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

• ∂t−

→ v +

−

→ v · ∇

−

→ v + ∇p = −g− → k in Ωt div − → v = 0 in Ωt − → v = u− → i + v− → j + w− → k Ωt ⊂ R3 domain occupied by the fluid

The water waves problem

        

∆φ + ∂2

z φ = 0

in −h ≤ z ≤ η (x, y, t) , ∂zφ = 0 at z = −h, ∂tη + ∇φ∇η − ∂zφ = 0 at z = η (x, y, t) , ∂tφ + 1

2

• |∇φ|2 + |∂zφ|2

+ gz = 0 at z = η (x, y, t) (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 = maxx,y,t |η| ◮ l the smallest wavelength for which the flow has significant

energy

◮

α = A h , β =

h

l

2

, S = α β , (2)

The Boussinesq regime

◮ A = maxx,y,t |η| ◮ l the smallest wavelength for which the flow has significant

energy

◮

α = A h , β =

h

l

2

, S = α β , (2)

◮

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) , η = A˜ η, t = ˜ tl/

• gh, φ = gAl

˜ φ √gh

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) , η = A˜ η, t = ˜ tl/

• gh, φ = gAl

˜ φ √gh

◮

φ (t, x, y, z) =

∞

• k=0

fk (t, x, y) zk

The abcd systems

Set for simplicity α = β = ε

• (I − εb∆) ∂tη + div V + aε div ∆V + ε div (ηV ) = 0,

(I − εd∆) ∂tV + ∇η + cε∇∆η + ε 1

2∇ |V |2 = 0.

(3)

The abcd systems

Set for simplicity α = β = ε

• (I − εb∆) ∂tη + div V + aε div ∆V + ε div (ηV ) = 0,

(I − εd∆) ∂tV + ∇η + cε∇∆η + ε 1

2∇ |V |2 = 0.

(3)

◮

• η = η (t, x) ∈ R,

V = V (t, x) ∈ R2, with (t, x) ∈ [0, ∞) × R2

◮

a + b + c + d = 1 3

The abcd systems

Set for simplicity α = β = ε

• (I − εb∆) ∂tη + div V + aε div ∆V + ε div (ηV ) = 0,

(I − εd∆) ∂tV + ∇η + cε∇∆η + ε 1

2∇ |V |2 = 0.

(3)

◮

• η = η (t, x) ∈ R,

V = V (t, x) ∈ R2, with (t, x) ∈ [0, ∞) × R2

◮

a + b + c + d = 1 3

◮ The zeros on the RHS of (3) are in fact the O(ε2)-terms

ignored in establishing the models.

The abcd systems

Set for simplicity α = β = ε

• (I − εb∆) ∂tη + div V + aε div ∆V + ε div (ηV ) = 0,

(I − εd∆) ∂tV + ∇η + cε∇∆η + ε 1

2∇ |V |2 = 0.

(3)

◮

• η = η (t, x) ∈ R,

V = V (t, x) ∈ R2, with (t, x) ∈ [0, ∞) × R2

◮

a + b + c + d = 1 3

◮ The zeros on the RHS of (3) are in fact the O(ε2)-terms

ignored in establishing the models.

◮ The error would accumulate like O(ε2t)

The long time existence problem

◮ The error would accumulate like O(ε2t): on time scales of

• rder 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(ε2t): on time scales of

• rder O(1/ε) the error would still remain small i.e. of order

O(ε).

◮ On time scales of order O(1/ε2) and larger, the models stop

being relevant as approximations.

The long time existence problem

◮ The error would accumulate like O(ε2t): on time scales of

• rder O(1/ε) the error would still remain small i.e. of order

O(ε).

◮ On time scales of order O(1/ε2) 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(ε2t): on time scales of

• rder O(1/ε) the error would still remain small i.e. of order

O(ε).

◮ On time scales of order O(1/ε2) 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(ε2t)

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.

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.

◮ Global existence results: available in dimension 1 for the cases

◮ a = b = c = 0, d > 0, M.E. Schonbek, Existence of solutions

for the Boussinesq system of equations, J. Differential Equations 42 (1981) 325–352

◮ b = d > 0 and a < 0, c < 0 under some smallness condition

• n the initial data, J.L. Bona, M. Chen, J.-C. Saut, Boussinesq

equations and other systems for small-amplitude long waves in nonlinear dispersive media. Part II. Nonlinear theory, Nonlinearity 17 (2004) 925–952.

Long time existence: where does the difficulty lie?

• (I − εb∆) ∂tη + div V + aε div ∆V + ε div (ηV ) = 0,

(I − εd∆) ∂tV + ∇η + cε∇∆η + ε 1

2∇ |V |2 = 0.

(4)

◮ The system is not symmetric(because of the η div V term) ◮ The presence of the dispersive terms, ε∆∂tη and ε∆∂tV

Littlewood-Paley theory

Theorem

Let C be the annulus {ξ ∈ Rn : 3/4 ≤ |ξ| ≤ 8/3}. There exist two radial functions χ ∈ D(B(0, 4/3)) and ϕ ∈ D(C) valued in the interval [0, 1] and such that: ∀ξ ∈ Rn, χ(ξ) +

• j≥0

ϕ(2−jξ) = 1, (5) 2 ≤

• j − j′

⇒ Supp(ϕ(2−j·)) ∩ Supp(ϕ(2−j′·)) = ∅,

(6) j ≥ 1 ⇒ Supp(χ) ∩ Supp(ϕ(2−j·)) = ∅. (7)

Dyadic blocks

Let us denote by h = F−1ϕ and ˜ h = F−1χ. For all u ∈ S′, the nonhomogeneous dyadic blocks are defined as follows: ∆ju = 0 if j ≤ −2, ∆−1u = χ (D) u = ˜ h ⋆ u, (8) ∆ju = ϕ

• 2−jD
• u = 2jd
• Rn h (2qy) u (x − y) dy if j ≥ 0.

Besov spaces

Definition

Let s ∈ R, r ∈ [1, ∞]. The Besov space Bs

2,r is the set of tempered

distributions u ∈ S′ such that: uBs

2,r :=

• 2js ∆juL2
• j∈Z
• ℓr(Z)

< ∞.

Besov spaces

Definition

Let s ∈ R, r ∈ [1, ∞]. The Besov space Bs

2,r is the set of tempered

distributions u ∈ S′ such that: uBs

2,r :=

• 2js ∆juL2
• j∈Z
• ℓr(Z)

< ∞. The classical Sobolev space Hs = Bs

2,2

Some definitions

Definition

Let T > 0 a positive real number. We will say that T is bounded from below by a O

• 1

ε

• order quantity if there exists another

positive real number C, independent of ε such that: T ≥ C ε .

Some definitions

Let us consider a Banach space (X, ·X) which is continuously imbedded in L2 (Rn) ×

L2 (Rn) n.

Definition

We say that we can establish long time existence and uniqueness

• f solutions for the equation (3) in X if for any (η0, V0) ∈ X there

exists a positive time T > 0, an unique solution (η, V ) ∈ C ([0, T] , X) of (3) and a function F : (0, +∞) → (0, +∞) independent of ε such that: T ≥ F ((η0, V0)X) ε .

The main result

• s1 = s + sgn (b) − sgn (c) ,

s2 = s + sgn (d) − sgn (a) . (9)

Theorem

Let a ≤ 0, b ≥ 0, c ≤ 0, d ≥ 0 with b + d > 0. Let us consider s such that s > n

2 + 1 with n ≥ 1. Let us also consider s1, s2 defined

by (9). Then, we can establish long time existence and uniqueness

• f solutions for the equation (3) in Hs1 × (Hs2)n. Moreover, if we

denote by T (η0, V0), the maximal time of existence then there exists some T ∈ [0, T (η0, V0)) which is bounded from below by an O

• 1

ε

• order quantity and a function G : R → R such that for all

t ∈ [0, T] we have: (η, V )s ≤ G ((η0, V0)s) , where G may depend on a, b, c, d, s, n but not on ε.

Method of proof

◮ For the sake of simplicity we will ilustrate how the method

works in the case when curl V0 = 0 and b = d > 0 and a = c = 0, the so called BBM-BBM case.

Method of proof

◮ For the sake of simplicity we will ilustrate how the method

works in the case when curl V0 = 0 and b = d > 0 and a = c = 0, the so called BBM-BBM case.

◮ Our approach is based on an energy method applied to

spectrally localized equations. We first derive a priori estimates and we establish local existence and uniqueness of solutions for the general abcd system.

Method of proof

◮ For the sake of simplicity we will ilustrate how the method

works in the case when curl V0 = 0 and b = d > 0 and a = c = 0, the so called BBM-BBM case.

◮ Our approach is based on an energy method applied to

spectrally localized equations. We first derive a priori estimates and we establish local existence and uniqueness of solutions for the general abcd system.

◮ Formally, the following identity holds true:

d dt

• η2 + (1 + εη) V 2 = 0

A priori estimates

By localizing equation (3) in Fourier we obtain that:

    

(I − εb∆) ∂tηj + div Vj + εV ∇ηj + εη div Vj = εR1j (I − εd∆) ∂tVj + ∇ηj + ε∇VjV = εR2j ηj|t=0 = ∆jη0 , Vj|t=0 = ∆jV0 (10) where (ηj, Vj) := (∆jη, ∆jV ).

A priori estimates

By localizing equation (3) in Fourier we obtain that:

    

(I − εb∆) ∂tηj + div Vj + εV ∇ηj + εη div Vj = εR1j (I − εd∆) ∂tVj + ∇ηj + ε∇VjV = εR2j ηj|t=0 = ∆jη0 , Vj|t=0 = ∆jV0 (10) where (ηj, Vj) := (∆jη, ∆jV ). d dt

• η2

j +εb |∇ηj|2+|Vj|2+εd (∇Vj : ∇Vj)+ε

• ηηj div Vj ≤ O (ε)

(11)

A priori estimates

U2

j (η, V ) =

• η2

j + εb |∇ηj|2 + V 2 j + εd (∇Vj : ∇Vj) ,

A priori estimates

U2

j (η, V ) =

• η2

j + εb |∇ηj|2 + V 2 j + εd (∇Vj : ∇Vj) ,

U2

s (η, V ) = η2 Bs

2,r + εb ∇η2

Bs

2,r + V 2

Bs

2,r + εd ∇V 2

Bs

2,r

A priori estimates

1 2∂t

• η2

j + εb |∇ηj|2 + (1 + εη) V 2 j + εd (1 + εη) (∇Vj : ∇Vj)

• ≤ εCUj
• Uj
• Us + U2

s

• + cj(t)(Us + U2

s )

• ,

(12)

A priori estimates

N2

j (η, V ) =

• (1 + ε ηL∞) η2

j + εb (1 + ε ηL∞) |∇ηj|2

+

• (1 + εη + ε ηL∞) V 2

j + εd (1 + εη + ε ηL∞) (∇Vj : ∇Vj)

(13)

A priori estimates

N2

j (η, V ) =

• (1 + ε ηL∞) η2

j + εb (1 + ε ηL∞) |∇ηj|2

+

• (1 + εη + ε ηL∞) V 2

j + εd (1 + εη + ε ηL∞) (∇Vj : ∇Vj)

(13) Ns (η, V ) =

• 2jsNj (η, V )
• j∈Z
• ℓr

A priori estimates

N2

j (η, V ) =

• (1 + ε ηL∞) η2

j + εb (1 + ε ηL∞) |∇ηj|2

+

• (1 + εη + ε ηL∞) V 2

j + εd (1 + εη + ε ηL∞) (∇Vj : ∇Vj)

(13) Ns (η, V ) =

• 2jsNj (η, V )
• j∈Z
• ℓr

which satisfies: Us (η, V ) Ns (η, V ) (1 + 2ε ηL∞)

1 2 Us (η, V )

(14)

A priori estimates

We finally obtain: Ns (t) ≤ Ns (0) + Cε

t

Us

• Us + U2

s

• dτ

A priori estimates

We finally obtain: Ns (t) ≤ Ns (0) + Cε

t

Us

• Us + U2

s

• dτ

Gronwall’s lemma finishes the proof.

Conclusions and further results

◮ The function spaces for which we obtain l.t.e. results are the

largest that are known so far (for some values of the parameters we require up to less than three derivatives on the initial data).

Conclusions and further results

◮ The function spaces for which we obtain l.t.e. results are the

largest that are known so far (for some values of the parameters we require up to less than three derivatives on the initial data).

◮ Our approach allows us to obtain in a unified manner l.t.e.

results for almost all the values of the parameters a, b, c, d.

Conclusions and further results

◮ The function spaces for which we obtain l.t.e. results are the

largest that are known so far (for some values of the parameters we require up to less than three derivatives on the initial data).

◮ Our approach allows us to obtain in a unified manner l.t.e.

results for almost all the values of the parameters a, b, c, d.

◮ The techniques employed may work when trying to establish

l.t.e. results for bore-type initial data.

Thank you for your attention!