On long-term existence of water wave models Alexandru Ionescu April 24, 2017 Alexandru Ionescu On long-term existence of water wave models
The water wave equations We consider the free boundary incompressible Euler equations v t + v · ∇ v = −∇ p − ge n , ∇ · v = 0 , x ∈ Ω t , where g is the gravitational constant. The free surface Γ t = { z ( α, t ) : α ∈ R } moves with the velocity, according to the kinematic boundary condition ( ∂ t z − v ) | Γ t tangent to Γ t . In the presence of surface tension the pressure on the interface is given by x ∈ Γ t , p ( x , t ) = σκ ( x , t ) , where κ is the mean-curvature of Γ t and σ > 0. Alexandru Ionescu On long-term existence of water wave models
Natural questions: • Local regularity • Global regularity and asymptotics • Dynamical formation of singularities Alexandru Ionescu On long-term existence of water wave models
Possible variants: Periodic conditions, finite bottom, two-fluid model. Local wellposedness: Nalimov (1974), Yosihara (1982), Craig (1985), Wu (1997, 1999), Beyer–Gunther (1998), Christodoulou–Lindblad (2000), Ambrose (2003), Ambrose–Masmoudi (2005), Lannes (2005), Lindblad (2005), Coutand–Shkoller (2007), Cheng–Coutand–Shkoller (2008), Christianson–Hur–Staffilani (2010), Alazard–Burq–Zuily (2011), Shatah–Zeng (2008, 2011). One has local regularity if σ > 0 or if the Rayleigh–Taylor condition is satisfied. The time of existence depends on two quantities: the smoothness, say in H 10 , of the interface and the fluid velocities, and the arc-chord constant of the interface. Alexandru Ionescu On long-term existence of water wave models
Formation of singularities: possible scenarios: (1) loss of regularity, and (2) self-intersection of the interface. The ”splash” singularity of Castro–Cordoba–Fefferman–Gancedo–Gomez-Serrano (new proof of Coutand–Shkoller). Interface at time t splash - ϵ Interface at time t splash Figure 2. Formation of “splash” singularities. • The splash singularity cannot form in the two-fluid model (Fefferman–I.–Lie). Alexandru Ionescu On long-term existence of water wave models
Global regularity Small irrotational global solutions, with either gravity or surface tension (but not both) in 2D or 3D: • (almost global) 2D gravity waves g > 0 , σ = 0: Wu (2009) • 3D gravity waves g > 0 , σ = 0: Wu, Germain–Masmoudi–Shatah; • 3D capillary waves g = 0 , σ > 0: Germain–Masmoudi–Shatah; • 2D gravity waves σ = 0 , g > 0: I.–Pusateri, Alazard–Delort (new proofs in different topologies by Hunter–Ifrim–Tataru (almost global regularity), Ifrim–Tataru (global regularity), Wang (removal of a momentum condition on the velocity field)); • 2D capillary waves g = 0 , σ > 0: I.–Pusateri in the general case, Ifrim–Tataru assuming one momentum condition on the Hamiltonian variables. • 3D gravity or capillary waves with finite bottom: Wang. • 3D gravity waves g > 0 , σ > 0: Deng–I.–Pausader–Pusateri. Alexandru Ionescu On long-term existence of water wave models
• In 2 dimensions (1D interface), there are no resonances if either g = 0 or σ = 0. An important piece of the proof is the quartic energy inequality (Wu) � t � � � � �∇� u · �∇� u · �∇� N u · �∇� N u dxds E N ( t ) − E N (0) � � . � 0 • Formally, it is similar to Shatah’s normal form method. It is important not to lose derivatives in the right-hand side. • The linearized and nonlinear solution have t − 1 / 2 pointwise decay, which leads to almost-global existence. Global existence relies on understanding the scattering theory, i.e. proving modified scattering (I.-Pusateri, Alazard-Delort). • Improvements: paradifferential energy estimates (Alazard-Delort), compatible vector-field structures (I.-Pusateri), modified energy method (Hunter-Ifrim-Tataru). • The quartic energy inequality was proved in other settings: gravity constant vorticity (Ifrim-Tataru), gravity finite bottom (Harrop-Griffith–Ifrim–Tataru). Alexandru Ionescu On long-term existence of water wave models
• In 2 dimensions (1D interface), there are no resonances if either g = 0 or σ = 0. An important piece of the proof is the quartic energy inequality (Wu) � t � � � � �∇� u · �∇� u · �∇� N u · �∇� N u dxds E N ( t ) − E N (0) � � . � 0 • Formally, it is similar to Shatah’s normal form method. It is important not to lose derivatives in the right-hand side. • The linearized and nonlinear solution have t − 1 / 2 pointwise decay, which leads to almost-global existence. Global existence relies on understanding the scattering theory, i.e. proving modified scattering (I.-Pusateri, Alazard-Delort). • Improvements: paradifferential energy estimates (Alazard-Delort), compatible vector-field structures (I.-Pusateri), modified energy method (Hunter-Ifrim-Tataru). • The quartic energy inequality was proved in other settings: gravity constant vorticity (Ifrim-Tataru), gravity finite bottom (Harrop-Griffith–Ifrim–Tataru). Alexandru Ionescu On long-term existence of water wave models
• In 3 dimensions (2D interface), if either g = 0 or σ = 0 then one has and 1 / t pointwise decay for both the linearized solution and the nonlinear solution. One can close the argument by letting the highest order energy grow slowly. Alexandru Ionescu On long-term existence of water wave models
The ”division” problem Consider a generic evolution problem of the type ∂ t u + i Λ u = N ( u , D x u ) where Λ is real and N is a quadratic nonlinearity. At first iteration u ( t ) = e − it Λ φ. At second iteration, assuming N = ∂ 1 ( u 2 ), u ( ξ, t ) = e − it Λ( ξ ) � � φ ( ξ ) � 1 − e it [Λ( ξ ) − Λ( η ) − Λ( ξ − η )] + Ce − it Λ( ξ ) φ ( ξ − η ) � � φ ( η ) i ξ 1 Λ( ξ ) − Λ( η ) − Λ( ξ − η ) d η. One has to understand the contribution of the set of (time) resonances: { ( ξ, η ) : ± Λ( ξ ) ± Λ( η ) ± Λ( ξ − η ) = 0 } . Alexandru Ionescu On long-term existence of water wave models
The ”division” problem Consider a generic evolution problem of the type ∂ t u + i Λ u = N ( u , D x u ) where Λ is real and N is a quadratic nonlinearity. At first iteration u ( t ) = e − it Λ φ. At second iteration, assuming N = ∂ 1 ( u 2 ), u ( ξ, t ) = e − it Λ( ξ ) � � φ ( ξ ) � 1 − e it [Λ( ξ ) − Λ( η ) − Λ( ξ − η )] + Ce − it Λ( ξ ) φ ( ξ − η ) � � φ ( η ) i ξ 1 Λ( ξ ) − Λ( η ) − Λ( ξ − η ) d η. One has to understand the contribution of the set of (time) resonances: { ( ξ, η ) : ± Λ( ξ ) ± Λ( η ) ± Λ( ξ − η ) = 0 } . Alexandru Ionescu On long-term existence of water wave models
The phases corresponding to bilinear interactions satisfy the following restricted nondegeneracy condition of the resonant hypersurfaces: if Φ( ξ, η ) := ± Λ( ξ ) ± Λ( η ) ± Λ( ξ − η ) and � � Υ( ξ, η ) := ∇ 2 ∇ ⊥ ξ Φ( ξ, η ) , ∇ ⊥ ξ,η Φ( ξ, η ) η Φ( ξ, η ) , then Υ( ξ, η ) � = 0 at (almost all) points on the time-resonant set Φ( ξ, η ) = 0. Alexandru Ionescu On long-term existence of water wave models
In the irrotational case curl v = 0, let Φ denote the velocity potential, v = ∇ Φ, and let φ ( x , t ) = Φ( x , h ( x , t ) , t ) denote its trace on the interface. Main Theorem. (Deng, I., Pausader, Pusateri) If g > 0, σ > 0, and � ( h 0 , φ 0 ) � Suitable norm ≤ ε 0 ≪ 1 then there is a unique smooth global solution of the gravity-capillary water-wave system in 3d, with initial data ( h 0 , φ 0 ), ∂ t h = G ( h ) φ, � � 2 |∇ φ | 2 + ( G ( h ) φ + ∇ h · ∇ φ ) 2 ∇ h − 1 ∂ t φ = − gh + σ div . 2(1 + |∇ h | 2 ) (1 + |∇ h | 2 ) 1 / 2 where G ( h ) is the (normalized) Dirichlet-Neumann map associated to the domain Ω t (the Zakharov-Craig-Sulem formulation). The solution ( h , φ )( t ) decays in L ∞ at t − 5 / 6+ rate as t → ∞ . Alexandru Ionescu On long-term existence of water wave models
For sufficiently smooth solutions, this is a Hamiltonian system which admits the conserved energy (Zakharov) � � H ( h , φ ) := 1 R n − 1 G ( h ) φ · φ dx + g R n − 1 h 2 dx 2 2 � |∇ h | 2 + σ � 1 + |∇ h | 2 dx 1 + R n − 1 � � � � � 2 � 2 � |∇| 1 / 2 φ � ( g − σ ∆) 1 / 2 h ≈ L 2 + L 2 . Model equation: ( ∂ t + i Λ) U = ∇ V · ∇ U + (1 / 2)∆ V · U , U (0) = U 0 , � | ξ | + | ξ | 3 , Λ( ξ ) := V := P [ − 10 , 10] ℜ U . which has the L 2 conservation law � U ( t ) � L 2 = � U 0 � L 2 , t ∈ [0 , ∞ ) . Alexandru Ionescu On long-term existence of water wave models
For sufficiently smooth solutions, this is a Hamiltonian system which admits the conserved energy (Zakharov) � � H ( h , φ ) := 1 R n − 1 G ( h ) φ · φ dx + g R n − 1 h 2 dx 2 2 � |∇ h | 2 + σ � 1 + |∇ h | 2 dx 1 + R n − 1 � � � � � 2 � 2 � |∇| 1 / 2 φ � ( g − σ ∆) 1 / 2 h ≈ L 2 + L 2 . Model equation: ( ∂ t + i Λ) U = ∇ V · ∇ U + (1 / 2)∆ V · U , U (0) = U 0 , � | ξ | + | ξ | 3 , Λ( ξ ) := V := P [ − 10 , 10] ℜ U . which has the L 2 conservation law � U ( t ) � L 2 = � U 0 � L 2 , t ∈ [0 , ∞ ) . Alexandru Ionescu On long-term existence of water wave models
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