The Cauchy problem for nonlocal evolution equations

Transcription

1 The Cauchy problem for nonlocal evolution equations Alex Himonas Department of Mathematics Mini-workshop non-local dispersive equations NTNU, Trondheim, Norway September 21 23, 2015

2 Abstract We shall consider the initial value problem for nonlocal and nonlinear evolution equations having peakon traveling wave solutions and discuss their well-posedness in Sobolev and analytic spaces. These include the Camassa-Holm (CH), the Degasperis-Procesi (DP), the Novikov (NE), and the Fokas-Olver-Rosenau-Qiao (FORQ) equations, which are integrable. Also, for these equations we shall discuss stability properties of the data-to-solution map. The talk is based on work with Carlos Kenig, Gerard Misio lek, Gustavo Ponce, Curtis Holliman, Dionyssis Mantzavinos, Gerson Petronilho and Rafael Barostichi.

3 Historical remarks 1973: G. B. Whitham, Linear and Nonlinear waves, p Breaking and Peaking It was remarked earlier that the nonlinear shallow water equations which neglect dispersion altogether lead to breaking of the typical hyperbolic kind, with the development of a vertical slope and a multivalued profile. It seems clear that the third derivative term in the Korteweg-deVries equation will prevent this ever happening in its solutions.... Although both breaking and peaking, as well as criteria for the occurrence of each, are without doubt contained in the equations of the exact potential theory, it is intriguing to know what kind of simpler mathematical equation could include all these phenomena.

4 Gerald B. Whitham

5 Historical remarks (cont.) Witham in this book continues by proposing a KdV-type equation with a weaker dispersion for modeling (hopefully) breaking and peaking shallow water waves. This equation is of the form: ( ) u t + N(u) + L(u) = 0 (1) where N = nonlinearity L = linear weak dispersion KdV(order of dispersion =3): Choosing N(u) = 1 2 u2, L(u) = x = x u t + uu x + u x xu = 0. (KdV ) (2) Whitham Equation(order of dispersion =1/2): Choosing N(u) = 1 ( tanh ξ ) 1/2 2 u2, Lf(ξ) = f(ξ) ξ 1/2 f(ξ) = ξ u t + uu x + x Lu = 0. (W E) (3)

6 u(x, t) = c 2 sech2 ( c 2 ) (x ct), u(x, t) = ce x ct Figure: KdV soliton & peaked soliton

7 KdV story 1834: Scott Russell reported his large wave of translation, nowadays known as soliton, that he observed in the Edinburgh channel while following a barge on horseback. 1877: Boussinesq, in his attempt to explain the soliton, derived approximations to the Navier-Stokes equations in several different regimes, among which the so-called KdV equation u t + u x + uu x + u xxx = : Korteweg and de Vries rederived the KdV.

8 Do Camassa & Holm answer Witham s question? 1993: Camassa & Holm derived the 0-order dispersion equation u t + uu x + x (1 x) 2 1[ u ] 2 u2 x = 0. (CH) (4) starting from Euler equations and showed that it admits peakon travelling waves that interact like solitons, thus providing an alternative model to Witham s equation. 1981: Fokas and Fuchssteiner obtained CH earlier based on hereditary symmetries of bi-hamiltonian systems. 1755: Euler derived the system of equations u + (u )u = p, u = (u, v, w), t known as Euler equations, modeling the flow of an inviscid fluid : Navier (1822), Cauchy (1828), Poisson (1829) and finally Stokes (1845) derived the Navier-Stokes equations u + (u )u = p + ν u, t modeling the flow of a viscous fluid.

9 Euler equations have 0-order dispersion like CH! The initial value problem for the Euler equations governing the motion of an incompressible fluid in an R n read tu + uu + p = 0, (5) div u = 0, u(0, x) = u 0(x), x Ω, t R, (6) where u : R Ω R n is the fluid velocity, p : R Ω R is the pressure function and u 0 : Ω R n is a divergence free initial condition. Pressure can be eliminated from (5). In fact, applying the divergence operator to the Euler equation and solving for p gives p = 1 div uu (7) Using (7) the first of the equations in (5) takes the form tu + uu 1 div uu = 0. (8) Note the nonlocal term in the above is more regular than it appears. In fact, since u = (u 1,..., u n) is divergence free it follows that n div uu = iu j ju i (9) i,j=1 involves only first order derivatives of u.

10 Generalized CH equation with peakon For k = 1, 2, and b any real number, the generalized Camassa-Holm equation (g-kbch or gch) (1 2 x)u t = u k u xxx + bu k 1 u x u xx (b + 1)u k u x. (10) has the fllowing important properties: It contains the following three integrable equations: k = 1 and b = 2 gives the well known Camassa-Holm (CH) equation. k = 1 and b = 3 gives the Degasperis-Procesi (DP) equation. CH and DP have quadratic nonlinearities. k = 2 and b = 3 gives the Novikov equation, which has cubic nonlinearities. It conserves H 1 when b = k + 1. It has peakon traveling wave solutions. It is well-posed in H s for s > 3/2 It satisfies a nonlocal Cauchy-Kovalevski theorem.

11 Integrable equations Integrable equations possess many special properties including: an infinite hierarchy of higher symmetries, infinitely many conserved quantities, a Lax pair, a bi-hamiltonian formulation, and they can be solved by the Inverse Scattering Method. Conserved quantities are useful for proving global in time solutions. A bi-hamiltonian formulation is used for finding conserved quantities. A Lax pair is used for decoupling the equation into two equations, one that describes the spatial structure of the equation and helps to solve it at the initial time and another that helps to compute the time evolution. This is implemented by the Inverse Scattering Method.

12 Integrable CH type equations [Novikov, 2009] In [J. Phys. A, 2009] Novikov investigated the question of integrability for equations of the form (1 2 x)u t = F (u, u x, u xx, u xxx, ), (11) where F is a polynomial of u and its x-derivatives. Definition of integrability: Existence of an infinite hierarchy of (quasi-) local higher symmetries.

13 Integrable CH type equations [Novikov, 2009] (cont.) He produced about 20 integrable equations with quadratic nonlinearities that include the Camassa-Holm (CH) equation (1 2 x)u t = uu xxx + 2u x u xx 3uu x (12) and the Degasperis-Procesi (DP) equation (1 2 x)u t = uu xxx + 3u x u xx 4uu x. (13) Moreover, he produced about 10 integrable equations with cubic nonlinearities that include the following new equation (1 2 x)u t = u 2 u xxx + 3uu x u xx 4u 2 u x, (14) which is now called the Novikov equation (NE).

14 Camassa and Holm approach Camassa and Holm derived CH from the Euler equations of hydrodynamics using asymptotic expansions. Also, they derived its peakon solutions. Concerning the hydrodynamical relevance of the Camassa-Holm equation as well as alternative derivations, we refers the reader to the works by Johnson [2002, 2003], Constantin and Lannes [2009], and Ionescu-Kruse [2007] and the references therein.

15 Fokas-Fuchssteiner approach Using the Fokas-Fuchssteiner approach one can derive the CH and the KdV in a unified way. This is based on the following fact. If for every n the operator θ 1 + nθ 2 is Hamiltonian, then q t = (θ 2 θ 1 1 )q x (15) is an integrable equation. Letting θ 1 = =. x, and θ 2 = + γ 3 + α (q + q), where 3 α and γ are constants, gives the celebrated KdV equation q t + q x + γq xxx + αqq x = 0. (16) Similarly, letting θ 1 = + ν 3 and θ 2 as above with q = u + νu xx gives the equation u t + u x + νu xxt + γu xxx + αuu x + αν 3 (uu xxx + 2u x u xx ) = 0, (17) which reduces to the CH equation by choosing the parameters appropriately and making a change of variables.

16 Degasperis and Procesi approach DP was discovered in 1998 by Degasperis and Procesi [1999] as one of the three equations to satisfy asymptotic integrability conditions in the following family of equations u t + c 0 u x + γu xxx α 2 u txx = (c 1,n u 2 + c 2,n u 2 x + c 3 uu xx ) x, (18) where α, c 0, c 1,n, c 2,n, c 3 R are constants. The other two integrable members are the CH and the KdV equations. Also, DP and CH are the only integrable members of the b-family (see Mikhailov and Novikov [2002]). The b-family is the g-kbch family with k = 1.

17 g-kbch conserved quantities Furthermore, for b = k + 1 the H 1 -norm of a solution u of g-kbch is conserved, that is d dt u(t) 2 H = d [ ] u 2 (t) + u 2 1 dt x(t) dx = 0. (19) In fact, we have 1 d 2 dt u(t) 2 H = 1 = = R or T R or T R or T R or T ] [uu t + u x u xt dx = R or T ] u [u t u xxt dx ] u [u k u xxx + (k + 1)u k 1 u x u xx (k + 2)u k u x dx [ (u k+1 u xx ) x ( u k+2) x ] dx = 0.

18 g-kbch peakon traveling waves The g-kbch family of equations has peakon traveling wave solution for all values of k and b [Grayshan-H, ADSA 2013]. On the line, these solutions are given by the formula while on the circle they are given by u c (x, t) = where c is any positive constant and [x ct] p u c (x, t) = c 1/k e x ct, (20) c1/k cosh(π) cosh ([x ct] p π), (21). [ = x ct 2π x ct ] 2π. More information about traveling wave solutions of CH type equations can be found in Lenells [2005], Constantin and Strauss [2002], Hone, Lundmark and Szmigielski [2009], Cao, Holm and Titi [204], Qiao [2006], and the references therein.

19 u(x, t) = c 2 sech2 ( c 2 ) (x ct), u(x, t) = c 1/k e x ct Figure: KdV soliton & g-kbch peakon

20 Multi-peakon solutions for g-kbch Lemma (Grayshan-H., ADSA 2013) The multi-peakon u(x, t) = n p j (t)e x qj(t) (22) i=1 is a solution of the g-kbch equation m t + u k m x + bu k 1 u x m = 0, where m. = (1 2 x)u. (23) if and only if the positions (q 1,..., q n ) and momenta (p 1,..., p n ) satisfy the following system of 2n ODE s { ṗ j = (b k)u k 1 (q j ) u x (q j ) p j q j = u k, (24) (q j ) where f(x) = 1 2 (f(x ) + f(x+)). Also, we have a periodic version of the Lemma.

21 Multi-peakon solutions for DP and CH It can be shown (see Degasperis, Holm, and Hone) that u(x, t) = n p j (t)e x qj(t), j=1 is a solution to CH or DP if the positions (q 1,..., q n ) and momenta (p 1,..., p n ) of the peakons satisfy the following system of 2n ODEs: q k = n p j e q k q j j=1 ṗ k = (b 1)p k n i=1 where b = 2 for CH and b = 3 for DP. p j sign(q k q j )e q k q j, Special two-peakon solutions (peakon-antipeakon) are used here to prove that these equations are not well-posed in Sobolev spaces H s when s < 3/2.

22 Non-uniform continuity of solution map Theorem (Grayshan-H., 2013) If s < 3/2, then the data-to-solution map u 0 u(t) of the g-kbch Cauchy problem is not uniformly continuous from H s into C([0, T ], H s ) for any T > 0. More precisely, on the non-periodic case there exist two sequences of peakon traveling wave solutions u c1,n (t) and u c2,n (t) such that in the non-periodic case, we have the estimates ( 1 ) 1/k, u c2,n (0) u c1,n (0) H s (R) C 1 (s) n = 1, 2, 3, n and sup u c2,n (t) u c1,n (t) H s (R) C 2 (s) n 2 s +s+1/2 T 3/2 s, n 2T 0<t T π, where ( ( C 1 (s) = ξ 2 ) ) s 2 1/2, dξ and C2 (s) = 2 5/2 s/2 3 s 2 π s 3/2. R In the periodic case, we have similar solution sequences.

23 Well-posedness in Sobolev spaces Hadamard well-posedness There exists a solution, it is unique and depends continuously on the initial data, i.e. the data-to-solution map u(0) u(t) is continuous. Well-posedness of Cauchy problem for CH type equations (i) 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 CH Cauchy problem. (ii) This solution u is unique in the space u C([0, T ]; H s ). (iii) 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 sup n 0 t T u n (t) u (t) H s = 0. (25)

24 Jacques Hadamard Jacques Hadamard (French mathematician) published Lectures on Cauchy s Problem in Linear Partial Differential Equations in The text was based on a series of lectures he had given at Yale University. In this book, Hadamard put forth the notion of a well-posed problem.

25 Hadamard s example (p. 33: Lectures on the Cauchy s problem in linear partial differential equations)... I have often maintained, against different geometers, the importance of this distinction. Some of them indeed argued that you may always consider any functions as analytic, as, in the contrary case, they could be approximated with any required precision by analytic ones. But, in my opinion, this objection would not apply, the question not being whether such an approximation would alter the data very little, but whether it would alter the solution very little. It is easy to see that, in the case we are dealing with, the two are not at all equivalent. Let us take the classic equation of two-dimensional potentials 2 u x u y 2 = 0 with the following data of Cauchy s u u(0, y) = 0, x (0, y) = A n sin(ny), n being a very large number, but A n a function of n assumed to be very small as n grows very large (for instance A n = 1/n p )....

26 These data differ from zero as little as can be wished. Nevertheless, such a Cauchy problem has for its solution u = A n n sin(ny)sh(nx), which, if An = 1 n, 1 n p, e n, is very large for any determinate value of x different from zero on account of the mode of growth of e nx and consequently Sh(nx). In this case, the presence of the factor sin ny produces a fluting of the surface, and we see that this fluting, however imperceptible in the immediate neighbourhood of the y-axis, becomes enormous at any given distance of it however small, provided the fluting be taken sufficiently thin by taking n sufficiently great.

27 Hadamard well-posedness for g-kbch Theorem (H-Holliman, ADE 2014) (1) 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 g-kbch which depends continuously on the initial data u 0. Furthermore, we have the estimate u(t) H s 2 u 0 H s, for 0 t T where c s > 0 is a constant depending on s. 1 2kc s u 0 k H s, (26) (2) Also, the data-to-solution map is not uniformly continuous from any bounded subset in H s into C([0, T ]; H s ). - CH (R: H-Kenig DIE 2009 & T: H-Kenig-Misiolek, CPDE 2010) - DP (H-Holliman, DCDS 2011, both line R and circle T) - DP (H-Holliman, Nonlinearity 2012, both line R and circle T)

28 The Proof of well-posedness Writing g-kbch in the following nonlocal form where t u + u k x u + F (u) = 0, (27) [ F (u) =(1 x) 2 1 b x k + 1 uk+1 + 3k b ] u k 1 u 2 x 2 + (1 x) 2 1[ (k 1)(b k) ] u k 2 u 3 x 2 (28) (29) and noticing that F maps H s into H s, we see that mollifying the local term u k x u we obtain the following ivp t u + J ε [ (J ε u) k J ε u x ] + F (u) = 0, u(x, 0) = u 0 (x), (30) which we solve by applying the fundamental ode theorem in an infinite dimensional Banach space......

29 The Proof of non-uniform continuity (more to come!) Choosing the approximate solutions u ω,n (x, t). = ωn 1/k + n s cos(nx ωt), (31) where ω = 1, 1 if k is odd and ω = 0, 1 if k is even, we see that they satisfy the three conditions of nonuniform continuity, that is: They are bounded. At t = 0 the H s distance of their difference goes to zero. And, at any later time t > 0 this distance it is bounded below by a positive constant depending on t. Then, solving the Cauchy problem with initial data the value of the approximate solutions at t = 0 we obtain two sequences of actual solutions with common lifespan and whose difference from the approximate solutions is negligible. This allow us to show that the sequences of these solutions satisfy the conditions of nonuniform dependence. We begin by substituting the approximate solutions (31) into g-kbch, giving rise to small error E, i.e. E. = t u ω,n + (u ω,n ) k x u ω,n + F (u ω,n ) H σ n rs. (32)

30 Holder continuity for NE Theorem (H.-Holmes 2013) If s > 3/2 and 0 r < s, then the solution map for NE is Hölder continuous on the space H s equipped with the H r norm. More precisely, for initial data u(0), w(0) in a ball B(0, ρ) of H s the corresponding NE solutions u(t), w(t) satisfy the inequality where α =... u(t) w(t) C([0,T ];H r ) c u(0) w(0) α Hr, (33) Remark. The CH has been done by Chen, Liu, and Zhang.

31 Ill-posedness for CH and DP Theorem (DP by H-Holliman-Grayshan, CPDE CH by Byers) For s < 3/2 the Cauchy problem for CH and DP on both the line and the circle, is not well-posed in H s in the sense of Hadamard. The Proof is based on: Conserved quantities (H 1 for CH and twisted L 2 for DP). The interaction of peakon-antipeakon traveling wave solutions. Open Problem: Is CH and DP well-posed at the critical Sobolev exponent s = 3/2?

32 The FORQ equation Not a member of g-kbch! The FORQ equation (1 2 x)u t = x ( u 3 + uu 2 x + u 2 u xx u 2 xu xx ) (34) was derived independently by Fokas (Phys. D 1995), as an integrable generalisation of the modified KdV equation via a bi-hamiltonian systems approach. Olver & Rosenau (Phys. Rev. E 1996), via a similar method but in a different form. This equation was also derived by Fuchssteiner. Qiao (J. Math. Phys. 2006), as an approximation to Euler equations. Qiao also showed that FORQ admits cusped and W/M soliton travelling waves. It can be written in the following nonlocal form t u = u 2 x u ( xu) 3 ( [ ] 1 x 2 ) ( xu) 3 [ ] ( ) x x 3 u3 + u ( x u) 2.

33 FORQ Cauchy problem Theorem (H-Mantzavinos, Nonlinear Anal. 2014) (1) If s > 5/2 and u 0 H s, then there exists T > 0 and a unique solution u C ( [0, T ]; H s) of the FORQ ivp, formulated either on the line or on the circle, which depends continuously on the initial data u 0. Moreover, we have the estimate 1 u(t) H s 2 u 0 H s for 0 t T 4c s u 0 2 H s where c s > 0 is a constant depending on s. (2) Also, the data-to-solution map is not uniformly continuous from any bounded subset in H s into C([0, T ]; H s ).

34 Peakon travelling waves For the purpose of finding travelling wave solutions of the form u(x, t) = f(x ct), we consider the FORQ in the local form (1 2 x)u t + ( x) x (u 3 ) x(u 3 x) + x (uu 2 x) = 0. Lemma 1. (H-Mantzavinos, 2013) The 2π periodic peakon u(x, t) = γ ( ) c cosh [x c t] p π, [x c t] p = x c t 2π is a weak solution of FORQ iff γ = ± ( sinh2 π ) 1 2. x c t 2. (Gui, Liu, Olver & Qu: CMP 2012) The non-periodic peakon is a weak solution iff α = ± 3 2. u(x, t) = α x c t c e 2π,

35 Non-uniform continuity below 3/2 The peakon solutions are used in proving non-uniform continuity of the data-to-solution map when s < 3/2. Theorem (H-Mantzavinos) If s < 3/2 then the data-to-solution map u 0 H s u C([0, T ], H s ) of the FORQ Cauchy problem both on the line and on the circle is not uniformly continuous for any T > 0.

36 Holder continuity for FORQ Theorem (H-Mantzavinos, JNLS 2014) If s > 5/2 and 0 r < s, then the solution map for FORQ is Hölder continuous on the space H s equipped with the H r norm. More precisely,...

37 Nonuniform continuity for CH

38 The Proof of non-uniform continuity for CH on T We shall prove that there exist two sequences of CH solutions u n (t) and v n (t) in C([0, T ]; H s (T)) such that: sup u n (t) H s + sup v n (t) H s 1, n n lim u n(0) v n (0) H s = 0 n u n (t) v n (t) H s sin t, 0 t < T 1. lim inf n

39 Approximate solutions of CH on T We consider approximate solutions of the form u ω,n (x, t) = ωn 1 + n s cos(nx ωt), (35) where ω = ±1 and n Z +. Substituting (46) into the CH equation gives the error F =. t u ω,n + u ω,n x u ω,n + x D 2[ (u ω,n ) ( xu ω,n ) 2 ] = ωn s cos(nx ωt) ωn s cos(nx ωt) 1 2 n 2s+1 sin 2(nx ωt) n 2s+1 D 2[ sin(2nx 2ωt) ] 2ωn s D 2[ sin(nx ωt) ] n 2s+3 D 2[ sin(2nx 2ωt) 0. Goal. H s -norm of error term F is small!

40 Sobolev norm of error F Since for σ R and n 1 cos(nx α) Hσ (T) n σ, and (36) sin(nx α) Hσ (T) n σ, α R, (37) we obtain the following estimate for the error F. Lemma If 1/2 < σ < min{1, s 1}, then the H σ norm of the error F can be estimated by { 2s σ 1, s 3, F H σ n rs, where r s = (38) s σ + 2, s 3. Remark. The cancelled term ωn s cos(nx ωt) in the error is bad since ωn s cos(nx ωt) Hs (T) 1.

41 u ±,n (t) satisfy the 3 nonuniform continuity conditions Choosing ω = ±1 we get the two sequences solutions of approximate u 1,n (t) = n 1 +n s cos(nx t), and u 1,n (t) = n 1 +n s cos(nx+t), having difference Then we have sup n u 1,n (t) u 1,n (t) = 2n 1 + 2n s sin(nx) sin t. u 1,n (t) H s + sup u 1,n (t) H s 1, n lim n u1,n (0) u 1,n (0) H s = lim n 2n n H s = 0 Furthermore, we have u 1,n (t) u 1,n (t) H s (T) 2n s sin(nx) H s (T) sin t 2n 1 1 H s (T) Now, letting n gives lim inf n u1,n (t) u 1,n (t) H s (T) sin t. (39)

42 Constructing actual solutions u ±,n (t) close to u ±,n (t) To prove nonuniform continuity, it suffices to construct actual CH solutions u ω,n (t) close to the approximate ones u ω,n (t), that is u ω,n (t) u ω,n (t) H s n ε, for some ε > 0. In fact, by adding and subtracting and then applying the triangle inequality we have u 1,n (t) u 1,n (t) H s u 1,n (t) u 1,n (t) H s Therefore, we will have u 1,n (t) u 1,n (t) H s u 1,n (t) u 1,n (t) H s lim inf u 1,n (t) u 1,n (t) H s lim inf u 1,n (t) u 1,n (t) H s n n sin t.

43 Actual CH solutions We construct the actual CH solutions by solving its ivp with initial data the value of the approximate solutions at t = 0, that is u ω,n (x, t) are defined by t u ω,n + u ω,n x u ω,n + D 2 x [u 2 ω,n ( xu ω,n ) 2] = 0, (40) u ω,n (x, 0) = u ω,n 0 (x) = ωn 1 + n s cos(nx). (41) Note that u ω,n (x, 0) belong in H and u ω,n (t) Hs (R) 1, (42) Therefore, applying the well-posednes results stated earlier we conclude that there is a T > 0 such that for any ω in a bounded set and n 1 this i.v.p. has a unique solution u ω,n (t) in C([ T, T ]; H s (T)).

44 Difference between approximate and actual solutions The difference between approximate and actual solutions satisfies the Cauchy problem v. = u ω,n u ω,n (43) t v = F 1 2 x[(u ω,n + u ω,n )v] D 2 x [(u ω,n + u ω,n )v x(u ω,n + u ω,n ) x v] (44) v(x, 0) = 0, x T, t R, (45) where F satisfies the H σ -estimate (38). Proposition If s > 3/2 and 1/2 < σ < min{1, s 1}, then v(t) H σ n rs. (46) Proof. It is based on energy estimates and the error estimate F H σ n rs.

45 H s estimate for the difference v Also, using the well-posedness estimates (26) we have that v(t) H s+1 u ω,n (0) H s+1 n, t [0, T ]. (47) Now, interpolating between σ and s + 1 and using estimates (46) and (47) we get v(t) H s v(t) 1/(s+1 σ) H σ v(t) (s σ)/(s+1 σ) H n 1 s+1 s+1 σ (rs s+σ). Finally, from this inequality and the definition of r s (49) we obtain where v(t) Hs (T) n ρs, t [0, T ], (48) ρ s = { (s 1)/(s + 1 σ), if s 3 2/(s + 1 σ), if s 3, which shows that the the H s norm of the difference between actual and approximate solutions is small! (49)

46 Approximate solutions for more equations The DP equation has the same approximate solutions as CH u ω,n (x, t) = ωn 1 + n s cos(nx ωt), ω = ±1. The Euler equations on T 2 have approximate solutions u ω,n (t, x)= ( ωn 1 +n s cos(nx 2 ωt), ωn 1 +n s cos(nx 1 ωt) ), ω = ±1, which are solutions! (H. Misio lek, CMP 2010) Novikov and FORQ equations have approximate solutions u ω,n = ωn 1/2 + n s cos(nx ωt), ω = 0, 1. The Benjamin-Ono (BO) equation u t + uu x + Hu xx = 0 where Hilbert transform H is defined by Ĥf(ξ) = isgn(ξ) f(ξ) has approximate solutions u ω,n (x, t) = ωn 1 + n s cos( n 2 t + nx ωt), ω = ±1. The nonperiodic version of these were introduced by Koch and Tzvetkov, IMRN 2005.

47 Approximate solutions of CH in nonperiodic case They are of the form u ω,n = u l + u h, with u h the high frequency u h = u h,ω,n (x, t) = n δ 2 s ϕ( x ) cos(nx ωt), ω = ±1, nδ where ϕ is in C and such that { ϕ(x) = 1, if x < 1, 0, if x 2. u l = u l,ω,n (x, t) is the solution of the Cauchy problem for the CH equation with the low frequency initial data ωn 1 ϕ( x n δ ) t u l + u x u l + F (u l ) = 0, u l (x, 0) = ωn 1 ϕ( x ), x R, t R, nδ where ϕ is a C 0 (R) function such that ϕ(x) = 1, if x supp ϕ.

48 Analytic theory

49 Cauchy Problem for CH equations Multiplying by the inverse of (1 2 x), we write the Cauchy problem for CH, DP, NE and FORQ equations in the following unified way u t = (1 2 x) 1 P (u). = F (u), u(0) = u 0. (50) where, P (u) is given by the right hand-sides of equations (12), (13), (14) and (34).

50 Analytic spaces G δ,s Furthermore, for analytic initial data, to obtain precise information about the uniform radius of analyticity of the solution to the Cauchy problem (54) we introduce the following scale of analytic (Banach) spaces. For δ > 0 and s 0, in the periodic case we define G δ,s (T) = {ϕ L 2 (T) : ϕ 2 G δ,s = k Z(1+k 2 ) s e 2δ k ϕ(k) 2 < }, (51) while in the nonperiodic case we define G δ,s (R) = {ϕ L 2 (R) : ϕ 2 G = (1 + ξ 2 ) s e 2δ ξ ϕ(ξ) 2 dξ < }. δ,s (52) Note that if ϕ G δ,s (T) then ϕ has an analytic extension to a symmetric strip around the real axis with width δ. This δ is called the radius of analyticity of ϕ. R

51 Well-posedness of CH equations in analytic spaces Theorem (Barostichi-H.-Petronilho, JFA 2015) Let s > 1 2. If u 0 G 1,s+2 on the circle or the line, then there exists a positive time T, which depends on the initial data u 0 and s, such that for every δ (0, 1), the Cauchy problem (54) has a unique solution u which is a holomorphic function in the disc D(0, T (1 δ)) valued in G δ,s+2. Furthermore, the analytic lifespan T satisfies the estimate T 1 u 0 k G 1,s, (53) where k = 1 for the CH and DP equations and k = 2 for the NE and FORQ equations. Theorem (Barostichi-H.-Petronilho, JFA 2015) If s > 1 2, then the data-to-solution map u(0) u(t) of the Cauchy problem (54) for the CH equations is continuous from G δ,s+2 into the solutions space.

52 Ovsyannikov theorem for g-kbch equation Theorem (Barostichi-H.-Petronilho, Baouendi Proceedings 2015) Let s > 1 2. If u 0 G 1,s+2 on the circle or the line, then there exists a positive time T, which depends on the initial data u 0 and s, such that for every δ (0, 1), the Cauchy problem for the generalized Camassa-Holm equation (g-kbch) u t = (1 x) 2 1[ u k u xxx + bu k 1 u x u xx (b + 1)u k ] u x, u(0) = u0, (54) has a unique solution u which is a holomorphic function in D(0, T (1 δ)) valued in G δ,s+2. Furthermore, the analytic lifespan T satisfies the estimate 1 T u 0 k. (55) 1,s+2 Proof. It is based on a Power Series Method for an autonomous Ovsyannikov theorem following Treves work.

53 Thanks!