the corresponing solution to (1.1) with the initial ata u 0. If there exists an M > 0 such that u x (t, x) L () M, t [0, T ), then the H s () norm of u(t, ) oes not blow up for t [0, T ). Let u be the solution in C([0, T ); H s ()) C 1 ([0, T ); H s 1 ()), it suffices to verify how u(t, ) s epens on u x (t, ). Here we coul carry out a careful energy estimate to obtain a ifferential inequality of the form t u(t, ) s C u x (t, ) u(t, ) s. The claim then follows from the Gronwall inequality. A etaile illustration of such a proceure for the case θ = 0 will be given in 5. emark.1. This result is funamental for us to prove or isprove the global existence of strong solutions. More precisely, global existence follows from a priori estimate on u x (t, ), an the finite time blow up of u x (t, ) uner certain initial conitions reveals the wave breaking phenomena. 3. Global regularity 3.1. Key invariants an favorable sign conitions. Let T be the life span of the strong solution u C([0, T ); H 3/+ ()) C 1 ([0, T ); H 1/+ ()). We now look at some key estimates vali for t [0, T ). First since the θ-equation is in conservative form, so ux = u 0 x. (3.1)

6 6 HAILIANG LIU AND ZHAOYANG YIN Let m = (1 x)u, then which implies u = (1 x) 1 m = Q m, (3.) mx = ux = u 0 x = Moreover the equation (1.1) can be reformulate as m 0 x. (3.3) m t + θum x + (1 θ)mu x = 0. (3.4) For any α, let x = x(t, α) be the curve etermine by x = θu(t, x), t x(0, α) = α for t [0, T ). Then F = x α solves t F = θu xf as long as u remains a strong solution. Along the curve x = x(t, α) we also have t m = (θ 1)u xm. These together when canceling the common factor u x leas to the following global invariant: m(t, x(t, α))f 1 θ 1 = m 0 (α), α. (3.5) From this Lagrangian ientity we see that m has a efinite sign once m 0 has. Corresponingly it follows from (3.) that u has a efinite sign provie that m 0 has a efinite sign on. From (3.5) it follows θ m 1 θ (t, x(t, α))f α = which yiels the following estimate: t sign(m) = sign(m 0 ) = sign(u) (3.6) θ θ m 1 θ x = m 0 θ 1 θ x, m 1 θ x = 0. (3.7) Inspire by [17] we ientify another conservation laws as follows ) ((1 θ) m θ θ 1 m t x + θ m θ θ 1 x = 0. (3.8) This conserve quantity will be use for some cases in the range θ > 1. Multiplying (3.4) by m = u u xx, an integrating by parts, we obtain t which suggests that θ = 3 m x = (3θ ) u x m x, (3.9) is a critical point for the blow-up scenario. Note that u(t, ) m(t, ) L u(t, ). (3.10) Both (3.9) an (3.10) together enable us to conclue the following Theorem 3.1. Assume u 0 H 3/+ (). If θ = 3, then every solution to (1.1)-(1.) remains regular globally in time. If θ < 3, then the solution will blow up in finite time if an only if the slope of the solution becomes unboune from below in finite time. If θ > 3, then the solution will blow up in finite time if an only if the slope of the solution becomes unboune from above in finite time.

7 ON NONLOCAL DISPESIVE EQUATIONS 7 emark 3.1. This result not only covers the corresponing results for the Camassa- Holm equation in [6, 45] an the Degasperis-Procesi equation in [46], but also presents another ifferent possible blow-up mechanism, i.e., if θ > 3, then the solution to (1.1) blows up in finite time if an only if the slope of the solution becomes unboune from above in finite time. 3.. Global existence: proof of Theorem 1.1. Let T be the maximum existence time of the solution u with initial ata u 0 H s. Using a simple ensity argument we can just consier the case s = 3. Base on Theorem. an Theorem.3 it suffices to show the uniform boun of u x (t, ) for all cases presente in Theorem 1.1. The proof of the first assertion i) is base on the global invariant (3.5), which implies (3.6), i.e., m has a efinite sign for t > 0 as long as m 0 has a efinite sign. Then for any (t, x) [0, T ), u x (t, x) = Q x m Q x m L 1 = 1 mx = 1 ux = 1 u 0 x 1 u 0 L 1. Then T =. The secon assertion ii) follows from the use of (3.7), i.e., θ θ m 1 θ x = m 0 1 θ x u0 W,p, p = θ 1 θ [1, ]. From m L p () an u u xx = m it follows that u W,p (). By the Sobolev imbeing theorem, we see that W,p () C 1 (). Thus T =. The last assertion iii) follows from the use of (3.8) with θ = n n 1, which upon integration leas to (m n m x + 4n m n )x = (m n 0 m 0x + 4n m n 0 )x. From this we see that m L, for m n = x nm n 1 m x x 1 (m n m x + 4n m n )x. Using u = Q m we obtain that u W, ; that is u x is uniformly boune. Thus T =. 4. Blow up phenomena: proof of Theorem 1. For the blow up analysis, one nees to fin a way to show that = u x will become unboune in finite time. ewriting (.1) as u t + θuu x = Q x [ (1 4θ)u x + (θ 1)u ]. Notice that Q xx = Q δ(x); a irect ifferentiation in x of the above equation leas to t + θu x + ( 1 θ ) = 1 θ u + Q [ (1 4θ) + (θ 1)u ]. For θ < 1 there is no control on u term while we track ynamics of. The iea here, motivate by that use in [4], is to focus on a curve x = h(t) such that u(t, h(t)) = 0 an h(0) = x. On this curve + ( 1 θ ) = Q [ (1 4θ)u x + (θ 1)u ] (t, h(t)). (4.1)

8 8 HAILIANG LIU AND ZHAOYANG YIN Two cases are istinguishe: (i) 1 4 θ < 1. In this range of θ, the right-han sie of (4.1) is non-positive. We thus have ( ) 1 + θ 0, for which will become unboune from below in finite time as long as (0, h(0)) = u x (0, x ) < 0. (ii) 0 < θ < 1 4. In this range of θ we also nee control the nonlocal term. If we can ientify some initial ata such that Q [ (1 4θ)u x + (θ 1)u ] (t, h(t)) (1 4θ) [ u x u ] (t, h(t)) = (1 4θ)(t). (4.) Then we have ( ) ( ) θ θ. That is + θ 0. Again in this case will become unboune from below in finite time once (0, h(0)) = u x (0, x ) < 0. Now we verify that the assumptions in Theorem 1. are sufficient for claim (4.) to hol. From u 0 (x + x) = u 0 (x x) for any x, it follows that u 0 (x ) = 0, an u(t, x + x) = u(t, x x) ue to symmetry of the equation. We then have u(t, x ) = 0, leaing to the case h(t) = x. We further assume that (x x )m 0 (x) 0, which combine with (3.5) yiels (x x )m(t, x) 0. This relation enables one to use a similar argument as (5.3)-(5.10) in [4] to obtain Hence Q [u x u ](t, x ) (u x u )(t, x ). Q [ (1 4θ)u x + (θ 1)u ] (t, x ) = (1 4θ)Q [u x u ](t, x ) 3θQ [u ](t, x ) which leas to (4.) as esire. (1 4θ)Q [u x u ](t, x ) (1 4θ)(u x u )(t, x ), 5. A etaile account of the case θ = 0 In this section, we establish the local well-poseness an present the precise blowup scenario an global existence results for the θ equation with θ = 0, i.e., u t u txx = u x u xx uu x. (5.1) Note that (1 x) 1 f = Q f for all f L () an Q m = u for m = u u xx. Using this relation, we can rewrite (5.1) as follows: { ut = x Q ( 1 u x 1 u ), t > 0, x, (5.) u(0, x) = u 0 (x), x, or in the equivalent form: { ut = x (1 x) 1 ( 1 u x 1 u ), t > 0, x, u(0, x) = u 0 (x), x. (5.3)

9 ON NONLOCAL DISPESIVE EQUATIONS 9 Theorem 5.1. Given u 0 H s (), s > 3, there exists a T = T ( u 0 s ) > 0, an a unique solution u to (5.1) such that u = u(, u 0 ) C([0, T ); H s ()) C 1 ([0, T ); H s ()). The solution epens continuously on the initial ata, i.e. the mapping u 0 u(, u 0 ) : H s () C([0, T ); H s ()) C 1 ([0, T ); H s ()) is continuous. Moreover, if T < then lim t T u(t, ) s =. Proof. Set f(u) = x Q ( 1 u x 1 u ) = x (1 x) 1 ( 1 u x 1 u ). Let u, v H s, s > 3. Note that H s 1 is a Banach algebra. Then, we have f(u) f(v) s ( 1 = x (1 x) 1 (u v ) + 1 ) (u x vx) s 1 (u v)(u + v) s (u x v x )(u x + v x ) s 1 (5.4) 1 u v s u + v s + 1 x(u v) s 1 u x + v x s 1 ( u s + v s ) u v s. This implies that f(u) satisfies a local Lipschitz conition in u, uniformly in t on [0, ). Next we show that for every t 0 0, u(t 0 ) H s (), the Cauchy problem (5.1) has a unique mil solution u on an interval [t 0, t 1 ] whose length is boune below by δ( u(t 0 ) s ) = u(t 0) s r (t 0 ) = 1 r(t 0 ), where r(t 0 ) = u(t 0 ) s. Set t 1 = t 0 + δ( u(t 0 ) s ). Let us efine by u C([t0,t 1 ];H s ()) := sup u s t [t 0,t 1] the norm of u as an element of C([t 0, t 1 ]; H s ()). For a given u(t 0 ) H s we efine a mapping F : C([t 0, t 1 ]; H s ()) C([t 0, t 1 ]; H s ()) by t (F u)(t) = u(t 0 ) + f(u(s)) s, t 0 t t 1. (5.5) t 0 The mapping F efine by (5.5) maps the ball of raius r(t 0 ) centere at 0 of C([t 0, t 1 ]; H s ()) into itself. This follows from the following estimate t (F u)(t) s u(t 0 ) s + f(u(s)) f(0) s s t 0 u(t 0 ) s + r (t 0 )(t t 0 ) u(t 0 ) s = r(t 0 ), (5.6) where we have use the relations (5.4)-(5.5), f(0) = 0 an the efinition of t 1. By (5.4) an (5.5), we have (F u)(t) (F v)(t) s r(t 0 )(t t 0 ) u v C([t0,t 1];H s ()). (5.7)

10 10 HAILIANG LIU AND ZHAOYANG YIN Using (5.5) an (5.7) an inuction on n, we obtain (F n u)(t) (F n v)(t) s (r(t 0)(t t 0 )) n u v C([t0,t n! 1];H s ()) (r(t 0)δ( u(t 0 ) s )) n u v C([t0,t n! 1 ];H s ()) 1 n! u v C([t 0,t 1 ];H s ()). (5.8) For n we have 1 n! < 1. Thus, by a well known extension of the Banach contraction principle, we know that F has a unique fixe point u in the ball of C([t 0, t 1 ]; H s ()). This fixe point is the mil solution of the following integral equation associate with Eq.(5.1): t u(t, x) = u(t 0, x) + x Q ( 1 t 0 u x 1 u )(τ, x)τ. (5.9) Next, we prove the uniqueness of u an the Lipschitz continuity of the map u(t 0 ) u. Let v be a mil solution to (5.1) on [t 0, t 1 ) with initial ata v(t 0 ). Note that u s u(t 0 ) s an v s v(t 0 ) s. Then u(t) v(t) s u(t 0 ) v(t 0 ) s + t t 0 f(u) f(v) s τ u(t 0 ) v(t 0 ) s + ( u(t 0 ) s + v(t 0 ) s ) An application of Gronwall s inequality yiels Therefore t t 0 u(t) v(t) s τ. u(t) v(t) s e ( u(t0) s+ v(t0) s)(t1 t0) u(t 0 ) v(t 0 ) s. (5.10) u v C([t0,t 1 ];H s ()) e ( u(t 0) s + v(t 0 ) s )(t 1 t 0 ) u(t 0 ) v(t 0 ) s, (5.11) which implies both the uniqueness of u an the Lipschitz continuity of the map u(t 0 ) u. From the above we know that if u is a mil solution of (5.1) on the interval [0, τ], then it can be extene to the interval [0, τ + δ] with δ > 0 by efining on [τ, τ + δ], u(t, x) = v(t, x) where v(t, x) is the solution of the following integral equation v(t, x) = u(τ) + t τ f(v(s, x))s, τ t τ + δ, where δ epens only on u(τ, ) s. Let T be the maximal existence time of the mil solution u of (5.1). If T < then lim t T u(t, s ) =. Otherwise there is a sequence t n T such that u(t n, ) s C for all n. This woul yiel that for each t n, near enough to T, u efine on [0, t n ] can be extene to [0, t n + δ] where δ > 0 is inepenent of t n. Thus u can be extene beyon T. This contraicts the efinition of T. Note that u C([0, T ); H s ()) an f(u) satisfies locally Lipschitz conitions in u, uniformly in t on [0, T ). Then we have that f(u(t, x)) is continuous in t. Thus it follows from (5.7)-(5.9) that u(t, x) C([0, T ); H s ()) C 1 ([0, T ); H s ()) is the solution to (5.1). This completes the proof of the theorem. Next, we present the precise blow-up scenario for solutions to Eq.(5.1). We first recall the following two useful lemmas.

11 ON NONLOCAL DISPESIVE EQUATIONS 11 Lemma 5.1. [31] If r > 0, then H r () L () is an algebra. Moreover fg r c( f L () g r + f r g L ()), where c is a constant epening only on r. Lemma 5.. [31] For Λ = (1 x) 1/. If r > 0, then [Λ r, f]g L () c( x f L () Λ r 1 g L () + Λ r f L () g L ()), where c is a constant epening only on r. Then we prove the following useful result. Theorem 5.. Let u 0 H s (), s > 3 be given an assume that T is the existence time of the corresponing solution to Eq.(5.1) with the initial ata u 0. If there exists M > 0 such that u x (t, x) L () M, t [0, T ), then the H s () norm of u(t, ) oes not blow up on [0, T ). Proof. Let u be the solution to Eq.(5.1) with initial ata u 0 H s (), s > 3, an let T be the maximal existence time of the solution u, which is guarantee by Theorem 5.1. Throughout this proof, c > 0 stans for a generic constant epening only on s. Applying the operator Λ s to Eq.(5.), multiplying by Λ s u, an integrating over, we obtain t u s = (u, f 1 (u)) s + (u, f (u)) s, (5.1) where f 1 (u) = x (1 x) 1 ( 1 u ) = (1 x) 1 (uu x ) an f (u) = x (1 x) 1 ( 1 u x). Let us estimate the first term of the right-han sie of Eq.(5.1). (f 1 (u), u) s = (Λ s (1 x) 1 (u x u), Λ s u) 0 (Λ s 1 (u x u), Λ s 1 u) 0 ([Λ s 1, u] x u, Λ s 1 u) 0 + (uλ s 1 x u, Λ s 1 u) 0 [Λ s 1, u] x u 0 Λ s 1 u (u xλ s 1 u, Λ s 1 u) 0 (c u x L () + 1 u x L ()) u s 1 c u x L () u s. (5.13) Here, we applie Lemma 5. with r = s 1. Then, let us estimate the secon term of the right-han sie of (5.1). (f (u), u) s f (u) s u s 1 u x s 1 u s c( u x L () u s 1 ) u s c u x L () u s, (5.14) where we use Lemma 5.1 with r = s. Combining inequalities (5.13)-(5.14) with (5.1), we obtain t u s cm u s. An application of Gronwall s inequality yiels This completes the proof of the theorem. u(t) s exp (cmt) u(0) s. (5.15)

12 1 HAILIANG LIU AND ZHAOYANG YIN We now present the precise blow-up scenario for Eq.(5.1). Theorem 5.3. Assume that u 0 H s (), s > 3. Then the solution to Eq.(5.1) blows up in finite time if an only if the slope of the solution becomes unboune from below in finite time. Proof. Applying Theorem 5.1 an a simple ensity argument, it suffices to consier the case s = 3. Let T > 0 be the maximal time of existence of the solution u to Eq.(5.1) with initial ata u 0 H 3 (). From Theorem 5.1 we know that u C([0, T ); H 3 ()) C 1 ([0, T ); H 3 ()). Multiplying Eq.(5.1) by u an integrating by parts, we get (u + u t x)x = uu x u xx x u u x x = u x u xx. (5.16) Differentiating Eq.(5.1) with respect to x, then multiplying the obtaine equation by u x an integrating by parts, we obtain (u x + u t xx)x = u x u xxx + uu x u xx x (5.17) = u x u xx u 3 xx. Summing up (5.16) an (5.17), we have (u + u x + u t xx)x = u x (u x + u xx)x. (5.18) If the slope of the solution is boune from below on [0, T ), then there exists M > 0 such that t u M u. By means of Gronwall s inequality, we have u(t, ) u(0, ) exp{mt}, t [0, T ). By Theorem 5., we see that the solution oes not blow up in finite time. On the other han, by Theorem 5.1 an Sobolev s imbeing theorem, we see that if the slope of the solution becomes unboune from below in finite time, then the solution will blow up in finite time. This completes the proof of the theorem. emark 5.1. Theorem 5.3 shows that (5.1) has the same blow-up scenario as the Camassa-Holm equation [6, 45] an the Degasperis-Procesi equation [46] o. Finally, we show that there exist global strong solutions to Eq.(5.1) provie the initial ata u 0 satisfies certain sign conitions. Lemma 5.3. Assume that u 0 H s (), s > 3. Let T > 0 be the existence time of the corresponing solution u to (5.1). Then we have m(t, x) = m 0 (x) exp t 0 u x(τ, x) τ, (5.19) where (t, x) [0, T ) an m = u u xx. Moreover, for every (t, x) [0, T ), m(t, x) has the same sign as m 0 (x) oes. Proof. Let T > 0 be the maximal existence time of the solution u with initial ata u 0 H s (). Due to u(t, x) C 1 ([0, T ); H s ()) an H s () C(), we see that the function u x (t, x) are boune, Lipschitz in the space variable x, an of class C 1 in time. For arbitrarily fixe T (0, T ), Sobolev s imbeing theorem implies that sup u x (s, x) <. (s,x) [0,T ]

13 ON NONLOCAL DISPESIVE EQUATIONS 13 Thus, we infer from the above inequality that there exists a constant K > 0 such that e t 0 ux(τ, x) τ e tk > 0 for (t, x) [0, T ]. (5.0) By Eq.(5.1) an m = u u xx, we have This implies that m t (t, x) = u x (t, x)m(t, x). (5.1) m(t, x) = m 0 (x) exp t 0 u x(τ, x) τ. By (5.0), we see that for every (t, x) [0, T ), m(t, x) has the same sign as m 0 (x) oes. This completes the proof of the lemma. Lemma 5.4. Let u 0 H s (), s > 3 be given. If m 0 := (u 0 u 0,xx ) L 1 (), then, as long as the solution u(t, ) to Eq.(5.1) with initial ata u 0 given by Theorem 5.1 exists, we have u(t, x)x = u 0 x = m 0 x = m(t, x)x. Proof. Again it suffices to consier the case s = 3. Let T be the maximal time of existence of the solution u to Eq.(5.1) with initial ata u 0 H 3 (). Note that u 0 = Q m 0 an m 0 = (u 0 u 0,xx ) L 1 (). By Young s inequality, we get u 0 L 1 () = Q m 0 L 1 () Q L 1 () m 0 L 1 () m 0 L 1 (). Integrating Eq.(3.) by parts, we get ux = x Q ( 1 t u x 1 u )x = 0. It then follows that u x = u 0 x. Due to m = u u xx, we have m x = u x u xx x = u x = u 0 x = This completes the proof of the lemma. We now present the first global existence result. u 0 x u 0,xx x = m 0 x. Theorem 5.4. Let u 0 H s () s > 3 be given. If m 0 := u 0 xu 0 L 1 () is nonnegative, then the corresponing solution to Eq.(5.) is efine globally in time. Moreover, I(u) = u x is a conservation law, an that for all (t, x) +, we have (i) m(t, x) 0, u(t, x) 0 an m 0 L 1 () = m(t) L 1 () = u(t, ) L 1 () = u 0 L 1 (). (ii) u x (t, ) L () u 0 L 1 () an u(t, ) L () 1 u(t, ) 1 e t u0 L1 () u0 1.

14 14 HAILIANG LIU AND ZHAOYANG YIN Proof. As we mentione before that we only nee to prove the above theorem for s = 3. Let T > 0 be the maximal existence time of the solution u with initial ata u 0 H 3 (). If m 0 (x) 0, then Lemma 5.3 ensures that m(t, x) 0 for all t [0, T ). Noticing that u = Q m an the positivity of Q, we infer that u(t, x) 0 for all t [0, T ). By Lemma 5.4, we obtain u x (t, x) + x u(t, x)x = = x x (u u xx )x mx mx = m 0 x = u 0 x. (5.) Therefore, from (5.) we fin that u x (t, x) u 0 x = u 0 L 1 (), (t, x) [0, T ). (5.3) On the other han, by m(t, x) 0 for all t [0, T ), we obtain u x (t, x) x u x = x (u u xx ) x = x m x 0. By the above inequality an u(t, x) 0 for all t [0, T ), we get x u x (t, x) u x u x = u 0 x = u 0 L 1 (). (5.4) Thus, (5.3) an (5.4) imply that u x (t, x) u x (t, ) L () u 0 L 1 () (t, x) [0, T ). (5.5) By Theorem 5. an the above inequality, we euce that T =. ecalling finally Lemma 5.4, we get assertion (i). Multiplying (5.1) by u an integrating by parts, we obtain 1 ( u (t, x) + u t x(t, x) ) x = (uu x u xx u u x )x = 1 u 3 xx 1 (5.6) u x(t, ) L () u xx. An application of Gronwall s inequality leas to ( u (t, x) + u x(t, x) ) x e t u 0 L 1 () Consequently, On the other han, u (t, x) = x uu x x ( u 0 + u ) 0,x x. (5.7) u(t, ) 1 e t u0 L 1 () u0 1. (5.8) x uu x x 1 (u + u x)x = 1 u(t, ) 1. (5.9) Combining (5.8) with (5.9), we obtain assertion (ii). This completes the proof of the theorem. In a similar way to the proof of Theorem 5.4, we can get the following global existence result.

15 ON NONLOCAL DISPESIVE EQUATIONS 15 Theorem 5.5. Let u 0 H s () s > 3 be given. If m 0 := u 0 xu 0 L 1 () is non-positive, then the corresponing solution to Eq.(5.1) is efine globally in time. Moreover, I(u) = u x is invariant in time, an that for all (t, x) +, we have (i) m(t, x) 0, u(t, x) 0 an m 0 L1 () = m(t) L1 () = u(t, ) L1 () = u 0 L1 (). (ii) u x (t, ) L () u 0 L 1 () an u(t, ) L () u(t, ) 1 6. Global weak solutions e t u0 L1 () u0 1. In this section, we present some results on global weak solutions to characterize peakon solutions to (1.1) for any θ provie initial ata satisfy certain sign conitions. Let us first introuce some notations to be use in the sequel. We let M() enote the space of aon measures on with boune total variation. The cone of positive measures is enote by M + (). Let BV () stan for the space of functions with boune variation an write V(f) for the total variation of f BV (). Finally, let {ρ n } n 1 enote the mollifiers ( 1 ρ n (x) := ρ(ξ)ξ) nρ(nx), x, n 1, where ρ C c () is efine by ρ(x) := {e 1 x 1, for x < 1, 0, for x 1. Note that the b equation for any b has peakon solutions with corners at their peaks, cf. [17, 8, 9]. Thus, the θ equation for any θ has also peakon solutions, see Example 6.1 below. Obviously, such solutions are not strong solutions to (1.1) for any θ. In orer to provie a mathematical framework for the stuy of peakon solutions, we shall first give the notion of weak solutions to (1.1). Equation (1.1) can be written as u t + θuu x + x (1 x) 1 B(u, u x ) = 0, B = (1 θ) u + (4θ 1)u x. If we set F (u) := [(1 θu + Q θ) u + (4θ 1)u x then the above equation takes the conservative form u t + F (u) x = 0, u(0, x) = u 0, t > 0, x. (6.1) In orer to introuce the notion of weak solutions to (6.1), let ψ C 0 ([0, T ) ) enote the set of all the restrictions to [0, T ) of smooth functions on with compact support containe in ( T, T ). Definition 6.1. Let u 0 H 1 (). If u belongs to L loc ([0, T ); H1 ()) an satisfies the following ientity T (uψ t + F (u)ψ x )xt + u 0 (x)ψ(0, x)x = 0 0 for all ψ C 0 ([0, T ) ), then u is calle a weak solution to (6.1). If u is a weak solution on [0, T ) for every T > 0, then it is calle a global weak solution to (6.1). ],

16 16 HAILIANG LIU AND ZHAOYANG YIN The following proposition is stanar. Proposition 6.1. (i) Every strong solution is a weak solution. (ii) If u is a weak solution an u C([0, T ); H s ()) C 1 ([0, T ); H s 1 ()), s > 3, then it is a strong solution. eferring to an approximation proceure use first for the solutions to the Camassa- Holm equation [11], a partial integration result in Bochner spaces [39] an Helly s theorem [40] together with the obtaine global existence results an two useful a priori estimates for strong solutions, e.g., Theorem 1.1 an Theorems , we may obtain the following uniqueness an existence results for the global weak solution to (6.1) for any θ provie the initial ata satisfy certain sign conitions. Theorem 6.1. Let u 0 H 1 () be given. Assume that (u 0 u 0,xx ) M + (). Then (6.1) for any θ has a unique weak solution with initial ata u(0) = u 0 an is uniformly boune for all t +. u W 1, ( + ) L loc( + ; H 1 ()) (u(t, ) u xx (t, )) M + () In the following, we only present main steps of the proof of the theorem, a refine analysis coul be one following those given in [4] for the b equation. A sketch of existence proof of weak solutions Step 1. Given u 0 H 1 () an m 0 := u 0 u 0,xx M + (). Then one can show that u 0 L 1 () m 0 M + (). Let us efine u n 0 := ρ n u 0 H () for n 1. Note that for all n 1, m n 0 := u n 0 u n 0,xx = ρ n (m 0 ) 0. By Theorem 1.1 an Theorem 5.4, we obtain that there exists a unique strong solution to (6.1), u n = u n (., u n 0 ) C([0, ); H s ()) C 1 ([0, ); H r 1 ()), s 3. Step. By a priori estimates in Theorem 1.1 an Theorem 5.4, Young s inequality an energy estimate for (6.1), we may get T ([u n (t, x)] + [u n x(t, x)] + [u n t (t, x)] )xt M, (6.) 0 where M is a positive constant epening only on θ, T, Q x L (), an u 0 1. It then follows from (6.) that the sequence {u n } n 1 is uniformly boune in the space H 1 ((0, T ) ). Thus, we can extract a subsequence such that an u n k u weakly in H 1 ((0, T ) ) for n k (6.3) u n k u a.e. on (0, T ) for n k, (6.4) for some u H 1 ((0, T ) ). From Theorem 1.1 (i) an the fact u n 0 1 u 0 1 we see that for any fixe t (0, T ), the sequence u n k x (t, ) BV () satisfies V[u n k x (t, )] m 0 M().

17 ON NONLOCAL DISPESIVE EQUATIONS 17 Step 3. This, when applying Helly s theorem, cf. [40], enables us to conclue that there exists a subsequence, enote still by {u n k x (t, )}, which converges to the function u x (t, ) for a.e. t (0, T ). A key energy estimate is of the form B(u n, u n x) L () C( u 0 1 ), which ensures B amits a weak limit. This when combine with the fact that (u n, u n x) converges to (u, u x ) as well as Q x L leas to the assertion that u satisfies (6.1) in istributional sense. Step 4. From equation (6.1) we see that u n k t (t, ) is uniformly boune in L (), an u n k (t, ) 1 is uniformly boune for all t (0, T ). This implies that the map t u n k t (t, ) H 1 () is weakly equi-continuous on [0, T ]. ecalling the Arzela- Ascoli theorem an a priori estimates in Theorem 1.1 an Theorem 5.4, we may prove u L loc( + ) L loc( + ; H 1 ()) an u x L ( + ). Step 5. Since u solves (6.1) in istributional sense, we have ρ n u t + ρ n (θuu x ) + ρ n x Q ((1 θ) u + (4θ 1)u x ) = 0, for a.e. t +. Integrating the above equation with respect to x on an then integrating by parts, we obtain ρ n u x = 0. t By a partial integration result in Bochner spaces [39] an Young s inequality, we may prove that u(t, )x = lim ρ n u(t, )x = lim ρ n u 0 x = u 0 x. n n Using the above conservation law, we get u(t, ) u xx (t, ) M() u(t, ) L 1 () + u xx (t, ) M() u 0 L 1 () + m 0 M() 3 m 0 M(), for a.e. t +. Note that u n k (t, x) u n k xx(t, x) 0 for all (t, x) +. Then the above inequality implies that (u(t, ) u xx (t, )) M + () for a.e. t +. Since u(t, x) = Q (u(t, x) u xx (t, x)), it follows that u(t, x) = Q (u(t, x) u xx (t, x)) Q L () u(t, ) u xx (t, ) M() 3 m 0 M(). This shows that u(t, x) W 1, ( + ) in view of Step 4. This proves the existence of global weak solutions to (6.1). Uniqueness of the weak solution Let u, v W 1, ( + ) L loc( + ; H 1 ()) be two global weak solutions of (6.1) with initial ata u 0. Set N := sup t + { u(t, ) u xx (t, ) M() + v(t, ) v xx (t, ) M() }. From Step 5, we know that N <. Let us set w(t, ) = u(t, ) v(t, ), (t, x) +,

18 18 HAILIANG LIU AND ZHAOYANG YIN an fix T > 0. Convoluting Eq.(6.1) for u an v with ρ n an with ρ n,x respectively, using Young s inequality an following the proceure escribe on page in [11], we may euce that t an t ρ n w x = C ρ n w x + C ρ n w x x + n (t), (6.5) ρ n w x x = C ρ n w x + C ρ n w x x + n (t), (6.6) for a.e. t [0, T ] an all n 1, where C is a generic constant epening on θ an N, an that n (t) satisfies { lim n n(t) = 0 n (t) K(T ), n 1, t [0, T ]. Here K(T ) is a positive constant epening on θ, T, N an the H 1 ()-norms of u(0) an v(0). Summing (6.5) an (6.6) an then using Gronwall s inequality, we infer that ( ρn w + ρ n w x ) t (t, x) x e C(t s) n (s)s+ 0 e Ct ( ρn w + ρ n w x ) (0, x) x, for all t [0, T ] an n 1. Note that w = u v W 1,1 (). Using Lebesgue s ominate convergence theorem, we may euce that for all t [0, T ] ( w + wx ) (t, x) x e Ct ( w + wx ) (0, x) x. Since w(0) = w x (0) = 0, it follows from the above inequality that u(t, x) = v(t, x) for all (t, x) [0, T ]. This proves the uniqueness of the global weak solution to (6.1). In a similar way to the proof of Theorem 6.1, we can get the following result. Theorem 6.. Let u 0 H 1 () be given. Assume that (u 0,xx u 0 ) M + (). Then (6.1) for any θ has a unique weak solution with initial ata u(0) = u 0 an is uniformly boune for all t +. u W 1, ( + ) L loc( + ; H 1 ()) (u xx (t, ) u(t, )) M + () emark 6.1. Theorems cover the recent results for global weak solutions of the Camassa-Holm equation in [11] an the Degasperis-Procesi equation in [47]. Example 6.1. (Peakon solutions) Consier (1.1) for any θ. Given the initial atum u 0 (x) = ce x, c. A straightforwar computation shows that an u 0 u 0,xx = c δ(x) M + () if c 0 u 0,xx u 0 = c δ(x) M + () if c < 0.

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21 ON NONLOCAL DISPESIVE EQUATIONS 1 [46] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (003), [47] Z. Yin, Global solutions to a new integrable equation with peakons, Iniana Univ. Math. J., 53 (004), [48] Z. Yin, Global weak solutions to a new perioic integrable equation with peakon solutions, J. Funct. Anal., 1 (004), Iowa State University, Mathematics Department, Ames, IA aress: hliu@iastate.eu Department of Mathematics, Sun Yat-sen University, 51075, Guangzhou, China aress: mcsyzy@mail.sysu.eu.cn