Waves, diffusion and variational principles1
Variational Principles for Travelling Waves of
Reaction-diffusion Equations
Oct 19-24, 2009
Workshop on Kinetic Theory and Fluid Dynamics
National Institute for Mathematical Sciences
and
Seoul National University and NIMS
Variational Principles for Travelling Waves
Reaction-diffusion equations
let u be a vector-valued function, A be a diagonal
nonnegative matrix.
∂u
= A ∆u + f (u).
∂t
(RD)
Reaction-diffusion equations describe phenomena in
chemical kinetics, physics, and biology
such as combustion, phase transitions,
propagation of nerve impulses, skin patterns of animals, etc.
Variational Principles for Travelling Waves
Travelling Waves:
(
∂u
∂t
= ∆u + f (u) on R
u(x, t) = u(x − ct)
Travelling waves usually describe transition process from one
equilibrium to another.
Bistable nonlinearity f (u) = −u(u − a)(u − 1).
Let z = x − ct.
ut = −cuz
∆z u + cuz + f (u) = 0 on R,
limz→−∞ u(z) = 1, limz→∞ u(z) = 0
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
◮
v ∈K z∈R
Vol’pert et al. (1994): monotone systems.
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
◮
Assumptions: monotonicity of the wave, uniqueness, stability
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
◮
Assumptions: monotonicity of the wave, uniqueness, stability
◮
Question: Can we use this principle to find the travelling
wave solution?
Variational Principles for Travelling Waves
Variational formulation for the solution
c = 0:
Let F (u) = −
Ru
0
f (s) ds.
vz2
+ F (v )] dz
R 2
0 < η(z) < 1, lim η(z) = 1, lim η(z) = 0
E (v ) =
Z
[
z→∞
z→−∞
1
S = {v = w + η : w ∈ H (R)}
Problem: minv ∈S E (v )
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
◮
Question: Which c should we choose?
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
◮
Question: Which c should we choose?
◮
Translation in space:
v (z) is a travelling wave solution ⇒ So is v (z − p).
Variational Principles for Travelling Waves
Variational formulation for the solution
◮
Translation in space:
Z
v 2 (z − p)
Ec (v (z − p)) =
e cz [ z
+ F (v (z − p))] dz
2
R
Z
v 2 (z ′ )
′
+ F (v (z ′ )] dz ′ = e cp Ec (v (z)).
=
e cz +cp [ z
2
R
(
∃Ec (v̄ ) < 0 ⇒ inf Ec = limp→∞ Ec (v̄ (z − p)) = −∞
∀Ec (v ) ≥ 0 ⇒ inf Ec = limp→−∞ Ec (v̄ (z − p)) = 0
Variational Principles for Travelling Waves
Variational formulation for the solution
Fife-McLeod (1977): cut-off of Ec to prove stability of travelling
wave
Ec as a variational problem: Heinze (2000), Muratov-Lucia-Novaga
(2004 -), Xinfu Chen (2006), Chen.
Theorem (Lucia-Muratov-Novaga, 2008)
Gradient system on a infinite cylinders: ∃ c∗ such that inf Ec∗ = 0
and it can be attained by a travelling wave solution.
Proof: Constraint
Z
e cz |∇v |2 = 1.
Gallay-Risler (preprint): Use Ec to prove the stability of travelling
waves for gradient systems
Variational Principles for Travelling Waves
Min-max formulation
z
To kill the translation: h(z) = 1 + tanh .
2
Z
v2
Eh,c (v ) =
h(cz)[ z + F (v )] dz
2
R
S = {v = w + η : w ∈ H 1 (R)}, lim η(z) = 1, lim η(z) = 0
z→−∞
z→∞
Problem: βc = inf v ∈S supp∈R Eh,c (v (z − p)) a critical value?
vzz +
hz (cz)
vz + f (v ) = 0
h(cz)
Theorem (Chen)
∃ c1 such that:
(1) βc > 0 if c > c1 ; βc = 0 if 0 < c ≤ c1 ,
(2) for c > c1 , ∃ critical point vc of Eh,c , Eh,c (vc ) = βc ,
(3) vck → a travelling wave solution v̄ .
Variational Principles for Travelling Waves
Gradient system
Gradient system on a infinite cylinders with Neumann boundary
condition:
∂u
= A ∆u − ∇u F (u).
∂t
(1)
Theorem
(1)coercive: F (u) → ∞ as |u| → ∞;
(2) w non-degenerate local min. point of F , F (w ) > inf F .
Then ∃ a travelling wave solution v such that v → w at ∞.
Variational Principles for Travelling Waves
Lyapunov Fuction
(1) Infinitely many Lyapunov fuctions
(2) Instantaneous propagation speed
Variational Principles for Travelling Waves
Open Problems
(1) Different diffusion rates
(2) Skew-gradient system
Variational Principles for Travelling Waves
Reaction-diffusion Equations
Oct 19-24, 2009
Workshop on Kinetic Theory and Fluid Dynamics
National Institute for Mathematical Sciences
and
Seoul National University and NIMS
Variational Principles for Travelling Waves
Reaction-diffusion equations
let u be a vector-valued function, A be a diagonal
nonnegative matrix.
∂u
= A ∆u + f (u).
∂t
(RD)
Reaction-diffusion equations describe phenomena in
chemical kinetics, physics, and biology
such as combustion, phase transitions,
propagation of nerve impulses, skin patterns of animals, etc.
Variational Principles for Travelling Waves
Travelling Waves:
(
∂u
∂t
= ∆u + f (u) on R
u(x, t) = u(x − ct)
Travelling waves usually describe transition process from one
equilibrium to another.
Bistable nonlinearity f (u) = −u(u − a)(u − 1).
Let z = x − ct.
ut = −cuz
∆z u + cuz + f (u) = 0 on R,
limz→−∞ u(z) = 1, limz→∞ u(z) = 0
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
◮
v ∈K z∈R
Vol’pert et al. (1994): monotone systems.
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
◮
Assumptions: monotonicity of the wave, uniqueness, stability
Variational Principles for Travelling Waves
Min-max principles for the wave speed
Hadeler and Rothe (1975): K = {v ∈ C 2 : vz < 0, 0 < v < 1}
ψ(v (z)) =
∆v (z) + f (v (z))
vz (z)
Theorem
sup inf ψ(v (z)) = c = inf sup ψ(v (z)).
v ∈K z∈R
v ∈K z∈R
◮
Vol’pert et al. (1994): monotone systems.
◮
Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
◮
Assumptions: monotonicity of the wave, uniqueness, stability
◮
Question: Can we use this principle to find the travelling
wave solution?
Variational Principles for Travelling Waves
Variational formulation for the solution
c = 0:
Let F (u) = −
Ru
0
f (s) ds.
vz2
+ F (v )] dz
R 2
0 < η(z) < 1, lim η(z) = 1, lim η(z) = 0
E (v ) =
Z
[
z→∞
z→−∞
1
S = {v = w + η : w ∈ H (R)}
Problem: minv ∈S E (v )
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
◮
Question: Which c should we choose?
Variational Principles for Travelling Waves
Variational formulation for the solution
c 6= 0: F (1) < F (0) = 0 ⇒ c > 0
Ec (v ) =
Z
R
Hc1
e cz [
vz2
+ F (v )] dz
2
= {v : ||v ||L2c + ||∇v ||L2c <
∞}, ||v ||2L2c
=
Z
e cz |v |2 dz
R
Variational problem: minv ∈Hc1 Ec (v )
◮
Question: Which c should we choose?
◮
Translation in space:
v (z) is a travelling wave solution ⇒ So is v (z − p).
Variational Principles for Travelling Waves
Variational formulation for the solution
◮
Translation in space:
Z
v 2 (z − p)
Ec (v (z − p)) =
e cz [ z
+ F (v (z − p))] dz
2
R
Z
v 2 (z ′ )
′
+ F (v (z ′ )] dz ′ = e cp Ec (v (z)).
=
e cz +cp [ z
2
R
(
∃Ec (v̄ ) < 0 ⇒ inf Ec = limp→∞ Ec (v̄ (z − p)) = −∞
∀Ec (v ) ≥ 0 ⇒ inf Ec = limp→−∞ Ec (v̄ (z − p)) = 0
Variational Principles for Travelling Waves
Variational formulation for the solution
Fife-McLeod (1977): cut-off of Ec to prove stability of travelling
wave
Ec as a variational problem: Heinze (2000), Muratov-Lucia-Novaga
(2004 -), Xinfu Chen (2006), Chen.
Theorem (Lucia-Muratov-Novaga, 2008)
Gradient system on a infinite cylinders: ∃ c∗ such that inf Ec∗ = 0
and it can be attained by a travelling wave solution.
Proof: Constraint
Z
e cz |∇v |2 = 1.
Gallay-Risler (preprint): Use Ec to prove the stability of travelling
waves for gradient systems
Variational Principles for Travelling Waves
Min-max formulation
z
To kill the translation: h(z) = 1 + tanh .
2
Z
v2
Eh,c (v ) =
h(cz)[ z + F (v )] dz
2
R
S = {v = w + η : w ∈ H 1 (R)}, lim η(z) = 1, lim η(z) = 0
z→−∞
z→∞
Problem: βc = inf v ∈S supp∈R Eh,c (v (z − p)) a critical value?
vzz +
hz (cz)
vz + f (v ) = 0
h(cz)
Theorem (Chen)
∃ c1 such that:
(1) βc > 0 if c > c1 ; βc = 0 if 0 < c ≤ c1 ,
(2) for c > c1 , ∃ critical point vc of Eh,c , Eh,c (vc ) = βc ,
(3) vck → a travelling wave solution v̄ .
Variational Principles for Travelling Waves
Gradient system
Gradient system on a infinite cylinders with Neumann boundary
condition:
∂u
= A ∆u − ∇u F (u).
∂t
(1)
Theorem
(1)coercive: F (u) → ∞ as |u| → ∞;
(2) w non-degenerate local min. point of F , F (w ) > inf F .
Then ∃ a travelling wave solution v such that v → w at ∞.
Variational Principles for Travelling Waves
Lyapunov Fuction
(1) Infinitely many Lyapunov fuctions
(2) Instantaneous propagation speed
Variational Principles for Travelling Waves
Open Problems
(1) Different diffusion rates
(2) Skew-gradient system
Variational Principles for Travelling Waves