14th Panhellenic Conference of Mathematical Analysis University of Patras May 18-19, 2012

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1 14th Panhellenic Conference of Mathematical Analysis University of Patras May 18-19, 2012 The Initial Value Problem for integrable Water Wave Equations in Sobolev and Analytic Spaces Alex Himonas University of Notre Dame

2 Abstract Focusing on the Korteweg-de Vries (KdV) and the Camassa- Holm (CH) equations we shall present a few ideas and techniques for studying the initial value problem in Sobolev and analytic space of such type of equatios. KdV was derived in 1895 as a model of long water waves propagating in a channel. Since then it has reappeared in many other areas of mathematics and its applications and has been a favorite subject of study by mathematicians of all kinds. It possesses traveling wave solutions (solitons) having many interesting properties. However they never break! The search for a water wave equation having traveling wave solutions that break led to the discovery of the CH equation. 1

3 Talk Outline Integrable equations Camassa-Holm type equations Well-posedness in Sobolev Spaces Non-uniform dependence on initial data The Euler Equations Ill-posedness in Sobolev Spaces Analytic initial data References 2

4 Question. What is the time evolution of this wave? 3

5 Can the motions of such a wave be modeled by a solution to an evolution equation like the Korteweg-de Vries (KdV) t u + u x u + 3 xu = 0, or the Camassa-Holm (CH) equation u t u xxt + 3uu x 2u x u xx uu xxx = 0? 4

6 Braking wave (Wikipedia) 5

7 Integrable equations Integrable equations possess special properties, like: Infinite hierarchy of higher symmetries (Novikov s test), Have a Lax Pair, Infinitely many conserved quantities, A bi-hamiltonian formulation, Can be solved by the Inverse Scattering Method. 6

8 Camassa-Holm type equations [Novikov: J. Phys. A, 2009] These are integrable equations of the form (1 2 x)u t = F (u, u x, u xx, u xxx, ) (1) where F is a polynomial of u and its x-derivatives. Definition of integrability: Existence of an infinite hierarchy of (quasi-) local higher symmetries. 7

9 CH equations with quadratic nonlinearities Theorem A. [Novikov, 2009] If the equation (1 ε x)u 2 t = c 1 uu x + ε[c 2 uu xx + c 3 u 2 x] + ε 2 [c 4 uu xxx + c 5 u x u xx ] + ε 3 [c 6 uu xxxx + c 7 u x u xxx + c 8 u 2 xx] + ε 4 [c 9 uu xxxxx + c 10 u x u xxxx + c 11 u xx u xxx ] is integrable then up to rescaling is one of the following 10: 1 Camassa-Holm (CH): (1 2 x )u t = 3uu x + 2u x u xx + uu xxx 2 Degasperis-Procesi (DP): (1 2 x)u t = 4uu x +3u x u xx +uu xxx

10 CH equations with cubic nonlinearities Theorem C. [Novikov, 2009] If the equation (1 ε x)u 2 t = c 1 u 2 u x + ε[c 2 u 2 u xx + c 3 uu 2 x] + ε 2 [c 4 u 2 u xxx + c 5 uu x u xx + c 6 u 3 x] + ε 3 [c 7 u 2 u xxxx + c 8 uu x u xxx + c 9 uu 2 xx + c 10 u 2 xu xx ] + ε 4 [c 11 u 2 u xxxxx + c 12 uu x u xxxx + c 13 uu xx u xxx + c 14 u 2 xu xxx + c 15 u x u 2 xx] is integrable then up to rescaling is one of the following 10: 1 Novikov equation: (1 2 x)u t = 4u 2 u x + 3uu x u xx + u 2 u xxx 2 Fokas equation: (1 2 x)u t = x ( u 3 + uu 2 x + u 2 u xx u 2 xu xx )

11 The Degasperis-Procesi Equation The Degasperis-Procesi (DP) equation, is also written as u t u xxt + 4uu x 3u x u xx uu xxx = 0, (2) This equation was introduced in the 1999 paper of Degasperis and Procesi, Asymptotic integrability symmetry and perturbation theory, as one of three equations to satisfy asymptotic integrability conditions for the following familty of third order dispersive PDE conservation laws u t + c 0 u x + γu xxx α 2 u txx = (c 1 u 2 + c 2 u 2 x + c 3 uu xx ) x, (3) where α, c 0, c 1, c 2, c 3 R are fixed constants. The other two equations arriving out of this machinery are the CH and KdV equations. 10

12 Nonlocal form of DP The Cauchy problem for the Degasperis-Procesi (DP) equation, written in its nonlocal form, is given by u t + uu x xd 2[ u 2] = 0, D 2. = (1 2 x) 1, (4) u(x, 0) = u 0 (x). (5) 11

13 CH and the Fokas-Fuchssteiner Algorithm If for every n the operator θ 1 + nθ 2 is Hamiltonian (skew symmetric and satisfies a properly defined Jacobi identity), then each member of the hierarchy q t = (θ 2 θ 1 1 )q x, (6) is an integrable equation. Letting θ 1 =. = x, and θ 2 = + γ 3 + α (q + q), 3 α, γ constants, we obtain the celebrated KdV equation q t + u x + γq xxx + αqq x = 0. (7) 12

14 Fokas-Fuchssteiner Algorithm (cont.) Similarly, letting θ 1 = + ν 3 and θ 2 = + γ 3 + α (q + q), 3 α, γ constants, and q = u + νu xx, we obtain the CH equation u t + u x + νu xxt + γu xxx + αuu x + αν 3 (uu (8) xxx + 2u x u xx ) = 0. This equation was derived physically as a water wave equation by Camassa and Holm (1993) who also studied its peakon solutions. 13

15 Nonlocal form of CH Choosing ν = 1, α = 3, γ = 0 and removing u x by a change of variables gives the standard local form of CH u t u xxt + 3uu x 2u x u xx uu xxx = 0, (9) which can be written in the form (1 2 x) t u + (1 2 x)[u x u] + x [ u ( xu) 2] = 0. Multiplying by (1 2 x) 1 gives CH as a nonlocal weakly dispersive perturbation of Burgers t u + u x u+(1 2 x) 1 x [ u ( xu) 2] = 0. (10) which is worth comparing it with the KdV t u + u x u+ 3 xu = 0. (11) thinking it as a strongly dispersive perturbation of Burgers. 14

16 Common phenomenon of CH, DP and NE They all have peakon solutions: u(x, t) = ce x ct The discovery of the CH equation by Camassa-Holm [1993] (Fokas-Fuchssteiner 1981) was partly driven by the desire to find a water wave equation which has traveling wave solutions that break. The Korteweg-de Vries equation (KdV), t u + 6u x u + 3 xu = 0, (12) which was derived in 1895 as a model of long water waves propagating in a channel has only smooth solitons. Also, CH, DP and NE have multipeakon solutions: u(x, t) = n j=1 p j (t)e x q j(t) 15

17 KdV Soliton: u(x, t) = f(x ct) 0.8 f(x) = c ( ) c 2 sech2 2 x

18 CH Peakon: u(x, t) = f(x ct) f(x) = ce x 17

19 Conserved Quantities CH, DP and NE have -many conserved quantities, which among other things are used for proving global solutions. Main conserved quantity by CH and NE is the H 1 -norm: u. = [u 2 + u 2 x]dx While, the main conserved quantity by DP is a twisted L 2 -norm: u 2 L 2. = (1 2 x)u (4 2 x) 1 u dx 18

20 The Cauchy problem for NE: Nonlocal form In its nonlocal form the Cauchy problem for NE is t u x(u 3 ) + F (u) = 0 (13) u(x, 0) = u 0 (x), u H s, (14) where F (u) =. [ D 2 x u )] [ ] (u( x u) 2 + D ( xu) 3 (15) and D 2 = (1 2 x) 1. Observe that F (u) H s for any u H s. Thus, applying a Galerkin-type approximation we prove the following local wellposedness result. 19

21 Local well-posedness for NE Theorem 1. [H. Holliman, Nonlinearity 2012] If s > 3/2 and u 0 H s then there exists T > 0 and a unique solution u C ( [0, T ]; H s) of the initial value problem for NE, which depends continuously on the initial data u 0. Furthermore, we have the estimate u(t) H s 2 u 0 H s, for 1 0 t T 4c s u 0 2, H s (16) where c s > 0 is a constant depending on s. Periodic case and s > 5/2 done by Tiglay [IMRN 2011]. Our Proof begins by solving the mollified i.v.p.... t u + J ε [(J ε u) 2 x J ε u ] + F (u) = 0, u(x, 0) = u 0 (x). (17) 20

22 Non-uniform dependence for NE Theorem 2. [H. Holliman, Nonlinearity 2012] If s > 3/2 then the solution map u 0 u(t) for the NE equation is not uniformly continuous from any bounded set of H s (R) into C([0, T ]; H s (R)). Remark. The analogous to Theorem 2 result has been prooved: For CH on the line by [H. Kenig, Diff. Int. Eqns 2009] For CH on the circle by [H. Kenig-Misiolek, CPDE 2010] For DP on the line and the circle by [H. Holliman, DCDS 2011] 21

23 Idea of Proof We shall prove that there exist two sequences of NE solutions u n (t) and v n (t) in C([0, T ]; H s (R)) such that: 1 sup n u n (t) H s + sup n v n (t) H s 1, 2 lim n u n (0) v n (0) H s = 0, 3 lim inf n u n (t) v n (t) H s sin t, 0 t < T 1, 22

24 Approximate solutions for CH and DP The approximate solutions u ω,n (x, t) = ωn 1 + n s cos(nx ωt), for ω = 1, 1, (18) where n Z +, satisfy conditions (1)-(3) for non-uniform dependence of the periodic CH and DP equations but they are not solutions. However, the error is small. E = CH(u ω,n ) or E = DP (u ω,n ) For the Benjamin-Ono equation this method was used by H. Koch and N. Tzvetkov in IMRN (2005). 23

25 Approximate solutions motivation (from Euler eqns) For any ω R and n Z + the divergence free vector field u ω,n (t, x) = ( ωn 1 + n s cos(nx 2 ωt), ωn 1 + n s cos(nx 1 ωt) ) is a solution! to the Euler equations on T 2. The corresponding to ω ± 1 sequences u +1,n (t, x) and u 1,n (t, x) Satisfy conditions (1)-(3) for non-uniform dependence of the periodic Euler equations in 2-D. Remark. Non-uniform dependence for the Euler equations in n-d was proved in [H. Misio lek, CMP 2010]. 24

26 They are NE approximate solutions on the circle u ω,n = ωn 1/2 + n s cos(nx ωt), for ω = 0, 1. (19) Note. One of them (ω = 0) has no low frequency. Unlike in the case of CH and DP they are asymmetric. 25

27 Approximate Solutions on the Line Again we will construct the u ω,n as a high-low frequency combination. We will take the cutoff function ϕ(x) = 1, if x < 1, 0 if x 2. (20) High Frequency Part: Is similar to the high frequency of the periodic approximate solution. We localize using a cutoff ϕ and introduce a parameter δ to help control decay in later estimates. u h = u h,ω,n = n δ/2 s ϕ( x nδ) cos(nx ωt). (21) Note: In periodic case δ = 0 and ϕ = 1 26

28 Approximate Solutions on the Line Low Frequency Part: While on the circle it is simply ωn 1/2, on the line it solves NE with initial data ωn 1/2 ϕ( x nδ). That is, u l = u l,ω,n is defined by where we have ϕ C 0 (u l ) t + (u l ) 2 (u l ) x + F (u l ) = 0, (22) u l (x, 0) = ωn 1/2 ϕ( x nδ), (23) (R) and ϕ(x) = 1, if x supp ϕ. (24) Now that we have approximate solutions, we can define actual solutions and follow the same program as in the periodic case. 27

29 Approximate Solutions and Error Defining the NE approximate solution by u ω,n = u l + u h ω = 0 or 1. (25) we show that the error E =. t u ω,n + (u ω,n ) 2 x u ω,n + F (u ω,n ). (26) satisfies the following estimate. Lemma. Let s > 3/2 and 1/4 < δ < 1. Then, with E(t) H σ n rs, for n 1, (27) r s. = s + 1 σ 2δ > 0. (28) Note: Later we will choose σ =

30 NE Actual Solutions and their difference Actual solutions solve the Novikov s equation with initial data u ω,n (0) = u ω,n (0). (29) Then the difference v =. u ω,n u ω,n. (30) satisfies the initial value problem v t = E 1 3 x(vw) F (u ω,n ) + F (u ω,n ), (31) v(0) = 0, (32) where w = (u ω,n ) 2 + u ω,n u ω,n + (u ω,n ) 2. (33) 30

31 Size of difference is small Proposition. If s > 3/2 and 1/4 < δ < 1, then where r s = s + 1 σ 2δ > 0. v(t) H σ n rs, 0 t T, (34) Proof. The last i.v.p. gives the following identity for v 1 d 2dt v(t) H R σ = Dσ E D σ vdx 1 3 R Dσ x (wv)d σ vdx (35) [ ] F (u ω,n ) F (u ω,n ) D σ vdx. R Dσ The first term on the right hand side of (35) is estimated by applying Cauchy-Schwarz R Dσ ED σ vdx E H σ v H σ (36) 31

32 Second Term of (35) We begin by rewriting this term by commuting w with D σ x to arrive at R Dσ x (vw)d σ vdx = R [Dσ x, w]vd σ vdx + R wdσ x vd σ vdx (37) The first integral can be handled by the following Calderon- Coifman-Meyer type commutator estimate that can be found in H.-Kenig-Misiolek [CPDE 2010]. Lemma. If σ then [D σ x, w]v L 2 C w H ρ v H σ (38) provided that ρ > 3/2 and σ + 1 ρ. 32

33 Second Term of (35) (cont.) Applying this lemma and the solution size estimate of the wellposedness theorem we have R [Dσ x, w]vd σ vdx w H s v 2 H σ v 2 H σ. (39) Next integrating by parts and using the Sobolev lemma we have R wdσ x vd σ vdx x w L R (Dσ v) 2 dx v 2 H σ. (40) 33

34 Third Term of (35) We use the following multiplier estimate that can be found in H.-Kenig-Misiolek [CPDE 2010]. Lemma. If σ (1/2, 1) then fg H σ 1 f H σ 1 g H σ, (41) Then, one can prove that for any u and w in H σ, we have F (u) F (w) H σ ( u H σ+1 + w H σ+1) 2 u w H σ. Therefore R Dσ [ ] F (u ω,n ) F (u ω,n ) D σ vdx v 2 H σ. (42) 34

35 End of proposition s proof Putting these results together we arrive at the differential inequality 1 d 2dt v(t) 2 H σ E H σ v H σ + v 2 H σ. (43) Solving this inequality gives v H σ E H σ n r. (44) 35

36 H s norm of the difference is small By the well-posedness theorem obtain the following estimate for the H k -norm of the difference of u ω,n and u ω,n v H k = u ω,n (t) u ω,n (t) H k n k s, 0 t T. (45) Interpolating between with s 1 = σ and s 2 = [s] + 2 = k gives v(t) H s v(t) (k s)/(k σ) H σ v(t) (s σ)/(k σ) H k n ( rs)[(k s)/(k σ)] n (k s)[(s σ)/(k σ)] n (1 2δ)[(k s)/(k σ)]. From the last inequality we obtain that where ε s is given by u ω,n (t) u ω,n (t) H s (R) n εs, 0 t T, (46) ε s = (1 2δ)/(s + 2) > 0, if δ < 1 2. (47) 36

37 End of Proof of Theorem 2 on the Line Behavior at time zero. We have u 1,n (0) u 0,n (0) H s n ( 1+δ)/2 ϕ H s 0 as n. Behavior at time t > 0. We have u 1,n (t) u 0,n (t) H s u 1,n (t) u 0,n (t) H s u 1,n (t) u 1,n (t) H s u 0,n (t) u 0,n (t) H s. Observe that for the second and third terms we have (48) lim n u1,n (t) u 1,n (t) H s = lim u 0,n (t) u n 0,n (t) H s = 0 (49) 37

38 For the remaining term observe lim inf n u 1,n(t) u 0,n (t) H s lim inf n u1,n (t) u 0,n (t) H s. (50) Therefore, the inequality lim inf n u1,n (t) u 0,n (t) H s 1 ϕ 2 L 2 (R) sin t, (51) completes the proof of Theorem 2 on the line. 38

39 Euler Equations 39

40 Non-uniform dependence for the Euler equations Theorem (H. Misio lek, CMP 2010) Let s R. The solution map u 0 u of the Cauchy problem for the Euler equations t u + u u + p = 0 (52) div u = 0 u(0, x) = u 0 (x), x T n, t R (53) is not uniformly continuous from any bounded subset in H s (T n, R n ) into C ( [0, T ], H s (T n, R n ) ). (Here n = 2 or 3 and T is the lifespan.) The same is true R n if s > 0. 40

41 Non-local form Applying the divergence operator to the first equation and then solving for p gives p = 1 div u u = (1 P ) u u (54) where P is the L 2 -orthogonal projection onto divergence free part in the Hodge decomposition of vector fields on Ω into divergence free fields and gradients of functions in Ω. Using (54) the first of the equations in (52) takes the form t u + u u 1 div u u = 0. (55) 41

42 Notation For any s R we shall equip the Sobolev space of vector-valued distributions H s (Ω, R n ) with the norm n Z u H s (Ω,R n ) = n j=1 u j H s (Ω) (56) where f H s (Ω) is the standard Sobolev norm for functions on Ω defined either as ( ( 1+n 2 ) s ˆf(n) 2) 1/2 ( ( or 1+ ξ 2 ) s ˆf(ξ) 2 ) 1/2 dξ depending on the context. We shall also often use the symbols and to denote estimates that hold up to some universal constant. R n 42

43 Local Well-posedness for Euler equations Theorem [Established by many authors: Ebin and Marsden,...] If s > n/2 + 1 and u 0 H s (Ω, R n ) is a divergence free vector field then there exists T > 0 and a unique solution u C ( [0, T ], H s (Ω, R n ) ) of the Cauchy problem (52) (53) which depends continuously on the initial data u 0. Furthermore, we have the estimate u(t) H s u 0 H s 1 Ct u 0 H s for where C > 0 is a constant depending on s. The proof can be found for example in: 0 t T < C 1 u 0 1 H s T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988),

44 Idea of Proof Construct two sequences of solutions such that: Their difference convergences to zero in H s -norm at time t = 0 Stay apart in H s -norm at later time t > 0 Remain in bounded set of H s for all 0 t T Remark. n = 2 suffices! 44

45 Proof in T 2 : Explicit construction For any ω R and n Z + the divergence free vector field u ω,n (t, x) = ( ωn 1 + n s cos(nx 2 ωt), ωn 1 + n s cos(nx 1 ωt) ) is a solution! to the Euler equations on T 2. Pick two sequences corresponding to ω ± 1: u +1,n (t, x) and u 1,n (t, x) Convergence at t=0: u 1,n (0) u 1,n (0) H s (T 2,R 2 ) 2n

46 Proof in T 2 (cont.) Boundedness t: since and u ω,n (t) H s (T 2,R 2 ) = uω,n 1 (t) H s (T 2 ) + uω,n 2 (t) H s (T 2 ) = ωn 1 + n s cos(n ωt) H s (dx 2 ) + ωn 1 + n s cos(n ωt) H s (dx 1 ) n n s cos(n ωt) H s (T) 1 n s sin(n ωt) H s (T) 1. 46

47 Proof in T 2 (cont.) Separation at t > 0: u 1,n (t) u 1,n (t) H s (T 2,R 2 ) = 2 2 cos(n t) cos(n +t) + n n s cos(n t) cos(n +t) n s = 2n s sin t sin n( ) H s 4 n H s 1 n H s sin t 1 n sin t 47

48 Proof in R 2 The method of proof is based on: Approximate solutions and Well-posdness estimates but Is technical due to localization. 48

49 The Degasperis-Procesi equation 49

50 Ill-posedness for Degasperis-Procesi equation Theorem 3 [H. Holliman-Grayshan] For s < 3/2 the Cauchy problem for DP u t + uu x (1 2 x) 1 x [ u 2 ] = 0, (57) u(x, 0) = u 0 (x), x R or T and t R, (58) is not well-posed in H s in the sense of Hadamard. Remark. This is true for CH too. 50

51 Well-posedness in the sense of Hadamard: 1.) For any initial data u(0) H s there exists T = T u(0) > 0 and a solution u C([0, T ]; H s ) to the DP IVP (57) (58). 2.) This solution u is unique in the space u C([0, T ]; H s ). 3.) The data-to-solution map u(0) u(t) is continuous. More precisely, if u n (0) is a sequence of initial data converging to u (0) in H s and if u n (t) C([0, T n ]; H s ) is the solution to the Cauchy problem with initial data u n (0), then there is T (0, T ) such that the solutions u n (t) can be extended to the interval [0, T ] for all sufficiently large n and lim n sup 0 t T u n (t) u (t) H s = 0. (59) 51

52 Key ingredients Norm inflation: Let s [ 2 1, 2 3 ). Then for any ε > 0 there exists T > 0 such that the DP Cauchy problem (57) (58) has a solution u C([0, T ); H s ) satisfying the following three properties: 1.) Lifespan T < ε, 2.) u 0 H s < ε, 3.) lim u(t) t T Hs = (norm inflation). DP solutions u conserve the following twisted L 2 -norm: u 2 L 2. = (1 2 x)u (4 2 x) 1 u dx 1 + ξ ξ 2 û(ξ) 2 dx, (60) which is equivalent to the L 2 -norm. More precisely, we have 1 2 u L 2 u L 2 u L 2. (61) 52

53 Peakon-antipeakon solutions Lemma The peakon-antipeakon traveling wave u(x, t) = p(t)e x+q(t) p(t)e x q(t) (62) is a solution to the DP equation if the amplitude p = p(t) and the speed q = q(t) solve q = p ( e 2q 1 ), and p = 2p 2 e 2q. (63) Furthermore, if for given ε > 0 we choose p 0 and q 0 so that p 0 1 and 0 < q 0 < ln 2 ε 8, (64) then there exist 0 < T < ε such that the system (63) has a unique smooth solution (q(t), p(t)) in the interval [0, T ) with p(t) increasing, q(t) decreasing, and lim t T p(t) = and lim q(t) = 0. (65) t T 53

54 DP peakon-antipeakon collision [Lundmark, 2006] Before collision x After collision 54 x

55 The Fundamental Estimate Lemma Let u(t) be the the peakon-antipeakon solution, then for all t [0, T ) we have the estimates u(t) H s p(t)q(t) 3/2 s, 1/2 < s < 3/2, p(t)q(t) ln(1/q(t)), s = 1/2, p(t)q(t), s < 1/2. (66) Note. Interestingly, below 1/2 there is no dependence on s beyond determining the constant associated with. Also, keep in mind that we have lim t T p(t) = and lim q(t) = 0. (67) t T which will be important in our next step. 55

56 Proof of DP Ill-Posedness Theorem: 1/2 s < 3/2 Let u n (t) be the peakon-antipeakon DP solution corresponding to the choice of ε = 1/n and let u (t) = 0. Then, by property (2) in Norm Inflation result we have u n (0) H s < 1/n. Therefore, u n (0) converges to u (0) = 0 in H s. Furthermore, by property (1) the lifespan T n of each solution u n (t) satisfies the inequality T n < 1/n, whereas the lifespan T of u (t) is equal to. Since, by property (3) in Norm Inflation result there is no T > 0 such that the solutions u n (t) can be extended to the interval [0, T ] for all sufficiently large n we see that continuity condition (iii) of well-posedness fails. 56

57 Proof of Ill-Posedness Theorem: s < 1/2 Now, the H s limit of u(t) as t T exists and is given by u(x, T ) = σ(x)p 0 (1 e 2q 0)e x, (68) where σ(x) is the sign function. Also, the shock peakon v(x, t) = σ(x) (T t) [p 0 (1 e 2q 0)] 1e x (69) is a solution to the DP equation for t T [p 0 (1 e 2q 0)] 1, and v(t) H s if s < 1/2. Furthermore, we have T = q 0 [p 0 (1 e 2q 0)] 1, (70) which implies that T [p 0 (1 e 2q 0)] 1 = (q 0 1)[p 0 (1 e 2q 0)] 1 < 0. Therefore, the shock peakon solution v(t) is welldefined for all t 0 and an element of C([0, T ]; H s ). This together with the fact that u(t ) = v(t ) shows that uniqueness of solutions for DP fails. 57

58 DP peakon-antipeakon collision [Lundmark, 2006] Before collision x After collision 58 x

59 Norm Inflation: 1/2 < s < 3/2 We use the fact that u conserves the L 2 -norm, that is to say for some C we have C = u(t) L 2. Therefore, using the fundamental estimate and the conserved quantity L 2 C u(t) H s = u(t) L 2 u(t) H s u(t) H 0 u(t) H s Since s > 1/2 and q 0 we obtain 0 lim t T or lim t T u(t) H s =. C u(t) H s c 0 pq c 1 s pq 3/2 s = c 0c s q s 1/2. (71) c 0 c s lim t T q s 1/2 = 0, (72) 59

60 Norm Inflation: s = 1/2 The case for s = 1/2 follows in the same way C u(t) H 1/2 u(t) H 0 u(t) H 1/2 c 0 pq c 1 1/2 pq ln(1/q) = c 0c 1/2 ln(1/q), (73) which yields 0 lim t T C u(t) H 1/2 lim t T c 0 c 1/2 ln(1/q) = 0, (74) or lim t T u(t) H 1/2 =. 60

61 CH and KdV with analytic initial data 61

62 Analyticity for CH in both variables Theorem [H.-Misio lek, Math Annalen, 2003] Let u o be a (real) analytic function on T. There exist an ɛ > 0 and a unique solution u of the initial value problem for the CH equation { t u + u x u + (1 x) 2 1 x u } 2 ( xu) 2 = 0, that is analytic on ( ɛ, ɛ) T. u(x, 0) = u 0 (x), t R, x T, This result can be viewed as a nonlocal Cauchy-Kowalevski type theorem for CH. 62

63 Analyticity for KdV only in space variable For the Cauchy problem of KdV t u + xu 3 + u x u = 0, x T, t R, (75) u(x, 0) = ϕ(x), (76) we have the following regularity result. Theorem If the initial data are analytic then the solution to i.v.p. (75) (76): Is analytic in x [Trubowitz] but it may not be analytic in t. 63

64 Outline of Proof of CH analyticity The Spaces: For any s > 0, we set E s = { u C (T) : u s = sup k N o s k k u H 2 k!/(k + 1) 2 < For 0 < s < 1, E s has the algebra property uv s c u s v s. (77) Letting u 1 = u, and u 2 = x u 1 we obtain the ode system: t u 1 = F 1 (u 1, u 2 ) and t u 2 = F 2 (u 1, u 2 ) u 1 (x, 0) = u o (x) and u 2 (x, 0) = x u o (x) }. 64

65 Defining and F (u 1, u 2 ) = (F 1 (u 1, u 2 ), F 2 (u 1, u 2 )), F (u 1, u 2 ) s = F 1 (u 1, u 2 ) s + F 2 (u 1, u 2 ) s. we have the following key result. Proposition Let R > 0. There is a constant C > 0 such that given arbitrary 0 < s < s 1, we have F (u 1, u 2 ) F (v 1, v 2 ) s C s s (u 1, u 2 ) (v 1, v 2 ) s, for any u j and v j in the ball B(0, R) E s. The analyticity of CH solutions follows from this proposition by applying an abstract ODE theorem [Ovsiannikov, Nirenberg, Nishida, Treves, Baouendi and Goulaouic]. 65

66 Evχαριστ ω! 66

67 References J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation. Discrete Contin. Dyn. Syst. 23 (2009), no. 4, R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6, O. Christov and S. Hakkaev, On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity 67

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