Boussinesq System for Water Waves Collaborators: Jerry Bona (UIC), Jean-Claude Saut (Paris-Orsay), Gerard Iooss (Nice, France), Olivier Goubet (Amiens, France), Nghiem Nguyen (Utah State), Shenghao Li (Purdue University) S. M. Sun (Virginia Tech), B. Deconinck (U. of Washington), Bingyu Zhang (Cincinnati Univ.) J. Albert (U. Oklahoma) J. Wu (Oklahoma State), H. Chen (Memphis) C. Curtis (San Diego State), Serge Dumont (Amiens, France) Y. Mammeri (Amiens, France), L. Dupaigne(Amiens, France) Crystal Lee, A.Alazman and others ... Thank you !!! May 17, 2017, ICERM Boussinesq System for Water Waves
Water waves (Pic from D. Henderson and etc. ) ◮ Unknowns: velocities ( u , v , w )( x , y , z , t ) , ◮ surface η ( x , y , t ) (with 0 being the still water position), ◮ domain Ω( t ) = ( 0 , L ) × ( 0 , H ) × (˜ h , η ( x , y , t )) , ◮ bottom topography ˜ h ( x , y , t ) , surface pressure P ( x , y , t ) . Boussinesq System for Water Waves
A Boussinesq system for 3D waves (Bona, C., Saut (2002) A Boussinesq system with moving bottom topography ˜ h ( x , y , t ) and the surface pressure P ( x , y , t ) , η t + ∇ · v + ∇ · ( h + η ) v − 1 6 ∆ η t = F ( h xxt , h xtt , ∇ P ) , (BBM 2 ) v t + ∇ η + 1 2 ∇| v | 2 − 1 6 ∆ v t = G ( h xxt , h xtt , ∇ P ) , ˜ h + h 0 where h = (flat means h =0) with h 0 = average water h 0 depth. ◮ The fluid is bounded by the bottom topography ˜ h ( x , y , t ) and the free surface η ( x , y , t ) , ◮ η ( x , y , t ) is a fundamental unknown of the problem, � 2 ◮ v ( x , y , t ) denotes the horizontal velocity at height 3 h 0 . There are investigations on other Boussinesq systems, such as systems with KdV terms. Boussinesq System for Water Waves
Justification (Bona, Colin, Lannes 2005) ◮ It is a first order approximation to Euler equations . Meaning: for any initial value ( η 0 , u 0 ) ∈ H σ ( R ) 2 with σ ≥ s ≥ 0 large enough, there exists a unique solution ( η euler , u euler ) of Euler equations, such that � u − u euler � L ∞ ( 0 , t ; H s ) + � η − η euler � L ∞ ( 0 , t ; H s ) = O ( ǫ 2 1 t , ǫ 2 2 t , ǫ 1 ǫ 2 t ) for 0 ≤ t ≤ O ( ǫ − 1 1 , ǫ − 1 2 ) . ◮ Other relevant works (Craig 1985,Schneider and Wayne 2000, Bona, Prichard, Scott 1981, Alazman, Albert, Bona, Chen, Wu (2003)... ◮ More justification needed, especially by comparing with data from the field and laboratory experiments. Boussinesq System for Water Waves
Advantages of Boussinesq system From practical side: it is ◮ Physically relevant, especially with BVP; ◮ Easy to analyze, to simulate, and to incorporate into a complex system when compared with Euler equation or Navier Stokes or ..., ◮ More accurate when compared with the Linear equation (lol). ◮ It has many desired and helpful properties, such as the existence of solitary waves, conservation of mass, · · · . Example, taking h = P = 0 and considering one-space dimension yields. η t + u x + ( η u ) x − 1 6 η xxt = 0 , u t + η x + uu x − 1 6 u xxt = 0 . Boussinesq System for Water Waves
Properties of the system (Bona, C., Saut (2002)) ◮ (Bona, C., Saut 2003) The linearized system is globally well posed in L p and in W k p for 1 ≤ p ≤ ∞ and k = 0 , 1 , 2 , · · · . ◮ It has the invariant functionals � ∞ H ( η, u ) = 1 u 2 ( 1 + η ) + η 2 dx , 2 −∞ � ∞ u η + 1 I ( η, u ) = 6 η x u x dx , −∞ � ∞ � ∞ I u = u dx I η = η dx . and −∞ −∞ ◮ there is a Hamiltonian structure based on H and I , namely ∂ t ∇ ( η, u ) I ( η, u ) + ∂ x ∇ ( η, u ) H ( η, u ) = 0 . Boussinesq System for Water Waves
Wellposedness of the Boussinesq system ◮ These invariants are useful, but none of these invariants are composed only of positive terms, so they do not on their own provide the a priori information one needs to conclude global existence of solutions of initial-value problems, except when 1 + η ≥ α > 0. ◮ Global well-posedness in time for the nonlinear problem is proved in [BCS] and [AABCW] under the condition that if there is an α > 0 such that the solution satisfies 1 + η ( x , t ) ≥ α for all t ≥ 0 . Note: Not a perfect result because it is based on an assumption about the solution η which is not known. In physical terms, the condition simply means that the bed does not run dry at any time, or what is the same, the free surface never touches the bottom. ◮ For some initial data, solution exists for all time. Examples, the exact nontrivial solution. For others? ◮ (Bona, C., Saut 2003) The linearized system is globally Boussinesq System for Water Waves
Numerical study on global existence (Bona and C(2016)) We tested initial data of the form η ( x , 0 ) = a e − x 2 , u ( x , 0 ) = b xe − x 2 Solution blows up with (a,b) at "x" and stays bounded at "o" 8 6 4 2 b 0 −2 −4 −6 −8 −5 0 5 a Figure: Blowup map for values of the parameters a and b . x blows up and the circles are values where the solution appears global. Boussinesq System for Water Waves
Conjectures on global solutions in time ◮ a close to 0, small amplitude; a > − 1, no dry up ; ◮ When a and b are small (and a > − 1), namely in the physically relevant modeling range, it seems the global solution exists; ◮ η ( x , 0 ) + 1 ≥ α > 0 does not guarantee η ( x , t ) + 1 positive for all t ; ◮ theoretical proof OPEN ◮ it seems to have a local similarity structure near the blowup point. Theoretical proof is also open Boussinesq System for Water Waves
Head-on collision of solitary waves ◮ the solitary waves are generated numerically; ◮ there is a small phase shift after the collision; ◮ the amplitude at the middle of the interaction is larger than the combination of two incoming waves. − 0.2 0 (a) t=0 (c) t=47.06 (d) t=70.30 1.8 0.6 0.6 1.6 0.4 0.4 1.4 0.2 0.2 1.2 0 0 1 0 50 100 150 0 50 100 150 0.8 (b) t=23.42 (e) t=93.73 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 Boussinesq System for Water Waves
The 2-D Wave tank at PSU (Henderson and Hammack) The Boussinesq system has these double periodic solutions. Boussinesq System for Water Waves
Wave patterns: linear plane waves ◮ the 2D patterns are the oblique interaction of two plane waves; ◮ parameters involved in describing a plane wave: ◮ traveling direction: c = c 0 ( 1 , 0 ) , so the direction is in the y -direction; ◮ the angle of the plane wave and the wave length of the plane wave k 1 = l 1 ( 1 , τ 1 ) ; ◮ to search for this plane wave means to find solutions in the form of η ( x ) = η k 1 e i k 1 · ( x − c t ) , v ( x ) = v k 1 e i k 1 · ( x − c t ) ; ◮ substitute this ansatz into the linear part of the equations η t + ∇ · v + ∇ · η v − 1 6 ∆ η t = 0 , (BBM 2 ) v t + ∇ η + 1 2 ∇| v | 2 − 1 6 ∆ v t = 0 , Boussinesq System for Water Waves
Wave patterns: linear plane waves ◮ k 1 , c , η k 1 and v k 1 satisfy − ( 1 + 1 6 | k | 2 )( c · k ) η k + k · v k = 0 , k η k − ( 1 + 1 6 | k | 2 )( c · k ) v k = 0 . ◮ For the nontrivial ( ( η k , v k ) � = 0 ) solutions (plane waves) to exist, k 1 and c are such that the determinant is zero, i.e. satisfy dispersion relation ∆( k , c ) = ( 1 + 1 6 | k | 2 ) 2 ( c · k ) 2 − | k | 2 = 0 . (Det) ◮ Similarly, for the other plane wave, η ( x ) = η k 2 e i k 2 · ( x − c t ) , v ( x ) = v k 2 e i k 2 · ( x − c t ) , where k 2 = l 2 ( 1 , − τ 2 ) , k 2 and c have to satisfy (Det). Boussinesq System for Water Waves
Sketch of the linear study on wave patterns Assume k 1 and k 2 are the solutions to (Det), then we have wave patterns with parameters consist of 5 parameters c 0 , l 1 , l 2 , τ 1 and τ 2 ◮ 3 parameter families of patterns because (Det) has to be satisfied by ( k 1 , c ) and ( k 2 , c ) , and ◮ amplitude η k 1 , η k 2 , v k 2 , v k 1 ; ◮ symmetric lattice: τ 1 = τ 2 , l 1 = l 2 ; ◮ symmetric pattern: symmetric lattice plus η k 1 = η k 2 , v k 2 = v k 1 . For symmetric patterns, two parameters for the lattice and half of the number of parameters for the amplitudes. Boussinesq System for Water Waves
Results of the nonlinear study on wave patterns (C. and Iooss (2006)) Idea: ◮ add the nonlinear term in, using a perturbation approach (Lyapunov Schmidt); ◮ invert the linear operator around the kernel and find the bound for the pseudo-inverse; ◮ perturbation parameter: w in c = c 0 ( 1 , w ) and amplitudes of the plane waves; Results: ◮ existence of symmetric patterns with almost all parameters; ◮ existence of asymmetric patterns with symmetric lattice with almost all parameters; ◮ existence of asymmetric patterns with asymmetric lattice for a large set of parameters (small devisor problem occurs). Boussinesq System for Water Waves
Theorem on symmetric wave patterns. For symmetric lattice (2 parameters) with symmetric pattern, we have Theorem (Chen and Iooss 2006) For almost every ( k , τ ) , k represents the wave number in y direction and τ represents the ratio between the periods in y and x directions, One example of the form of the free surface, even in y , is given by 2 ( 1 + τ 2 ) { 1 − τ 2 ε 2 η = ε cos ky cos k τ x − cos 2 k τ x 4 + c 1 cos 2 ky + d 1 cos 2 ky cos 2 k τ x } + O ( ε 3 ) . Boussinesq System for Water Waves
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