New long time existence results for a class of Boussinesq-type - PowerPoint PPT Presentation

New long time existence results for a class of Boussinesq-type systems Cosmin Burtea Université Paris-Est Créteil, LAMA - CNRS UMR 8050 Mathflows 2015, Porquerolles September 13-18th, 2015 September 13, 2015

Outline Introduction The Euler system The Boussinesq regime Derivation of the abcd systems The long time existence problem Motivation and some references Littlewood-Paley theory Our result Conclusions and further results

The Euler system ◮ We consider a layer of incompressible, irrotational, perfect fluid flowing through a canal with flat bottom.

The Euler system ◮ We consider a layer of incompressible, irrotational, perfect fluid flowing through a canal with flat bottom. ◮ The bottom is represented by the plane { ( x , y , z ) : z = − h } , with h > 0.

The Euler system ◮ We consider a layer of incompressible, irrotational, perfect fluid flowing through a canal with flat bottom. ◮ The bottom is represented by the plane { ( x , y , z ) : z = − h } , with h > 0. ◮ The free surface can be described as being the graph of a function η over the flat bottom.

The Euler system � − v + ∇ p = − g − → ∂ t − → � − → → � v · ∇ v + k in Ω t div − → v = 0 in Ω t v = u − → i + v − → j + w − → − → k Ω t ⊂ R 3 domain occupied by the fluid

The water waves problem ∆ φ + ∂ 2 − h ≤ z ≤ η ( x , y , t ) ,  z φ = 0 in   ∂ z φ = 0 at z = − h ,   ∂ t η + ∇ φ ∇ η − ∂ z φ = 0 at z = η ( x , y , t ) ,  |∇ φ | 2 + | ∂ z φ | 2 �  � ∂ t φ + 1  + gz = 0 at z = η ( x , y , t )  2 (1) ◮ η the free surface ◮ φ the fluid’s velocity potential ◮ g is the acceleration of gravity

The water waves problem ◮ The above problem raises a significant number of problems both theoretically and numerically

The water waves problem ◮ The above problem raises a significant number of problems both theoretically and numerically ◮ A way to overcome this: according to the physical regime in question, derive some approximate models.

The Boussinesq regime ◮ A = max x , y , t | η | ◮ l the smallest wavelength for which the flow has significant energy ◮ � 2 α = A � h , S = α h , β = β , (2) l

The Boussinesq regime ◮ A = max x , y , t | η | ◮ l the smallest wavelength for which the flow has significant energy ◮ � 2 α = A � h , S = α h , β = β , (2) l ◮ α ≪ 1 , β ≪ 1 and S ≈ 1 .

The abcd systems ◮ J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory , J. Nonlinear Sci., 12 (2002), 283-318.

The abcd systems ◮ J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory , J. Nonlinear Sci., 12 (2002), 283-318. ◮ x = l ˜ x , y = l ˜ y , z = h (˜ z − 1) , ˜ φ η, t = ˜ � √ gh η = A ˜ tl / gh , φ = gAl

The abcd systems ◮ J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory , J. Nonlinear Sci., 12 (2002), 283-318. ◮ x = l ˜ x , y = l ˜ y , z = h (˜ z − 1) , ˜ φ η, t = ˜ � √ gh η = A ˜ tl / gh , φ = gAl ◮ ∞ � f k ( t , x , y ) z k φ ( t , x , y , z ) = k =0

The abcd systems Set for simplicity α = β = ε � ( I − ε b ∆) ∂ t η + div V + a ε div ∆ V + ε div ( η V ) = 0 , (3) 2 ∇ | V | 2 = 0 . ( I − ε d ∆) ∂ t V + ∇ η + c ε ∇ ∆ η + ε 1

The abcd systems Set for simplicity α = β = ε � ( I − ε b ∆) ∂ t η + div V + a ε div ∆ V + ε div ( η V ) = 0 , (3) 2 ∇ | V | 2 = 0 . ( I − ε d ∆) ∂ t V + ∇ η + c ε ∇ ∆ η + ε 1 ◮ � η = η ( t , x ) ∈ R , V = V ( t , x ) ∈ R 2 , with ( t , x ) ∈ [0 , ∞ ) × R 2 ◮ a + b + c + d = 1 3

The abcd systems Set for simplicity α = β = ε � ( I − ε b ∆) ∂ t η + div V + a ε div ∆ V + ε div ( η V ) = 0 , (3) 2 ∇ | V | 2 = 0 . ( I − ε d ∆) ∂ t V + ∇ η + c ε ∇ ∆ η + ε 1 ◮ � η = η ( t , x ) ∈ R , V = V ( t , x ) ∈ R 2 , with ( t , x ) ∈ [0 , ∞ ) × R 2 ◮ a + b + c + d = 1 3 ◮ The zeros on the RHS of (3) are in fact the O ( ε 2 )-terms ignored in establishing the models.

The abcd systems Set for simplicity α = β = ε � ( I − ε b ∆) ∂ t η + div V + a ε div ∆ V + ε div ( η V ) = 0 , (3) 2 ∇ | V | 2 = 0 . ( I − ε d ∆) ∂ t V + ∇ η + c ε ∇ ∆ η + ε 1 ◮ � η = η ( t , x ) ∈ R , V = V ( t , x ) ∈ R 2 , with ( t , x ) ∈ [0 , ∞ ) × R 2 ◮ a + b + c + d = 1 3 ◮ The zeros on the RHS of (3) are in fact the O ( ε 2 )-terms ignored in establishing the models. ◮ The error would accumulate like O ( ε 2 t )

The long time existence problem ◮ The error would accumulate like O ( ε 2 t ): on time scales of order O (1 /ε ) the error would still remain small i.e. of order O ( ε ).

The long time existence problem ◮ The error would accumulate like O ( ε 2 t ): on time scales of order O (1 /ε ) the error would still remain small i.e. of order O ( ε ). ◮ On time scales of order O (1 /ε 2 ) and larger, the models stop being relevant as approximations.

The long time existence problem ◮ The error would accumulate like O ( ε 2 t ): on time scales of order O (1 /ε ) the error would still remain small i.e. of order O ( ε ). ◮ On time scales of order O (1 /ε 2 ) and larger, the models stop being relevant as approximations. ◮ The global existence problem is an interesting problem only from a mathematical point of view.

The long time existence problem ◮ The error would accumulate like O ( ε 2 t ): on time scales of order O (1 /ε ) the error would still remain small i.e. of order O ( ε ). ◮ On time scales of order O (1 /ε 2 ) and larger, the models stop being relevant as approximations. ◮ The global existence problem is an interesting problem only from a mathematical point of view. ◮ From a practical/physical point of view, long time existence results, i.e. on, time scales of order 1 /ε are sufficient

Justification of the models ◮ The proof of the well-posedness of the Euler system on the time scale 1 ε .

Justification of the models ◮ The proof of the well-posedness of the Euler system on the time scale 1 ε . ◮ B. Alvarez-Samaniego, D. Lannes, Large time existence for 3D water-waves and asymptotics , Invent. Math. 171 (2008) 485–541.

Justification of the models ◮ The proof of the well-posedness of the Euler system on the time scale 1 ε . ◮ B. Alvarez-Samaniego, D. Lannes, Large time existence for 3D water-waves and asymptotics , Invent. Math. 171 (2008) 485–541. ◮ Establishing long time existence of solutions to the Boussinesq systems satisfying uniform bounds.

Justification of the models ◮ The proof of the well-posedness of the Euler system on the time scale 1 ε . ◮ B. Alvarez-Samaniego, D. Lannes, Large time existence for 3D water-waves and asymptotics , Invent. Math. 171 (2008) 485–541. ◮ Establishing long time existence of solutions to the Boussinesq systems satisfying uniform bounds. ◮ Proving (optimal)error estimates.

Justification of the models ◮ The proof of the well-posedness of the Euler system on the time scale 1 ε . ◮ B. Alvarez-Samaniego, D. Lannes, Large time existence for 3D water-waves and asymptotics , Invent. Math. 171 (2008) 485–541. ◮ Establishing long time existence of solutions to the Boussinesq systems satisfying uniform bounds. ◮ Proving (optimal)error estimates. ◮ J.L. Bona, T. Colin, D. Lannes, Long wave approximations for water-waves, Arch. Ration. Mech. Anal. 178 (2005) 373–410. ◮ Error estimate between the solution Euler with free surface and solutions of the abcd models: O ( ε 2 t )

Long time existence results ◮ M. Ming, J.-C. Saut, P. Zhang, Long-Time Existence of Solutions to Boussinesq Systems , SIAM J. Math. Anal., 44(6) (2012), 4078–4100. The case a < 0 , c < 0 , b > 0 , d > 0 and a = c = 0 , b > 0 , d > 0. Proof based on a Nash-Moser theorem: loss of derivatives.

Long time existence results ◮ M. Ming, J.-C. Saut, P. Zhang, Long-Time Existence of Solutions to Boussinesq Systems , SIAM J. Math. Anal., 44(6) (2012), 4078–4100. The case a < 0 , c < 0 , b > 0 , d > 0 and a = c = 0 , b > 0 , d > 0. Proof based on a Nash-Moser theorem: loss of derivatives. ◮ J.-C. Saut, Li Xu, The Cauchy problem on large time for surface waves Boussinesq systems , J. Math. Pures Appl. 97 (2012) 635–662.