Discontinuous Solutions for the Degasperis-Procesi Equation. - - PDF document

Discontinuous Solutions for the Degasperis-Procesi Equation.

Giuseppe Maria Coclite

Department of Mathematics University of Bari Via Orabona 4, 70125 Bari, Italy e-mail: coclitegm@dm.uniba.it joint work with Professor

• K. H. Karlsen (University of Oslo and C.M.A. - Oslo, Norway)

Consider the equation ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 , (DP) where t ≥ 0, x ∈ I R, u(t, x) ∈ I R. (DP) is termed Degasperis - Procesi Equation.

Deduction.

Degasperis and Procesi in 1999 studied the following family of third order dispersive nonlinear equations, indexed over six constants c0, γ, α, c1, c2, c3 ∈ I R: ∂u ∂t +c0 ∂u ∂x+γ ∂3u ∂x3 −α2 ∂3u ∂t∂x2 = ∂ ∂x

• c1u2 + c2

∂u ∂x 2 + c3u∂2u ∂x2

• .

Using the method of asymptotic integrability, they found that only three equations from this family were asymptotically integrable up to third order:

• the KdV equation: α = c2 = c3 = 0;
• the Camassa-Holm equation: c1 = − 3c3

2α2 , c2 = c3 2 ;

• one new equation: c1 = −2c3

α2 , c2 = c3 (after a proper scaling) ∂u ∂t +∂u ∂x+6u∂u ∂x+∂3u ∂x3 −α2 ∂3u ∂t∂x2 + 9 2 ∂u ∂x ∂2u ∂x2 + 3 2u∂3u ∂x3

• = 0.

1

After

• rescaling
• shifting the dependent variable
• applying a Galilean boost

the three equations read

• the KdV equation:

∂u ∂t + u∂u ∂x + ∂3u ∂x3 = 0 (KdV )

• the Camassa-Holm equation:

∂u ∂t − ∂3u ∂t∂x2 + 3u∂u ∂x = 2∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 (CH)

• the Degasperis - Procesi equation:

∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 (DP)

Physics: Shallow Water Waves.

Shallow Water Waves ⇐ ⇒ depth of the water length of the wave < < 1

• Unidirectional shallow water waves
• u ≡ wave height (KdV) / velocity (CH - DP) above the bottom
• Flat bottom

2

Essential Literature.

Degasperis - Holm - Khon (2002):

• exact integrability (by constructing a Lax pair)
• bi-Hamiltonian structure
• two infinite sequences of conserved quantities
• “non-smooth” solutions: superpositions of multipeakons
• some special explicit solution

Lundmark - Szmigielski (2003):

• n-peakon solutions (via inverse scattering)

Mustafa (2005):

• smooth solutions have infinite speed of propagation

Yin (2003-2003-2004-2004):

• Cauchy Problem: local and global well-posedness

u0 ∈ Hr(S1), r > 3 2

• r

u0 ∈ Hs(I R), s ≥ 3 sign

• u0 − ∂2u0

∂x2

• constant
• Remark. The signum of the vorticity u − ∂2u

∂x2 is conserved sign

• u0 − ∂2u0

∂x2

• > 0 =

⇒ sign

• u(t, ·) − ∂2u

∂x2 (t, ·)

• > 0

3

Conservation Law Approach.

This third order equation ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 , (DP) is formally equivalent to the elliptic - hyperbolic system        ∂u ∂t + u∂u ∂x + ∂P ∂x = 0, −∂2P ∂x2 + P = 3 2u2, to the integro - differential system        ∂u ∂t + u∂u ∂x + ∂P ∂x = 0, P(t, x) = 3 4

• I

R

e−|x−y|u2(t, y)dy, and, finally, to the integro - differential equation ∂u ∂t + u∂u ∂x + 3 4

• I

R

e−|x−y|sign(y − x)u2(t, y)dy = 0. 4

Comparison with the Camssa - Holm Equation.

Degasperis - Procesi Equation. ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 (DP)

• 

      ∂u ∂t + u∂u ∂x + ∂P ∂x = 0 −∂2P ∂x2 + P = 3 2u2 (wDP) Functional setting L∞

loc

• I

R+; L2

loc(I

R)

• : discontinuous solutions.

Camassa - Holm Equation. ∂u ∂t − ∂3u ∂t∂x2 + 3u∂u ∂x = 2∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 (CH)

• 

      ∂u ∂t + u∂u ∂x + ∂P ∂x = 0 −∂2P ∂x2 + P = 1 2 ∂u ∂x 2 + u2 (wCH) Functional setting L∞

loc

• I

R+; H1

loc(I

R)

• : continuous solutions.

5

Definitions of Solutions.

Weak Solutions.

We call u : I R+ × I R → I R weak solution of the Cauchy problem      ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 , u(0, x) = u0(x), (CP) if and only if i) u ∈ L∞ I R+; L2(I R)

• ;

ii) u satisfies              ∂u ∂t + u∂u ∂x + ∂P ∂x = 0, −∂2P ∂x2 + P = 3 2u2, u(0, x) = u0(x), (wCP) in the sense of distributions. 6

entropy weak solutions.

We call u : I R+ × I R → I R entropy weak solution of the Cauchy problem      ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 , u(0, x) = u0(x), (CP) if and only if i) u ∈ L∞ [0, T]; BV (I R)

• ∩ L∞

I R+; L2(I R)

• , T > 0;

ii) u satisfies              ∂u ∂t + u∂u ∂x + ∂P ∂x = 0, −∂2P ∂x2 + P = 3 2u2, u(0, x) = u0(x), (wCP) in the sense of distributions; iii) for any convex C2 entropy η : I R → I R with corresponding entropy flux q : I R → I R defined by q′(u) = η′(u) u there holds ∂η(u) ∂t + ∂q(u) ∂x + η′(u)∂P ∂x ≤ 0, in the sense of distributions on I R+ × I R. Remark. u entropy weak solution = ⇒ u weak solution. 7

• Theorem. (G.M.C. - K. H. Karlsen (JFA - 2006))
• (Existence) If

u0 ∈ L1(I R) ∩ BV (I R), then there exists an entropy weak solution to the Cauchy prob- lem (CP).

• (Stability and uniqueness) Fix any T > 0, and let u, v be

two entropy weak solutions to (CP) with initial data u0, v0 ∈ L1(I R) ∩ BV (I R),

• respectively. Then for any t ∈ (0, T)

u(t, ·) − v(t, ·)L1(I

R) ≤ eMT tu0 − v0L1(I R),

MT := 3 2

• uL∞((0,T )×I

R) + vL∞((0,T )×I R)

• < ∞.

In particular, there exists at most one entropy weak solution to (CP).

• (Time L1-continuity)For any T > 0:

u(t2, ·) − u(t1, ·)L1(I

R) ≤ CT |t2 − t1|,

∀t1, t2 ∈ [0, T], CT :=

• u0L1(I

R) + 12Tu02 L2(I R)

2 + 12u02

L2(I R).

• (Oleinik type estimate) For a.e. (t, x) ∈ (0, T] × I

R, ∂u ∂x(t, x) ≤ 1 t + KT , KT :=

• 6u02

L2(I R) + 3

2

• TV (u0) + 24Tu02

L2(I R)

21/2 . 8

Vanishing Viscosity Approximation.

We approximate (wCP) with the following elliptic - parabolic system              ∂uε ∂t + uε ∂uε ∂x + ∂Pε ∂x = ε∂2uε ∂x2 , −∂2Pε ∂x2 + Pε = 3 2u2

ε,

uε(0, x) = uε,0(x), that is equivalent to the fourth order problem              ∂uε ∂t − ∂3uε ∂t∂x2 + 3uε ∂uε ∂x = 3∂uε ∂x ∂2uε ∂x2 + u∂3uε ∂x3 + ε∂2uε ∂x2 − ε∂4uε ∂x4 , uε(0, x) = uε,0(x).

• Existence and Uniqueness of smooth solutions if uε,0 ∈ H2(I

R).

• Lipschitz continuity with respect to the viscosity coefficient ε

and the initial condition uε,0.

• The Lipschitz constant depends on ε.

(see G.M.C. - H. Holden - K. H. Karlsen (DCDS - 2005)) 9

L2- estimate. ∂uε ∂t + uε ∂uε ∂x + 3 4

• I

R

e−|x−y|sign(y − x)u2

ε(t, y)dy = ε∂2uε

∂x2 Viscous Burgers equation with a nonlocal source. Estimate on the nonlocal term: L2 bound on uε.

• multiplying by uε and integrating on I

R is not working.

• Hamiltonian structure: conserved quantities for (DP).

Let v = v(t, x) be defined by the equation −∂2v ∂x2 + 4v = u

• I

R

v

• u − ∂2u

∂x2

• dx

is a conserved quantity for (DP) Observe that

• I

R

v

• u − ∂2u

∂x2

• dx =
• I

R

• 4 − ∂2

∂x2 −1 u

• 1 − ∂2

∂x2

• u dx =

=

• I

R

• 4v2 + 5

∂v ∂x 2 + ∂2v ∂x2 2 dx ≃ v2

H2(I R) ≃ u2 L2(I R)

10

At the viscus level, let vε = vε(t, x) be defined by the equation −∂2vε ∂x2 + 4vε = uε. Equation for vε: 4∂vε ∂t − 5 ∂3vε ∂t∂x2 + ∂5vε ∂t∂x4 + 43vε ∂vε ∂x = 42 4∂vε ∂x ∂2vε ∂x2 + 2vε ∂3vε ∂x3

• − 4
• 5∂2vε

∂x2 ∂3vε ∂x3 + 3∂vε ∂x ∂4vε ∂x4 + vε ∂5vε ∂x5

• + 3

∂3vε ∂x3 ∂4vε ∂x4 + ∂2vε ∂x2 ∂5vε ∂x5

• + 4ε∂2vε

∂x2 − 5ε∂4vε ∂x4 + ε∂6vε ∂x6 . Multiplying by vε, integrating on I R and playing the game of the integrations by parts

• vε(t, ·)
• 2
• H2(I

R) + 2ε

t

• ∂vε

∂x (τ, ·)

• 2
• H2(I

R)dτ =

• vε(0, ·)
• 2
• H2(I

R),

where f2

• H2(I

R) := 4f2 L2(I R) + 5

• ∂f

∂x

• 2

L2(I R) +

• ∂2f

∂x2

• 2

L2(I R).

In particular

• uε(t, ·)
• L2(I

R)

√ε

• ∂uε

∂x

• L2(I

R+×I R) ≤ 2

√ 2u0L2(I

R)

11

Bounds on the Nonlocal Term Pε.

Since −∂2Pε ∂x2 + Pε = 3 2u2

ε

⇓ Pε(t, x) = 3 4

• I

R

e−|x−y|u2

ε(t, y)dy

∂Pε ∂x (t, x) = 3 4

• I

R

e−|x−y|sign(y − x)u2

ε(t, y)dy

playing with the H¨

• lder Inequality
• uε(t, ·)
• L2(I

R) ≤ 2

√ 2u0L2(I

R)

⇓ Pε(t, ·)L1(I

R),

• ∂Pε

∂x (t, ·)

• L1(I

R) ≤ 12u02 L2(I R)

PεL∞(I

R+×I R),

• ∂Pε

∂x

• L∞(I

R+×I R) ≤ 6u02 L2(I R)

• ∂2Pε

∂x2 (t, ·)

• L1(I

R) ≤ 24u02 L2(I R)

12

L1, L∞, BV - estimates on uε. ∂uε ∂t + uε ∂uε ∂x + ∂Pε ∂x = ε∂2uε ∂x2 estimates on the nonlocal term Pε and ∂Pε ∂x ⇓ uε(t, ·)L1(I

R) ≤ u0L1(I R) + 12tu02 L2(I R)

TV (uε)(t, ·) =

• ∂uε

∂x (t, ·)

• L1(I

R) ≤ TV (u0) + 24tu02 L2(I R)

uε(t, ·)L∞(I

R) ≤ TV (u0) + 24tu02 L2(I R)

• ∂uε

∂t (t, ·)

• L1(I

R) ≤

• TV (u0) + 24tu02

L2(I R)

2 + 12u0|2

L2(I R)

• ∂2Pε

∂x2 (t, ·)

• L∞(I

R) ≤ 6u02 L2(I R) + 3

2

• TV (u0) + 24tu02

L2(I R)

2 In addition uε(t2, ·) − uε(t1, ·)L1(I

R) ≤ CT |t2 − t1|

0 ≤ t1, t2 ≤ T CT :=

• u0L1(I

R) + 12Tu02 L2(I R)

2 + 12u02

L2(I R)

13

Oleinik Type Estimate.

∂uε ∂x (t, x) ≤ 1 t + KT 0 < t ≤ T where KT :=

• 6u02

L2(I R) + 3

2

• TV (u0) + 24Tu02

L2(I R)

21/2

• only decreasing discontinuities (shocks), see Burgers equation
• KT depends on TV (u0)
• KT depends on T

Equation for the First Derivative. qε := ∂uε ∂x ∂qε ∂t + uε ∂qε ∂x + q2

ε − ε∂2qε

∂x2 = −∂2Pε ∂x2 ≤ K2

T

Comparing qε with the solution f of the ODE d f dt + f2 = K2

T

we find ∂uε ∂x (t, x) ≤ 1 t + KT 0 < t ≤ T 14

Existence of Entropy Weak Solutions.

L1 and BV bounds ⇓ ∃ {εk}k∈I

N s.t. εk > 0, εk −

→ 0 ∃ u ∈ L∞ [0, T]; BV (I R)

• ∩ L∞

I R+; L2(I R)

• , T > 0

uεk − → u a.e. in I R+ × I R uεk − → u strongly in Lp

loc(I

R+ × I R) 1 ≤ p < ∞ Convergence of the Nonlocal Term. −∂2Pεk ∂x2 + Pεk = 3 2u2

εk

− ∂2P u ∂x2 + P u = 3 2u2 ⇓ Pεk(t, x) = 3 4

• I

R

e−|x−y|u2

ε(t, y)dy

P u(t, x) = 3 4

• I

R

e−|x−y|u2(t, y) dy Pεk−P uLp((0,T )×I

R),

• ∂Pεk

∂x −∂P u ∂x

• Lp((0,T )×I

R) ≤ CT uεk−uLp((0,T )×I R)

• Pεk −

→ P u, ∂Pεk ∂x − → ∂P u ∂x in Lp

loc(I

R+ × I R) 1 ≤ p < ∞ 15

Stability and Uniqueness

• f Entropy Weak Solutions.

Let u, v be two entropy weak solutions to (CP) with initial data u0, v0 ∈ L1(I R) ∩ BV (I R) Doubling of Variables ⇓ u(t, ·)−v(t, ·)L1(I

R) ≤ u0−v0L1(I R)+

t P u(s, ·)−P v(s, ·)L1(I

R)ds

−∂2P u ∂x2 + P u = 3 2u2 − ∂2P v ∂x2 + P v = 3 2v2 Since P u(s, ·)−P v(s, ·)L1(I

R) ≤

≤3 2

• uL∞((0,T )×I

R) + vL∞((0,T )×I R)

• ×

× u(s, ·) − v(s, ·)L1(I

R)

0 ≤ s ≤ T ⇓ u(t, ·) − v(t, ·)L1(I

R) ≤

≤ u0 − v0L1(I

R) + CT

t u(s, ·) − v(s, ·)L1(I

R)ds

0 < t < T ⇓ u(t, ·) − v(t, ·)L1(I

R) ≤ eCT tu0 − v0L1(I R)

0 < t < T 16

Uniqueness via an Oleinik type Estimate.

• Theorem. (G.M.C. - K. H. Karlsen (2006))

Suppose u0 ∈ L∞(I R). Then there exists at most one weak solution to the Cauchy problem      ∂u ∂t − ∂3u ∂t∂x2 + 4u∂u ∂x = 3∂u ∂x ∂2u ∂x2 + u∂3u ∂x3 , u(0, x) = u0(x), (CP) such that

• for each T > 0 there exists a constant ΛT > 0 such that the

estimate u(t, x) − u(t, y) x − y ≤ ΛT 1 t + 1

• holds for any x, y ∈ I

R, x = y, 0 < t < T. 17

Comparison with the Burgers Equation.

Burgers Equation. u entropy weak solution and u0 ∈ L∞(I R) ⇓ u(t, x) − u(t, y) x − y ≤ 1 t ⇓ u(t, ·) ∈ BV (I R) t > 0 Degasperis - Procesi Equation. u entropy weak solution and u0 ∈ BV (I R) ⇓ ΛT exists and ΛT depends on TV (u0), T 18

The Duality Argument.

Let u, u be two weak solutions of (CP) satisfying an Oleinik type estimate. Claim: u = u a.e. in I R+ × I R. Define ω := u − u b := u + u 2                ∂ω ∂t + ∂ ∂x(bω) + ∂ P ∂x = 0 −∂2 P ∂x2 + P = 3bω ω(0, x) = 0 The Nonlocal Adjoint Problem.              ∂ϕ ∂t + b∂ϕ ∂x + 3b∂Φ ∂x = ψ −∂2Φ ∂x2 + Φ = ϕ ϕ(τ, x) = 0

• linear hyperbolic-elliptic terminal value problem
• ψ ∈ C∞

c (I

R+ × I R), supp(ψ) ⊂]0, τ[×I R 19

The Approximate Nonlocal Adjoint Problem.

• due to the low regularity of the coefficient b, we cannot solve

directly the problem Regularization.

• smoothing the coefficient b by convolution:

bε := b ∗ ρε ε > 0. where {ρε(t, x)}ε>0 is a sequence of standard mollifiers

• adding an artificial viscosity term

             ∂ϕε ∂t + bε ∂ϕε ∂x + 3bε ∂Φε ∂x = ψ −∂2Φε ∂x2 + Φε = ϕε ϕ(τ, x) = 0

• linear parabolic-elliptic terminal value problem

20

A priori estimates. ϕε(t, ·)2

H1(I R) + 2ε

τ

t

• ∂ϕε(s, ·)

∂x

• 2

H1(I R)ds

≤ eCτ τ τ t Cτ τ

t

ψ(s, ·)2

H1(I R)ds

Φε(t, ·)2

H3(I R) + 2ε

τ

t

• ∂Φε(s, ·)

∂x

• 2

H3(I R)ds

≤ eCτ τ τ t Cτ τ

t

ψ(s, ·)2

H1(I R)ds

0 < t < τ Cτ is a constant independent of ε but dependent of τ Remark. Existence of distributional solutions to the nonlocal adjoint problem ϕ ∈ L∞((δ, τ); H1(I R)) Φ ∈ L∞((δ, τ); H3(I R)) ∀δ > 0 21