On the highest wave for the Whitham equation

Transcription

1 On the highest wave for the Whitham equation Erik Wahlén Lund, Sweden Joint with Mats Ehrnström Trondheim, September 23, 2015

2 Stokes conjecture for the water wave problem [Irrotational water wave problem without surface tension] Conjecture Stokes 1880: There is a highest periodic travelling water wave with a corner at the crest. The angle enclosed by the crest is 120. The wave is convex between successive crests.

3 Stokes conjecture for the water wave problem [Irrotational water wave problem without surface tension] Conjecture Stokes 1880: There is a highest periodic travelling water wave with a corner at the crest. The angle enclosed by the crest is 120. The wave is convex between successive crests. 1st part proved by Amick, Fraenkel, & Toland and Plotnikov around nd by Plotnikov and Toland 04.

4 Stokes conjecture for the water wave problem [Irrotational water wave problem without surface tension] Conjecture Stokes 1880: There is a highest periodic travelling water wave with a corner at the crest. The angle enclosed by the crest is 120. The wave is convex between successive crests. 1st part proved by Amick, Fraenkel, & Toland and Plotnikov around nd by Plotnikov and Toland 04. What about model equations?

5 The Whitham equation u t + 2uu x + (Lu) x = 0, ( ) 1 ( Lu)(k) tanh ξ 2 = û(ξ) ξ ( ) 1 tanh ξ 2 c W (ξ) =, ckdv (ξ) = 1 1 ξ 6 ξ2. c Water waves KdV

6 Travelling waves u = ϕ(x µt) µϕ x + 2ϕϕ x + (Lϕ) x = 0

7 Travelling waves u = ϕ(x µt) µϕ + ϕ 2 + Lϕ = 0 (Galiliean invariance)

8 Travelling waves u = ϕ(x µt) µϕ + ϕ 2 + Lϕ = 0 (Galiliean invariance) Conjecture (Whitham 67) The Whitham equation has a highest, cusped travelling wave.

9 Travelling waves u = ϕ(x µt) µϕ + ϕ 2 + Lϕ = 0 (Galiliean invariance) Conjecture (Whitham 67) The Whitham equation has a highest, cusped travelling wave. Heuristics: ( µ ) 2 2 ϕ µ 2 = 4 Lϕ L order 1/2, so ϕ C if ϕ(x) < µ/2 for all x Highest : ϕ = µ/2 at crest

10 Main result Theorem P 2π, a curve s (ϕ s (x), µ s ), s 0, of even, P -periodic, C solutions of the Whitham equation, such that ϕ 0 (x) 0 and µ 0 = m(2π/p ), while ϕ s(x) > 0 for P/2 < x < 0 and ϕ s (0) < µ/2 for s > 0.

11 Main result Theorem P 2π, a curve s (ϕ s (x), µ s ), s 0, of even, P -periodic, C solutions of the Whitham equation, such that ϕ 0 (x) 0 and µ 0 = m(2π/p ), while ϕ s(x) > 0 for P/2 < x < 0 and ϕ s (0) < µ/2 for s > 0. There is a sequence s n such that (ϕ sn (x), µ sn ) (ϕ(x), µ) where µ 0 and ϕ is a solution with ϕ C (( P, 0)), ϕ(0) = µ/2 and ϕ (x) > 0 in ( P/2, 0).

12 Main result Theorem P 2π, a curve s (ϕ s (x), µ s ), s 0, of even, P -periodic, C solutions of the Whitham equation, such that ϕ 0 (x) 0 and µ 0 = m(2π/p ), while ϕ s(x) > 0 for P/2 < x < 0 and ϕ s (0) < µ/2 for s > 0. There is a sequence s n such that (ϕ sn (x), µ sn ) (ϕ(x), µ) where µ 0 and ϕ is a solution with ϕ C (( P, 0)), ϕ(0) = µ/2 and ϕ (x) > 0 in ( P/2, 0). ϕ C 1/2 \ C 1/2+ and has Hölder regularity exactly 1/2 at x = 0: with 0 < c 1 < c 2. c 1 x 1/2 µ 2 ϕ(x) c 2 x 1/2, x 0,

13 Conjecture ϕ C 1/2 per (R) with ϕ(x) = µ 2 π 2 x 1/2 + o( x 1/2 ), x 0. ϕ is convex between successive crests.

14 Previous results Ehrnström & Kalisch 09: Existence of small amplitude solutions by local bifurcation theory. Ehrnström & Kalisch 13: Existence of large (finite) amplitude solutions by global bifurcation theory. waveheight (a) φ (b) µ x Figure 6. AbranchofapproximatetravelingwavesfortheWhithamequation with k =2isshownin(a). Theprofileofthehighestwaveon[0, 2π] isshownin(b).

15 Ingredient 1: Global bifurcation theory Fix 1/2 < α < 1 and P 2π. Work in the set ϕ C α per(r) with ϕ(x) < µ/2.

16 Ingredient 1: Global bifurcation theory Fix 1/2 < α < 1 and P 2π. Work in the set ϕ C α per(r) with ϕ(x) < µ/2. Theorem (Global bifurcation) There is a curve s (ϕ s (x), µ s ) of solutions R 0 C α per(r) R, bifurcating from (0, m(2π/p )), that allows a local real-analytic reparameterisation around each s > 0. One of the following alternatives holds: (i) (ϕ s, µ s ) C α per (R) R as s. (ii) max x ϕ sn µ/2 for some sequence s n. (iii) The function s (ϕ s, µ s ) is periodic.

17 Ingredient 1: Global bifurcation theory Fix 1/2 < α < 1 and P 2π. Work in the set ϕ C α per(r) with ϕ(x) < µ/2. Theorem (Global bifurcation) There is a curve s (ϕ s (x), µ s ) of solutions R 0 C α per(r) R, bifurcating from (0, m(2π/p )), that allows a local real-analytic reparameterisation around each s > 0. One of the following alternatives holds: (i) (ϕ s, µ s ) C α per (R) R as s. (ii) max x ϕ sn µ/2 for some sequence s n. (iii) The function s (ϕ s, µ s ) is periodic. Remark: Uses the fact that (ϕ, µ) µϕ + ϕ 2 + Lϕ is analytic. Reference: Dancer, Buffoni & Toland

18 Ingredient 2: The kernel K(x) = 1 2π Lϕ = K ϕ m(ξ)e ixξ dξ, m(ξ) = tanh (ξ) ξ

19 Ingredient 2: The kernel K(x) = 1 2π Proposition Lϕ = K ϕ m(ξ)e ixξ dξ, m(ξ) = K(x) = 1 x + K reg (x), K reg C ω. tanh (ξ) ξ

20 Proposition s 0 (0, π/2), n 0 : D n xk(x) e s 0 x, x 1

21 Proposition s 0 (0, π/2), n 0 : D n xk(x) e s 0 x, x 1 Proposition K(x) = 2 π n=1 nπ (2n 1)π 2 e s x tan s s ds

22 Proposition s 0 (0, π/2), n 0 : D n xk(x) e s 0 x, x 1 Proposition K(x) = 2 π n=1 nπ (2n 1)π 2 e s x tan s s Corollary K is a completely monotone function on R + : ( 1) n D n xk(x) > 0, x > 0, n 0. ds

23 Proposition s 0 (0, π/2), n 0 : D n xk(x) e s 0 x, x 1 Proposition K(x) = 2 π n=1 nπ (2n 1)π 2 e s x tan s s Corollary K is a completely monotone function on R + : ( 1) n D n xk(x) > 0, x > 0, n 0. ds Remark: Can also be proved by using the fact that λ m( λ) is a Stieltjes function or (partly) using the theory of positive definite functions.

24 The periodic kernel K P (x) = K(x + np ) n=

25 The periodic kernel Proposition K P (x) = for 0 < x < P. 2 π K P (x) = n=1 nπ (2n 1)π 2 n= K(x + np ) cosh(s(x P 2 )) tan s sinh( P 2 s) ds, s

26 The periodic kernel Proposition K P (x) = for 0 < x < P. Corollary 2 π K P (x) = n=1 nπ (2n 1)π 2 n= K(x + np ) cosh(s(x P 2 )) tan s sinh( P 2 s) ds, s ( 1) n D n xk P (x) > 0, 0 < x < P/2, n 0.

27 The periodic kernel Proposition K P (x) = for 0 < x < P. Corollary Proposition 2 π K P (x) = n=1 nπ (2n 1)π 2 n= K(x + np ) cosh(s(x P 2 )) tan s sinh( P 2 s) ds, s ( 1) n D n xk P (x) > 0, 0 < x < P/2, n 0. K P (x) = 1 x + K P, reg (x), K reg C ω ( P, P ).

28 Ingredient 3: Maximum principles and nodal pattern Lemma Lf(x) > 0 on ( P/2, 0) for f odd and continuous with f 0 on ( P/2, 0). Proof. Lf(x) = = = P/2 P/2 0 P/2 0 P/2 K P (x y)f(y) dy K P (x y)f(y) dy + P/2 (K P (x y) K P (x + y)) f(y) dy. }{{} >0 0 K P (x y)f(y) dy

29 Theorem (i) Any non-constant and even solution ϕ C 1 with ϕ (x) 0 on ( P/2, 0) satisfies ϕ > 0, ϕ < µ 2 on ( P/2, 0). (ii) If ϕ C 2, then ϕ < µ 2 everywhere, with (iii) If in addition ϕ(0) > µ 4, then ϕ (0) < 0 and ϕ (±P/2) > 0. ϕ ( P 2 ) ϕ (0) K P ( P/4) 2.

30 Theorem Alternative (iii) in the global bifurcation theorem can t occur. Sketch of proof. 1. Show that for s > 0, ϕ(s) is either strictly increasing on ( P/2, 0) or constant. Uses the previous theorem. 2. Show that the constant value is This shows that the curve has to return to the bifurcation point ϕ = 0, µ = µ, which gives a contradiction. Remark: Step 2 is not completely trivial, but uses the Galilean invariance.

31 Ingredient 4: Estimates and regularity Lemma λ P > 0, such that if ϕ is an even, non-constant C 1 -solution with ϕ < µ 2 and ϕ 0 in ( P/2, 0), then µ 2 ϕ(x) λ P x 1/2, x [ P/2, 0]. Proposition (Regularity) Let ϕ µ 2 be a solution. Then: (i) ϕ C on any open set where ϕ < µ 2. If ϕ is furthermore even, non-constant, and non-decreasing on ( P/2, 0) with ϕ(0) = µ 2, then: (ii) ϕ C 1/2 (R) \ C 1/2+ (R). (iii) ϕ has Hölder regularity exactly 1 2 at x = 0.

32 Sketch of proof of (iii). Set u(x) = µ 2 ϕ(x) = ϕ(0) ϕ(x). u(x) 2 = 1 2 P/2 P/2 c, independent of α, such that P/2 P/2 (K P (x + y) + K P (x y) 2K P (y))u(y) dy. K P (x+y)+k P (x y) 2K P (y) y α 2 dy c x α, 0 α 1, for x < P/2. From this we get u(x) 2 c x α y α/2 u = x α/2 u c = u(x) α 1 c x 1/2.

33 Lemma Any sequence of Whitham solutions (ϕ n, µ n ) with {µ n } bounded has a subsequence which converges uniformly to a solution ϕ. Proof. ϕ 2 µ ϕ + K P 1 ϕ = ( µ + 1) ϕ + Arzela-Ascoli. Lemma µ P > 0 such that µ P µ 2 for any solution (ϕ, µ) with ϕ < µ 2, µ > 0 and ϕ 0 in ( P/2, 0). Theorem Alternative (i) in the global bifurcation theorem alternative (ii). Sketch of proof. ( µ 2 ϕ ) 2 = µ 2 4 Lϕ µ 2 ϕ = µ 2 4 Lϕ, µ 2 ϕ c 0 > 0.

34 Theorem There is a sequence of solutions (ϕ n, µ n ) with ϕ > 0 on ( P/2, 0), lim n ϕ n (0) = µ/2 and lim n ϕ n C α =. The limiting wave lim n ϕ n satisfies ϕ C 1/2 (R) \ C 1/2+ (R) and has Hölder regularity exactly 1 2 at x = 0.