On the existence of traveling waves for monomer chains with - - PowerPoint PPT Presentation

On the existence of traveling waves for monomer chains with pre-compression

Atanas Stefanov

Department of Mathematics The University of Kansas

2nd LENCOS conference July 11th 2012, Seville

The Fermi-Pasta-Ulam (FPU) problem - I

Infinite dimensional Hamiltonian system with H =

∞

• j=−∞

˙ q2

j

2 + V(qj+1 − qj) Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V(r) = ar 2 + br 3 cubic FPU V(r) = c((1 + r)−12 − 2(1 + r)−6 + 1) Lennard-Jones V(r) = α(eβr − βr − 1) Toda

The Fermi-Pasta-Ulam (FPU) problem - I

Infinite dimensional Hamiltonian system with H =

∞

• j=−∞

˙ q2

j

2 + V(qj+1 − qj) Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V(r) = ar 2 + br 3 cubic FPU V(r) = c((1 + r)−12 − 2(1 + r)−6 + 1) Lennard-Jones V(r) = α(eβr − βr − 1) Toda

The Fermi-Pasta-Ulam (FPU) problem - I

Infinite dimensional Hamiltonian system with H =

∞

• j=−∞

˙ q2

j

2 + V(qj+1 − qj) Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V(r) = ar 2 + br 3 cubic FPU V(r) = c((1 + r)−12 − 2(1 + r)−6 + 1) Lennard-Jones V(r) = α(eβr − βr − 1) Toda

The Fermi-Pasta-Ulam (FPU) problem - I

Infinite dimensional Hamiltonian system with H =

∞

• j=−∞

˙ q2

j

2 + V(qj+1 − qj) Models of (infinite) nonlinear mass-spring chains. V is the “interaction potential”, usually smooth. The particular form of V is the Hooke’s law that we assume. Prototypical examples: V has minimum at zero, convex at zero V(r) = ar 2 + br 3 cubic FPU V(r) = c((1 + r)−12 − 2(1 + r)−6 + 1) Lennard-Jones V(r) = α(eβr − βr − 1) Toda

The Fermi-Pasta-Ulam (FPU) problem - II

The equation of motion is ¨ qj(t) = V ′(qj+1 − qj) − V ′(qj − qj−1) (1) In terms of the displacement functions rj = qj+1 − qj, ¨ rj(t) = V ′(rj+1) − 2V ′(rj) + V ′(rj−1) (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.

The Fermi-Pasta-Ulam (FPU) problem - II

The equation of motion is ¨ qj(t) = V ′(qj+1 − qj) − V ′(qj − qj−1) (1) In terms of the displacement functions rj = qj+1 − qj, ¨ rj(t) = V ′(rj+1) − 2V ′(rj) + V ′(rj−1) (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.

The Fermi-Pasta-Ulam (FPU) problem - II

The equation of motion is ¨ qj(t) = V ′(qj+1 − qj) − V ′(qj − qj−1) (1) In terms of the displacement functions rj = qj+1 − qj, ¨ rj(t) = V ′(rj+1) − 2V ′(rj) + V ′(rj−1) (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.

The Fermi-Pasta-Ulam (FPU) problem - II

The equation of motion is ¨ qj(t) = V ′(qj+1 − qj) − V ′(qj − qj−1) (1) In terms of the displacement functions rj = qj+1 − qj, ¨ rj(t) = V ′(rj+1) − 2V ′(rj) + V ′(rj−1) (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.

The Fermi-Pasta-Ulam (FPU) problem - II

The equation of motion is ¨ qj(t) = V ′(qj+1 − qj) − V ′(qj − qj−1) (1) In terms of the displacement functions rj = qj+1 − qj, ¨ rj(t) = V ′(rj+1) − 2V ′(rj) + V ′(rj−1) (2) differential advance-delay system of infinitely many ODE’s Standard theories for existence and uniqueness fail.

Traveling waves for FPU systems

Substituting r(t, j) = rc(j − ct) yields c2r ′′

c (x)

= V ′(rc(x + 1)) − 2V ′(rc(x)) + V ′(rc(x − 1)) = ∆disc.V ′(rc(·))(x) Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞? Are the solutions r bell-shaped, i.e. positive, even and decaying in (0, ∞)?

Traveling waves for FPU systems

Substituting r(t, j) = rc(j − ct) yields c2r ′′

c (x)

= V ′(rc(x + 1)) − 2V ′(rc(x)) + V ′(rc(x − 1)) = ∆disc.V ′(rc(·))(x) Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞? Are the solutions r bell-shaped, i.e. positive, even and decaying in (0, ∞)?

Traveling waves for FPU systems

Substituting r(t, j) = rc(j − ct) yields c2r ′′

c (x)

= V ′(rc(x + 1)) − 2V ′(rc(x)) + V ′(rc(x − 1)) = ∆disc.V ′(rc(·))(x) Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞? Are the solutions r bell-shaped, i.e. positive, even and decaying in (0, ∞)?

Traveling waves for FPU systems

Substituting r(t, j) = rc(j − ct) yields c2r ′′

c (x)

= V ′(rc(x + 1)) − 2V ′(rc(x)) + V ′(rc(x − 1)) = ∆disc.V ′(rc(·))(x) Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞? Are the solutions r bell-shaped, i.e. positive, even and decaying in (0, ∞)?

Traveling waves for FPU systems

Substituting r(t, j) = rc(j − ct) yields c2r ′′

c (x)

= V ′(rc(x + 1)) − 2V ′(rc(x)) + V ′(rc(x − 1)) = ∆disc.V ′(rc(·))(x) Questions: For which speeds c are there solutions? Are the solutions r decaying at ±∞? Are the solutions r bell-shaped, i.e. positive, even and decaying in (0, ∞)?

Previous results

Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy K = ∞

−∞

V(u(x − 1) − u(x))dx Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. cs + ǫ > |c| ≥ cs :=

• V ′′(0).

Previous results

Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy K = ∞

−∞

V(u(x − 1) − u(x))dx Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. cs + ǫ > |c| ≥ cs :=

• V ′′(0).

Previous results

Toda - explicit formulas in the “Toda” case, due to the integrability. Friesecke-Wattis [Comm. Math. Phys.’94] - existence of waves with prescribed average potential energy K = ∞

−∞

V(u(x − 1) − u(x))dx Also work of Smets-Willem, [J. Funct. Anal.’97] clarifying FW about the set of speeds. Friesecke-Pego [Nonlinearity 99, 02, 04,04], Existence and stability for “slightly supersonic speeds”, i.e. cs + ǫ > |c| ≥ cs :=

• V ′′(0).

Starosvetsky-Vakakis - PRE’10, existence of periodic

• waves. More results for dimers and numerics.
• G. James - J. Nonl. Sci.’12, existence of periodic traveling

waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in

• scillatory chains with Hertzian interactions.

Starosvetsky-Vakakis - PRE’10, existence of periodic

• waves. More results for dimers and numerics.
• G. James - J. Nonl. Sci.’12, existence of periodic traveling

waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in

• scillatory chains with Hertzian interactions.

Starosvetsky-Vakakis - PRE’10, existence of periodic

• waves. More results for dimers and numerics.
• G. James - J. Nonl. Sci.’12, existence of periodic traveling

waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in

• scillatory chains with Hertzian interactions.

Starosvetsky-Vakakis - PRE’10, existence of periodic

• waves. More results for dimers and numerics.
• G. James - J. Nonl. Sci.’12, existence of periodic traveling

waves for monomers and the Newton’s cradle. Betti-Pelinovsky’ 12 - continuation of periodic waves close to the anti-continuum limit. James-Kevrekidis-Cuevas, Physica D’12 - breathers in

• scillatory chains with Hertzian interactions.

Results: Precompression free case

Together with P . Kevrekidis [J. Nonlinear Science’ 2012], Theorem For all p > 1, c = 0, the equation c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p, has a positive H∞(R1) solution, which is bell-shaped. Moreover, r is a compacton. That is, it decays at a double exponential rate, i.e. ∃N > 0, ∀x > N, r(x) ≤ 2−px−N The compacton property was observed numerically and proved by English-Pego’1995, under the assumption for bell-shapedness, which was not known till our result.

Results: Precompression free case

Together with P . Kevrekidis [J. Nonlinear Science’ 2012], Theorem For all p > 1, c = 0, the equation c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p, has a positive H∞(R1) solution, which is bell-shaped. Moreover, r is a compacton. That is, it decays at a double exponential rate, i.e. ∃N > 0, ∀x > N, r(x) ≤ 2−px−N The compacton property was observed numerically and proved by English-Pego’1995, under the assumption for bell-shapedness, which was not known till our result.

Results: Precompression free case

Together with P . Kevrekidis [J. Nonlinear Science’ 2012], Theorem For all p > 1, c = 0, the equation c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p, has a positive H∞(R1) solution, which is bell-shaped. Moreover, r is a compacton. That is, it decays at a double exponential rate, i.e. ∃N > 0, ∀x > N, r(x) ≤ 2−px−N The compacton property was observed numerically and proved by English-Pego’1995, under the assumption for bell-shapedness, which was not known till our result.

Results: Precompression free case

Together with P . Kevrekidis [J. Nonlinear Science’ 2012], Theorem For all p > 1, c = 0, the equation c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p, has a positive H∞(R1) solution, which is bell-shaped. Moreover, r is a compacton. That is, it decays at a double exponential rate, i.e. ∃N > 0, ∀x > N, r(x) ≤ 2−px−N The compacton property was observed numerically and proved by English-Pego’1995, under the assumption for bell-shapedness, which was not known till our result.

An alternative formulation - FT it!

c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p = ∆disc.[r p](x) (3) If Λ : ˆ Λ(ξ) = sin2(πξ)

π2ξ2 ,

c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy (4)

Question: What is Λ? Recall

• χ[− 1

2, 1 2 ](ξ) = sin(πξ)

πξ Λ(x) = χ[− 1

2 , 1 2 ] ∗ χ[− 1 2 , 1 2] = (1 − |x|)+ =

1 − |x| |x| ≤ 1, |x| > 1.

An alternative formulation - FT it!

c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p = ∆disc.[r p](x) (3) If Λ : ˆ Λ(ξ) = sin2(πξ)

π2ξ2 ,

c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy (4)

Question: What is Λ? Recall

• χ[− 1

2 , 1 2 ](ξ) = sin(πξ)

πξ Λ(x) = χ[− 1

2 , 1 2] ∗ χ[− 1 2 , 1 2 ] = (1 − |x|)+ =

1 − |x| |x| ≤ 1, |x| > 1.

An alternative formulation - FT it!

c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p = ∆disc.[r p](x) (3) If Λ : ˆ Λ(ξ) = sin2(πξ)

π2ξ2 ,

c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy (4)

Question: What is Λ? Recall

• χ[− 1

2 , 1 2 ](ξ) = sin(πξ)

πξ Λ(x) = χ[− 1

2 , 1 2] ∗ χ[− 1 2 , 1 2 ] = (1 − |x|)+ =

1 − |x| |x| ≤ 1, |x| > 1.

An alternative formulation - FT it!

c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p = ∆disc.[r p](x) (3) If Λ : ˆ Λ(ξ) = sin2(πξ)

π2ξ2 ,

c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy (4)

Question: What is Λ? Recall

• χ[− 1

2 , 1 2 ](ξ) = sin(πξ)

πξ Λ(x) = χ[− 1

2 , 1 2] ∗ χ[− 1 2 , 1 2 ] = (1 − |x|)+ =

1 − |x| |x| ≤ 1, |x| > 1.

An alternative formulation - FT it!

c2r ′′(x) = r(x + 1)p − 2r(x)p + r(x − 1)p = ∆disc.[r p](x) (3) If Λ : ˆ Λ(ξ) = sin2(πξ)

π2ξ2 ,

c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy (4)

Question: What is Λ? Recall

• χ[− 1

2 , 1 2 ](ξ) = sin(πξ)

πξ Λ(x) = χ[− 1

2 , 1 2] ∗ χ[− 1 2 , 1 2 ] = (1 − |x|)+ =

1 − |x| |x| ≤ 1, |x| > 1.

Double exponential rate of decay

English and Pego’95 proved Lemma If the TW r is bell-shaped, then it has double exponential decay at ±∞. Proof. WLOG c = 1. Let r ↓ in (0, ∞), x > 0. Since

• Λ(y)dy = 1,

r(x +1) = 1

−1

Λ(y)r p(x +1−y)dy ≤ r p(x) ⇒ r(x +n) ≤ r(x)pn, Recall that r ↓ in (0, ∞) was not known at the time.

Double exponential rate of decay

English and Pego’95 proved Lemma If the TW r is bell-shaped, then it has double exponential decay at ±∞. Proof. WLOG c = 1. Let r ↓ in (0, ∞), x > 0. Since

• Λ(y)dy = 1,

r(x +1) = 1

−1

Λ(y)r p(x +1−y)dy ≤ r p(x) ⇒ r(x +n) ≤ r(x)pn, Recall that r ↓ in (0, ∞) was not known at the time.

Double exponential rate of decay

English and Pego’95 proved Lemma If the TW r is bell-shaped, then it has double exponential decay at ±∞. Proof. WLOG c = 1. Let r ↓ in (0, ∞), x > 0. Since

• Λ(y)dy = 1,

r(x +1) = 1

−1

Λ(y)r p(x +1−y)dy ≤ r p(x) ⇒ r(x +n) ≤ r(x)pn, Recall that r ↓ in (0, ∞) was not known at the time.

Double exponential rate of decay

English and Pego’95 proved Lemma If the TW r is bell-shaped, then it has double exponential decay at ±∞. Proof. WLOG c = 1. Let r ↓ in (0, ∞), x > 0. Since

• Λ(y)dy = 1,

r(x +1) = 1

−1

Λ(y)r p(x +1−y)dy ≤ r p(x) ⇒ r(x +n) ≤ r(x)pn, Recall that r ↓ in (0, ∞) was not known at the time.

Double exponential rate of decay

English and Pego’95 proved Lemma If the TW r is bell-shaped, then it has double exponential decay at ±∞. Proof. WLOG c = 1. Let r ↓ in (0, ∞), x > 0. Since

• Λ(y)dy = 1,

r(x +1) = 1

−1

Λ(y)r p(x +1−y)dy ≤ r p(x) ⇒ r(x +n) ≤ r(x)pn, Recall that r ↓ in (0, ∞) was not known at the time.

Sketch of the proof in the precompression-free case

Recall c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy Set w = r p. c2w1/p = Λ ∗ w = χ[− 1

2, 1 2 ] ∗ χ[− 1 2, 1 2 ] ∗ w

This is formally the Euler-Lagrange equation of

• χ[− 1

2, 1 2 ] ∗ v2

L2 =

∞

−∞ |

x+ 1

2

x− 1

2 v(y)dy|2dx → max

subject to

• R1 |v(x)|1+ 1

p dx = 1,

(5) Note: c2 is the Lagrange multiplier. The utility functional is bounded.

Sketch of the proof in the precompression-free case

Recall c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy Set w = r p. c2w1/p = Λ ∗ w = χ[− 1

2 , 1 2 ] ∗ χ[− 1 2, 1 2 ] ∗ w

This is formally the Euler-Lagrange equation of

• χ[− 1

2, 1 2 ] ∗ v2

L2 =

∞

−∞ |

x+ 1

2

x− 1

2 v(y)dy|2dx → max

subject to

• R1 |v(x)|1+ 1

p dx = 1,

(5) Note: c2 is the Lagrange multiplier. The utility functional is bounded.

Sketch of the proof in the precompression-free case

Recall c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy Set w = r p. c2w1/p = Λ ∗ w = χ[− 1

2 , 1 2 ] ∗ χ[− 1 2, 1 2 ] ∗ w

This is formally the Euler-Lagrange equation of

• χ[− 1

2, 1 2 ] ∗ v2

L2 =

∞

−∞ |

x+ 1

2

x− 1

2 v(y)dy|2dx → max

subject to

• R1 |v(x)|1+ 1

p dx = 1,

(5) Note: c2 is the Lagrange multiplier. The utility functional is bounded.

Sketch of the proof in the precompression-free case

Recall c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy Set w = r p. c2w1/p = Λ ∗ w = χ[− 1

2 , 1 2 ] ∗ χ[− 1 2, 1 2 ] ∗ w

This is formally the Euler-Lagrange equation of

• χ[− 1

2, 1 2 ] ∗ v2

L2 =

∞

−∞ |

x+ 1

2

x− 1

2 v(y)dy|2dx → max

subject to

• R1 |v(x)|1+ 1

p dx = 1,

(5) Note: c2 is the Lagrange multiplier. The utility functional is bounded.

Sketch of the proof in the precompression-free case

Recall c2r(x) = Λ ∗ r p(x) = ∞

−∞

Λ(x − y)r p(y)dy Set w = r p. c2w1/p = Λ ∗ w = χ[− 1

2 , 1 2 ] ∗ χ[− 1 2, 1 2 ] ∗ w

This is formally the Euler-Lagrange equation of

• χ[− 1

2, 1 2 ] ∗ v2

L2 =

∞

−∞ |

x+ 1

2

x− 1

2 v(y)dy|2dx → max

subject to

• R1 |v(x)|1+ 1

p dx = 1,

(5) Note: c2 is the Lagrange multiplier. The utility functional is bounded.

Flavor of Proof

Recall the decreasing rearrangement function f ∗(t) = inf{s > 0 : |{x ∈ R1 : |f(x)| > s}| ≤ t}; f #(t) = f ∗(2|t|) fLq = f #Lq; f = f # ⇐ ⇒ f - bell-shaped

• R2 f(x)g(x − y)h(y)dxdy ≤
• R2 f #(x)g#(x − y)h#(y)dxdy

QvL2 = sup

hL2=1

• v(x)χ[− 1

2 , 1 2](x − y)h(y)dxdy ≤

≤ sup

hL2=1

• v#(x)χ#

[− 1

2 , 1 2](x − y)h#(y)dxdy = Qv#L2

v# (a bell-shaped fuct.) is a better choice ⇒ solution is bell-shaped.

Flavor of Proof

Recall the decreasing rearrangement function f ∗(t) = inf{s > 0 : |{x ∈ R1 : |f(x)| > s}| ≤ t}; f #(t) = f ∗(2|t|) fLq = f #Lq; f = f # ⇐ ⇒ f - bell-shaped

• R2 f(x)g(x − y)h(y)dxdy ≤
• R2 f #(x)g#(x − y)h#(y)dxdy

QvL2 = sup

hL2=1

• v(x)χ[− 1

2 , 1 2](x − y)h(y)dxdy ≤

≤ sup

hL2=1

• v#(x)χ#

[− 1

2, 1 2 ](x − y)h#(y)dxdy = Qv#L2

v# (a bell-shaped fuct.) is a better choice ⇒ solution is bell-shaped.

Flavor of Proof

Recall the decreasing rearrangement function f ∗(t) = inf{s > 0 : |{x ∈ R1 : |f(x)| > s}| ≤ t}; f #(t) = f ∗(2|t|) fLq = f #Lq; f = f # ⇐ ⇒ f - bell-shaped

• R2 f(x)g(x − y)h(y)dxdy ≤
• R2 f #(x)g#(x − y)h#(y)dxdy

QvL2 = sup

hL2=1

• v(x)χ[− 1

2 , 1 2](x − y)h(y)dxdy ≤

≤ sup

hL2=1

• v#(x)χ#

[− 1

2, 1 2 ](x − y)h#(y)dxdy = Qv#L2

v# (a bell-shaped fuct.) is a better choice ⇒ solution is bell-shaped.

Flavor of Proof

Recall the decreasing rearrangement function f ∗(t) = inf{s > 0 : |{x ∈ R1 : |f(x)| > s}| ≤ t}; f #(t) = f ∗(2|t|) fLq = f #Lq; f = f # ⇐ ⇒ f - bell-shaped

• R2 f(x)g(x − y)h(y)dxdy ≤
• R2 f #(x)g#(x − y)h#(y)dxdy

QvL2 = sup

hL2=1

• v(x)χ[− 1

2 , 1 2](x − y)h(y)dxdy ≤

≤ sup

hL2=1

• v#(x)χ#

[− 1

2, 1 2 ](x − y)h#(y)dxdy = Qv#L2

v# (a bell-shaped fuct.) is a better choice ⇒ solution is bell-shaped.

Extensions - the case with precompression

Recall c2r ′′(x) = (δ0 + r(x + 1))p − 2(δ0 + r(x))p + (δ0 + r(x − 1))p Same transformations yield c2((z(x) + δp

0)

1 p − δ0) = Λ ∗ z.

(6) Question: What is the appropriate constrained maximization problem?

Extensions - the case with precompression

Recall c2r ′′(x) = (δ0 + r(x + 1))p − 2(δ0 + r(x))p + (δ0 + r(x − 1))p Same transformations yield c2((z(x) + δp

0)

1 p − δ0) = Λ ∗ z.

(6) Question: What is the appropriate constrained maximization problem?

Extensions - the case with precompression

Recall c2r ′′(x) = (δ0 + r(x + 1))p − 2(δ0 + r(x))p + (δ0 + r(x − 1))p Same transformations yield c2((z(x) + δp

0)

1 p − δ0) = Λ ∗ z.

(6) Question: What is the appropriate constrained maximization problem?

Constrained optimization problem for the precompression case

Jmax

δ

= sup

v∈Z

{ ∞

−∞

| x+1/2

x−1/2

v(y)dy|2} where Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

Constrained optimization problem for the precompression case

Jmax

δ

= sup

v∈Z

{ ∞

−∞

| x+1/2

x−1/2

v(y)dy|2} where Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

Recent result

With P . Kevrekidis, we have proved (2012), Theorem Let δ0 > 0, p > 1. Then, there is a solution to the problem for every speed |c| > δ

p−1 2

ap, where ap = infδ>0

Jmax

δ

δp−1 . In addition, p ≤ ap ≤ 2p.

Recall that Friesecke-Pego showed that the speed of sound is cs = √pδ

p−1 2

Thus, if we can verify that ap = p, the theorem is sharp. OPEN QUESTION.

Recent result

With P . Kevrekidis, we have proved (2012), Theorem Let δ0 > 0, p > 1. Then, there is a solution to the problem for every speed |c| > δ

p−1 2

ap, where ap = infδ>0

Jmax

δ

δp−1 . In addition, p ≤ ap ≤ 2p.

Recall that Friesecke-Pego showed that the speed of sound is cs = √pδ

p−1 2

Thus, if we can verify that ap = p, the theorem is sharp. OPEN QUESTION.

Recent result

With P . Kevrekidis, we have proved (2012), Theorem Let δ0 > 0, p > 1. Then, there is a solution to the problem for every speed |c| > δ

p−1 2

ap, where ap = infδ>0

Jmax

δ

δp−1 . In addition, p ≤ ap ≤ 2p.

Recall that Friesecke-Pego showed that the speed of sound is cs = √pδ

p−1 2

Thus, if we can verify that ap = p, the theorem is sharp. OPEN QUESTION.

Recent result

With P . Kevrekidis, we have proved (2012), Theorem Let δ0 > 0, p > 1. Then, there is a solution to the problem for every speed |c| > δ

p−1 2

ap, where ap = infδ>0

Jmax

δ

δp−1 . In addition, p ≤ ap ≤ 2p.

Recall that Friesecke-Pego showed that the speed of sound is cs = √pδ

p−1 2

Thus, if we can verify that ap = p, the theorem is sharp. OPEN QUESTION.

Recent result

With P . Kevrekidis, we have proved (2012), Theorem Let δ0 > 0, p > 1. Then, there is a solution to the problem for every speed |c| > δ

p−1 2

ap, where ap = infδ>0

Jmax

δ

δp−1 . In addition, p ≤ ap ≤ 2p.

Recall that Friesecke-Pego showed that the speed of sound is cs = √pδ

p−1 2

Thus, if we can verify that ap = p, the theorem is sharp. OPEN QUESTION.

Idea of the proof

Much more involved than the previous one. The constraining set Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

is the unit sphere of a (reflexive) Orlicz space with Orlicz function Φδ(z) := p p + 1(|z| + δp)1+ 1

p − δ|z| −

p p + 1δp+1. The main challenge is, after picking a weakly convergent subsequence of a maximizing sequence vnk ⇀ v to show QvnkL2 → QvL2.

Idea of the proof

Much more involved than the previous one. The constraining set Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

is the unit sphere of a (reflexive) Orlicz space with Orlicz function Φδ(z) := p p + 1(|z| + δp)1+ 1

p − δ|z| −

p p + 1δp+1. The main challenge is, after picking a weakly convergent subsequence of a maximizing sequence vnk ⇀ v to show QvnkL2 → QvL2.

Idea of the proof

Much more involved than the previous one. The constraining set Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

is the unit sphere of a (reflexive) Orlicz space with Orlicz function Φδ(z) := p p + 1(|z| + δp)1+ 1

p − δ|z| −

p p + 1δp+1. The main challenge is, after picking a weakly convergent subsequence of a maximizing sequence vnk ⇀ v to show QvnkL2 → QvL2.

Idea of the proof

Much more involved than the previous one. The constraining set Z = {v :

• R1
• p

p + 1(|v(x)| + δp)1+ 1

p − δ|v(x)| −

p p + 1δp+1

• dx = 1.}

is the unit sphere of a (reflexive) Orlicz space with Orlicz function Φδ(z) := p p + 1(|z| + δp)1+ 1

p − δ|z| −

p p + 1δp+1. The main challenge is, after picking a weakly convergent subsequence of a maximizing sequence vnk ⇀ v to show QvnkL2 → QvL2.

Thank you for your attention.