Metastable and interface dynamics for the hyperbolic Jin-Xin system - - PowerPoint PPT Presentation

Metastable and interface dynamics for the hyperbolic Jin-Xin system in one space dimension

Marta Strani, Sapienza Universit` a di Roma, Dipartimento di Matematica 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications Universit` a di Padova, June 25-29 2012

Outline

1

Slow motion of internal shock layers for the Jin-Xin system in one space dimension Overview of the problem Spectral analysis Main results

The main problem

We describe slow motion for the Jin-Xin system, with Dirichlet boundary conditions in the bounded interval I = (−ℓ, ℓ), that is              ∂tu + ∂xv = 0 ∂tv + a2∂xu = 1 ε (f (u) − v) u(±ℓ, t) = u± t ≥ 0 u(x, 0) = u0(x), v(x, 0) = v0(x) ≡ f (u0(x)) x ∈ I (1) for some ε, ℓ, a > 0, u± ∈ R and flux function f that satisfies f ′′(s) ≥ c0 > 0, f ′(u+) < 0 < f ′(u−), f (u+) = f (u−) In vector form ∂tZ = F ε[Z], Z

• t=0 = Z0

where F ε[Z] :=

• Pε

1 [Z]

Pε

2 [Z]

• =

  − ∂xv −a2∂xu + 1 ε (f (u) − v)  

The main problem

We describe slow motion for the Jin-Xin system, with Dirichlet boundary conditions in the bounded interval I = (−ℓ, ℓ), that is              ∂tu + ∂xv = 0 ∂tv + a2∂xu = 1 ε (f (u) − v) u(±ℓ, t) = u± t ≥ 0 u(x, 0) = u0(x), v(x, 0) = v0(x) ≡ f (u0(x)) x ∈ I (1) for some ε, ℓ, a > 0, u± ∈ R and flux function f that satisfies f ′′(s) ≥ c0 > 0, f ′(u+) < 0 < f ′(u−), f (u+) = f (u−) In vector form ∂tZ = F ε[Z], Z

• t=0 = Z0

where F ε[Z] :=

• Pε

1 [Z]

Pε

2 [Z]

• =

  − ∂xv −a2∂xu + 1 ε (f (u) − v)  

Metastable dynamics

First time scale where solutions are close to some non stationary state. Exponentially long time convergence to the asymptotic limit. Presence of a first small eigenvalue of the linearized operator. Allen-Cahn: Carr, Pego, Comm. Pure Appl. Math. 1989 Fusco, Hale, J. Dyn. Diff. Eq. 1989 Cahn-Hilliard: Pego, Proc. Roy. Soc. London Ser. A 1989 Alikakos, Bates, Fusco, J. Diff. Eq. 1991 Burgers: Reyna, Ward, Comm. Pure Appl. Math. 1995 Laforgue, O’Malley, SIAM J. Appl. Math 1995 Mascia, S., submitted

Metastable dynamics

First time scale where solutions are close to some non stationary state. Exponentially long time convergence to the asymptotic limit. Presence of a first small eigenvalue of the linearized operator. Allen-Cahn: Carr, Pego, Comm. Pure Appl. Math. 1989 Fusco, Hale, J. Dyn. Diff. Eq. 1989 Cahn-Hilliard: Pego, Proc. Roy. Soc. London Ser. A 1989 Alikakos, Bates, Fusco, J. Diff. Eq. 1991 Burgers: Reyna, Ward, Comm. Pure Appl. Math. 1995 Laforgue, O’Malley, SIAM J. Appl. Math 1995 Mascia, S., submitted

Metastable dynamics

First time scale where solutions are close to some non stationary state. Exponentially long time convergence to the asymptotic limit. Presence of a first small eigenvalue of the linearized operator. Allen-Cahn: Carr, Pego, Comm. Pure Appl. Math. 1989 Fusco, Hale, J. Dyn. Diff. Eq. 1989 Cahn-Hilliard: Pego, Proc. Roy. Soc. London Ser. A 1989 Alikakos, Bates, Fusco, J. Diff. Eq. 1991 Burgers: Reyna, Ward, Comm. Pure Appl. Math. 1995 Laforgue, O’Malley, SIAM J. Appl. Math 1995 Mascia, S., submitted

Metastable dynamics

First time scale where solutions are close to some non stationary state. Exponentially long time convergence to the asymptotic limit. Presence of a first small eigenvalue of the linearized operator. Allen-Cahn: Carr, Pego, Comm. Pure Appl. Math. 1989 Fusco, Hale, J. Dyn. Diff. Eq. 1989 Cahn-Hilliard: Pego, Proc. Roy. Soc. London Ser. A 1989 Alikakos, Bates, Fusco, J. Diff. Eq. 1991 Burgers: Reyna, Ward, Comm. Pure Appl. Math. 1995 Laforgue, O’Malley, SIAM J. Appl. Math 1995 Mascia, S., submitted

The relaxation limit

In the relaxation limit (ε → 0+), system (1) can be approximated to leading

• rder by

∂tu + ∂xf (u) = 0, v = f (u) (2) The first equation is a quasi-linear equation of hyperbolic type, whose standard setting is given by the entropy formulation, hence possessing discontinuous solutions with speed of propagation s given by the Rankine–Hugoniot relation s[ [u] ] = [ [f (u)] ] together with appropriate entropy conditions. Concerning the stationary solutions From the entropy conditions follows that only a single jump from the value u− ≥ u+ is admitted, with speed s = 0.

The relaxation limit

In the relaxation limit (ε → 0+), system (1) can be approximated to leading

• rder by

∂tu + ∂xf (u) = 0, v = f (u) (2) The first equation is a quasi-linear equation of hyperbolic type, whose standard setting is given by the entropy formulation, hence possessing discontinuous solutions with speed of propagation s given by the Rankine–Hugoniot relation s[ [u] ] = [ [f (u)] ] together with appropriate entropy conditions. Concerning the stationary solutions From the entropy conditions follows that only a single jump from the value u− ≥ u+ is admitted, with speed s = 0.

The relaxation limit

In the relaxation limit (ε → 0+), system (1) can be approximated to leading

• rder by

∂tu + ∂xf (u) = 0, v = f (u) (2) The first equation is a quasi-linear equation of hyperbolic type, whose standard setting is given by the entropy formulation, hence possessing discontinuous solutions with speed of propagation s given by the Rankine–Hugoniot relation s[ [u] ] = [ [f (u)] ] together with appropriate entropy conditions. Concerning the stationary solutions From the entropy conditions follows that only a single jump from the value u− ≥ u+ is admitted, with speed s = 0.

Stationary solutions for ε = 0

Stationary solutions

We have a one-parameter family of stationary solutions Uhyp(x; ξ) = u−χ(−ℓ,ξ)(x) + u+χ(ξ,ℓ)(x) V hyp(x; ξ) = f (u−)χ(−ℓ,ξ)(x) + f (u+)χ(ξ,ℓ)(x) The dynamics determined by initial-value problem for (2) is very simple. Hypotheses: If f (u) is convex and such that f (u−) = f (u+), where u+ ≤ u− every entropy solution converges in finite time to an element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Stationary solutions for ε = 0

Stationary solutions

We have a one-parameter family of stationary solutions Uhyp(x; ξ) = u−χ(−ℓ,ξ)(x) + u+χ(ξ,ℓ)(x) V hyp(x; ξ) = f (u−)χ(−ℓ,ξ)(x) + f (u+)χ(ξ,ℓ)(x) The dynamics determined by initial-value problem for (2) is very simple. Hypotheses: If f (u) is convex and such that f (u−) = f (u+), where u+ ≤ u− every entropy solution converges in finite time to an element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Stationary solutions for ε = 0

Stationary solutions

We have a one-parameter family of stationary solutions Uhyp(x; ξ) = u−χ(−ℓ,ξ)(x) + u+χ(ξ,ℓ)(x) V hyp(x; ξ) = f (u−)χ(−ℓ,ξ)(x) + f (u+)χ(ξ,ℓ)(x) The dynamics determined by initial-value problem for (2) is very simple. Hypotheses: If f (u) is convex and such that f (u−) = f (u+), where u+ ≤ u− every entropy solution converges in finite time to an element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Metastable dynamics for ε > 0

Stationary solution

For ε > 0, the presence of the Laplace operator has the effect of a drastic reduction of the number of stationary solutions: in this case there exists a unique stationary solution that is asymptotically stable. Such solution, denoted here by (¯ Uε

rel (x), ¯

V ε

rel (x)), converges in the limit ε → 0+

to a specific element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Question

What happens to the dynamics generated by an initial datum localized that still presents a sharp transition from u− to u+, but it is localized far from the equilibrium solution?

Metastable dynamics for ε > 0

Stationary solution

For ε > 0, the presence of the Laplace operator has the effect of a drastic reduction of the number of stationary solutions: in this case there exists a unique stationary solution that is asymptotically stable. Such solution, denoted here by (¯ Uε

rel (x), ¯

V ε

rel (x)), converges in the limit ε → 0+

to a specific element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Question

What happens to the dynamics generated by an initial datum localized that still presents a sharp transition from u− to u+, but it is localized far from the equilibrium solution?

Metastable dynamics for ε > 0

Stationary solution

For ε > 0, the presence of the Laplace operator has the effect of a drastic reduction of the number of stationary solutions: in this case there exists a unique stationary solution that is asymptotically stable. Such solution, denoted here by (¯ Uε

rel (x), ¯

V ε

rel (x)), converges in the limit ε → 0+

to a specific element of the family {Uhyp(·; ξ), V hyp(·; ξ)}.

Question

What happens to the dynamics generated by an initial datum localized that still presents a sharp transition from u− to u+, but it is localized far from the equilibrium solution?

Figure: Profiles of (u, v), solutions to (1), with f (u) = u2/2, a =1 ε = 0.04 and u± = ∓1. The initial data is given by the couple (u0(x), f (u0(x))), with u0(x) a decreasing function connecting u+ and u−.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

The strategy

To describe the dynamics generated by an initial datum localized far from the position of the steady state, our strategy is: to build up a one parameter family of approximate stationary solutions Wε(x; ξ) = {Uε(x; ξ), V ε(x; ξ)}, parametrized by ξ that represent the location

• f the internal shock, such that F ε[Wε] → 0 as ε → 0.

(Uε(·; ¯ ξ), V ε(·, ¯ ξ)) := (¯ Uε

rel , ¯

V ε

rel ),

∃ ¯ ξ ∈ I to describe the dynamics of the system in a neighborhood of the family, by linearizing the equation around an element Wε(x; ξ), in order to obtain a coupled system for the shock layer location ξ(t) and the perturbation Y . Z(x, t) = Y (x, t) + Wε(x; ξ(t)) to determine spectral properties of the linearized operator at such states. to show that, under a control on how far is the approximate state from being an exact solution, a metastable behavior occur.

Example

In the case of Burgers flux, i.e. f (s) = 1

2 s2 and u± := ∓u∗, for some u∗ > 0,

an approximate solution Uε(x; ξ) is obtained by matching two different steady states satisfying, respectively, the left and the right boundary condition together with the request Uε(ξ) = 0; in formulas, Uε(x; ξ) =

• κ− tanh (κ−(ξ − x)/2ε)

in (−ℓ, ξ) κ+ tanh (κ+(ξ − x)/2ε) in (ξ, ℓ), where κ± = κ±(u∗). Moreover, by the condition v = C2

2 , we have

V ε(x; ξ) =

• κ2

−/2

in (−ℓ, ξ) κ2

+/2

in (ξ, ℓ) By direct substitution, we obtain the identity Pε

1 [Wε(·; ξ)] = [

[∂xUε] ]x=ξδx=ξ, Pε

2 [Wε(·; ξ)] = 0

[ [∂xUε] ]x=ξ = 2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

Example

In the case of Burgers flux, i.e. f (s) = 1

2 s2 and u± := ∓u∗, for some u∗ > 0,

an approximate solution Uε(x; ξ) is obtained by matching two different steady states satisfying, respectively, the left and the right boundary condition together with the request Uε(ξ) = 0; in formulas, Uε(x; ξ) =

• κ− tanh (κ−(ξ − x)/2ε)

in (−ℓ, ξ) κ+ tanh (κ+(ξ − x)/2ε) in (ξ, ℓ), where κ± = κ±(u∗). Moreover, by the condition v = C2

2 , we have

V ε(x; ξ) =

• κ2

−/2

in (−ℓ, ξ) κ2

+/2

in (ξ, ℓ) By direct substitution, we obtain the identity Pε

1 [Wε(·; ξ)] = [

[∂xUε] ]x=ξδx=ξ, Pε

2 [Wε(·; ξ)] = 0

[ [∂xUε] ]x=ξ = 2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

Example

In the case of Burgers flux, i.e. f (s) = 1

2 s2 and u± := ∓u∗, for some u∗ > 0,

an approximate solution Uε(x; ξ) is obtained by matching two different steady states satisfying, respectively, the left and the right boundary condition together with the request Uε(ξ) = 0; in formulas, Uε(x; ξ) =

• κ− tanh (κ−(ξ − x)/2ε)

in (−ℓ, ξ) κ+ tanh (κ+(ξ − x)/2ε) in (ξ, ℓ), where κ± = κ±(u∗). Moreover, by the condition v = C2

2 , we have

V ε(x; ξ) =

• κ2

−/2

in (−ℓ, ξ) κ2

+/2

in (ξ, ℓ) By direct substitution, we obtain the identity Pε

1 [Wε(·; ξ)] = [

[∂xUε] ]x=ξδx=ξ, Pε

2 [Wε(·; ξ)] = 0

[ [∂xUε] ]x=ξ = 2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

The equations

By linearizing around and element Wε of the family of approximate steady states, we obtain the following coupled system    dξ dt = θε(ξ)

• 1 + ∂ξψε

1, Y

• ,

∂tY = Hε(ξ) + (Lε

ξ + Mε ξ)Y

(3) to be complemented with initial conditions ξ(0) = ξ0 ∈ (−ℓ, ℓ) and Y (x, 0) = Y0(x) ∈ L2(I; R2). (4) Here Lε

ξ is the operator arising from the linearization, and Mε ξ is a linear

bounded operator.

Leading order term in the equation for ξ(t)

θε(ξ) := ψε

1, Fε[Wε],

θε(ξ) → 0 as ε → 0 When f (u) = u2/2 and Wε = (Uε, V ε), we have θε(ξ) ∼ 1 ε u∗ (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

The equations

By linearizing around and element Wε of the family of approximate steady states, we obtain the following coupled system    dξ dt = θε(ξ)

• 1 + ∂ξψε

1, Y

• ,

∂tY = Hε(ξ) + (Lε

ξ + Mε ξ)Y

(3) to be complemented with initial conditions ξ(0) = ξ0 ∈ (−ℓ, ℓ) and Y (x, 0) = Y0(x) ∈ L2(I; R2). (4) Here Lε

ξ is the operator arising from the linearization, and Mε ξ is a linear

bounded operator.

Leading order term in the equation for ξ(t)

θε(ξ) := ψε

1, Fε[Wε],

θε(ξ) → 0 as ε → 0 When f (u) = u2/2 and Wε = (Uε, V ε), we have θε(ξ) ∼ 1 ε u∗ (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

The equations

By linearizing around and element Wε of the family of approximate steady states, we obtain the following coupled system    dξ dt = θε(ξ)

• 1 + ∂ξψε

1, Y

• ,

∂tY = Hε(ξ) + (Lε

ξ + Mε ξ)Y

(3) to be complemented with initial conditions ξ(0) = ξ0 ∈ (−ℓ, ℓ) and Y (x, 0) = Y0(x) ∈ L2(I; R2). (4) Here Lε

ξ is the operator arising from the linearization, and Mε ξ is a linear

bounded operator.

Leading order term in the equation for ξ(t)

θε(ξ) := ψε

1, Fε[Wε],

θε(ξ) → 0 as ε → 0 When f (u) = u2/2 and Wε = (Uε, V ε), we have θε(ξ) ∼ 1 ε u∗ (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε)

Spectral analysis

Lε

ξY :=

  − ∂xv −a2∂xu+1 ε (f ′(Uε)u − v)   ⇒    λϕ = −∂xψ λψ = −a2∂xϕ + 1 ε (f ′(Uε)ϕ − ψ) By differentiating the second equation with respect to x, we obtain εa2∂xxϕ − ∂x(f ′(Uε)ϕ) = λ(1 + ελ)ϕ To study the eigenvalue problem for the differential linear diffusion-transport

• perator

Lε,vscϕ := ε∂xxϕ − ∂x(f ′(Uε(x; ξ))ϕ) ∂tu = ε ∂xxu − ∂xf (u) Kreiss, Kreiss, Appl. Numer. Math. 1986 Mascia, S. 2012 − → Metastability for scalar conservation laws in a bounded domain.

Spectral analysis

Lε

ξY :=

  − ∂xv −a2∂xu+1 ε (f ′(Uε)u − v)   ⇒    λϕ = −∂xψ λψ = −a2∂xϕ + 1 ε (f ′(Uε)ϕ − ψ) By differentiating the second equation with respect to x, we obtain εa2∂xxϕ − ∂x(f ′(Uε)ϕ) = λ(1 + ελ)ϕ To study the eigenvalue problem for the differential linear diffusion-transport

• perator

Lε,vscϕ := ε∂xxϕ − ∂x(f ′(Uε(x; ξ))ϕ) ∂tu = ε ∂xxu − ∂xf (u) Kreiss, Kreiss, Appl. Numer. Math. 1986 Mascia, S. 2012 − → Metastability for scalar conservation laws in a bounded domain.

Spectral analysis

Lε

ξY :=

  − ∂xv −a2∂xu+1 ε (f ′(Uε)u − v)   ⇒    λϕ = −∂xψ λψ = −a2∂xϕ + 1 ε (f ′(Uε)ϕ − ψ) By differentiating the second equation with respect to x, we obtain εa2∂xxϕ − ∂x(f ′(Uε)ϕ) = λ(1 + ελ)ϕ To study the eigenvalue problem for the differential linear diffusion-transport

• perator

Lε,vscϕ := ε∂xxϕ − ∂x(f ′(Uε(x; ξ))ϕ) ∂tu = ε ∂xxu − ∂xf (u) Kreiss, Kreiss, Appl. Numer. Math. 1986 Mascia, S. 2012 − → Metastability for scalar conservation laws in a bounded domain.

Spectral analysis

Lε

ξY :=

  − ∂xv −a2∂xu+1 ε (f ′(Uε)u − v)   ⇒    λϕ = −∂xψ λψ = −a2∂xϕ + 1 ε (f ′(Uε)ϕ − ψ) By differentiating the second equation with respect to x, we obtain εa2∂xxϕ − ∂x(f ′(Uε)ϕ) = λ(1 + ελ)ϕ To study the eigenvalue problem for the differential linear diffusion-transport

• perator

Lε,vscϕ := ε∂xxϕ − ∂x(f ′(Uε(x; ξ))ϕ) ∂tu = ε ∂xxu − ∂xf (u) Kreiss, Kreiss, Appl. Numer. Math. 1986 Mascia, S. 2012 − → Metastability for scalar conservation laws in a bounded domain.

Spectral analysis

Lε

ξY :=

  − ∂xv −a2∂xu+1 ε (f ′(Uε)u − v)   ⇒    λϕ = −∂xψ λψ = −a2∂xϕ + 1 ε (f ′(Uε)ϕ − ψ) By differentiating the second equation with respect to x, we obtain εa2∂xxϕ − ∂x(f ′(Uε)ϕ) = λ(1 + ελ)ϕ To study the eigenvalue problem for the differential linear diffusion-transport

• perator

Lε,vscϕ := ε∂xxϕ − ∂x(f ′(Uε(x; ξ))ϕ) ∂tu = ε ∂xxu − ∂xf (u) Kreiss, Kreiss, Appl. Numer. Math. 1986 Mascia, S. 2012 − → Metastability for scalar conservation laws in a bounded domain.

Theorem (Spectral analysis)

Under opportune hypotheses on the family of functions f ′(Uε(x; ξ)) the spectrum of the linearized operator Lε

ξ can be decomposed as follow

• 1. λJX

1

∈ R and −e−C′/ε ≤ λJX

1

< 0

• 2. All the remaining eigenvalues λJX

n

are such that Re[λJX

n ] ≤ −C/ε

First real small eigenvalue f (u) = u2/2

|λJX

1,+(ξ)| ∼ u∗2 ε

• e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

1 +

• 1 − 2u∗2

e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

Theorem (Spectral analysis)

Under opportune hypotheses on the family of functions f ′(Uε(x; ξ)) the spectrum of the linearized operator Lε

ξ can be decomposed as follow

• 1. λJX

1

∈ R and −e−C′/ε ≤ λJX

1

< 0

• 2. All the remaining eigenvalues λJX

n

are such that Re[λJX

n ] ≤ −C/ε

First real small eigenvalue f (u) = u2/2

|λJX

1,+(ξ)| ∼ u∗2 ε

• e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

1 +

• 1 − 2u∗2

e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

Theorem (Spectral analysis)

Under opportune hypotheses on the family of functions f ′(Uε(x; ξ)) the spectrum of the linearized operator Lε

ξ can be decomposed as follow

• 1. λJX

1

∈ R and −e−C′/ε ≤ λJX

1

< 0

• 2. All the remaining eigenvalues λJX

n

are such that Re[λJX

n ] ≤ −C/ε

First real small eigenvalue f (u) = u2/2

|λJX

1,+(ξ)| ∼ u∗2 ε

• e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

1 +

• 1 − 2u∗2

e−u∗ε−1(ℓ−ξ) + e−u∗ε−1(ℓ+ξ)

Hypotheses

• H1. The family {Wε(·, ξ)} is such that there exist two families of smooth

positive functions Ωε

1 = Ωε 1(ξ) and Ωε 2 = Ωε 2(ξ), uniformly convergent to zero

as ε → 0, such that |ψ(·), Pε

1 [Wε(·, ξ)]| ≤ Ωε 1(ξ)|ψ|L∞

∀ψ ∈ C(I) |ψ(·), Pε

2 [Wε(·, ξ)]| ≤ Ωε 2(ξ)|ψ|L∞

∀ψ ∈ C(I)

• H2. Ωε

1(ξ) + Ωε 2(ξ) ≤ C|λJX 1,+(ξ)|, for all ξ ∈ (−ℓ, ℓ).

For example, if f (u) = u2/2 and Wε = (Uε, V ε) Ωε(ξ) = (Ωε

1(ξ), Ωε 2(ξ)) ∼

2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε), 0

Hypotheses

• H1. The family {Wε(·, ξ)} is such that there exist two families of smooth

positive functions Ωε

1 = Ωε 1(ξ) and Ωε 2 = Ωε 2(ξ), uniformly convergent to zero

as ε → 0, such that |ψ(·), Pε

1 [Wε(·, ξ)]| ≤ Ωε 1(ξ)|ψ|L∞

∀ψ ∈ C(I) |ψ(·), Pε

2 [Wε(·, ξ)]| ≤ Ωε 2(ξ)|ψ|L∞

∀ψ ∈ C(I)

• H2. Ωε

1(ξ) + Ωε 2(ξ) ≤ C|λJX 1,+(ξ)|, for all ξ ∈ (−ℓ, ℓ).

For example, if f (u) = u2/2 and Wε = (Uε, V ε) Ωε(ξ) = (Ωε

1(ξ), Ωε 2(ξ)) ∼

2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε), 0

Hypotheses

• H1. The family {Wε(·, ξ)} is such that there exist two families of smooth

positive functions Ωε

1 = Ωε 1(ξ) and Ωε 2 = Ωε 2(ξ), uniformly convergent to zero

as ε → 0, such that |ψ(·), Pε

1 [Wε(·, ξ)]| ≤ Ωε 1(ξ)|ψ|L∞

∀ψ ∈ C(I) |ψ(·), Pε

2 [Wε(·, ξ)]| ≤ Ωε 2(ξ)|ψ|L∞

∀ψ ∈ C(I)

• H2. Ωε

1(ξ) + Ωε 2(ξ) ≤ C|λJX 1,+(ξ)|, for all ξ ∈ (−ℓ, ℓ).

For example, if f (u) = u2/2 and Wε = (Uε, V ε) Ωε(ξ) = (Ωε

1(ξ), Ωε 2(ξ)) ∼

2 u∗

2

ε (e−u∗(ℓ+ξ)/ε − e−u∗(ℓ−ξ)/ε), 0

Slow motion for the shock layer

The perturbation Y

The operator Lε

ξ(t) + Mε ξ(t) depends on time → Theory of Stable family of

generators. |Y |L2 ≤ C(|Ωε|L∞ + e−µεt|Y0|L2 ), µε = supξ λJX

1 (ξ) − C|Ωε|L∞ > 0

Theorem (Slow motion of the shock layer)

Let hypotheses H1-2 be satisfied. Assume also s θε(s) < 0 for any s ∈ I, s = 0 and θε′(¯ ξ) < 0. Then for ε and |Y0|L2 sufficiently small, than the solution ξ(t) converges to ¯ ξ as t → +∞. More precisely, the dynamics of ξ(t) is described by dξ dt = θε(ξ) , θε(ξ) = ψε

1, F[Wε]

so that |ξ(t) − ¯ ξ| ≤ |ξ0|eβεt, βε ∼ θε′(¯ ξ) this estimate shows the exponentially slow motion of the shock layer for small ε.

Slow motion for the shock layer

The perturbation Y

The operator Lε

ξ(t) + Mε ξ(t) depends on time → Theory of Stable family of

generators. |Y |L2 ≤ C(|Ωε|L∞ + e−µεt|Y0|L2 ), µε = supξ λJX

1 (ξ) − C|Ωε|L∞ > 0

Theorem (Slow motion of the shock layer)

Let hypotheses H1-2 be satisfied. Assume also s θε(s) < 0 for any s ∈ I, s = 0 and θε′(¯ ξ) < 0. Then for ε and |Y0|L2 sufficiently small, than the solution ξ(t) converges to ¯ ξ as t → +∞. More precisely, the dynamics of ξ(t) is described by dξ dt = θε(ξ) , θε(ξ) = ψε

1, F[Wε]

so that |ξ(t) − ¯ ξ| ≤ |ξ0|eβεt, βε ∼ θε′(¯ ξ) this estimate shows the exponentially slow motion of the shock layer for small ε.

Slow motion for the shock layer

The perturbation Y

The operator Lε

ξ(t) + Mε ξ(t) depends on time → Theory of Stable family of

generators. |Y |L2 ≤ C(|Ωε|L∞ + e−µεt|Y0|L2 ), µε = supξ λJX

1 (ξ) − C|Ωε|L∞ > 0

Theorem (Slow motion of the shock layer)

Let hypotheses H1-2 be satisfied. Assume also s θε(s) < 0 for any s ∈ I, s = 0 and θε′(¯ ξ) < 0. Then for ε and |Y0|L2 sufficiently small, than the solution ξ(t) converges to ¯ ξ as t → +∞. More precisely, the dynamics of ξ(t) is described by dξ dt = θε(ξ) , θε(ξ) = ψε

1, F[Wε]

so that |ξ(t) − ¯ ξ| ≤ |ξ0|eβεt, βε ∼ θε′(¯ ξ) this estimate shows the exponentially slow motion of the shock layer for small ε.

Slow motion for the shock layer-Numerical computations

This is a table containing the values of the different locations of the shock layer

• btained numerically for different values of the parameter ε.

The numerical location of the shock layer ξ(t) for different values of the parameter ε TIME t ξ(t), ε = 0.10 ξ(t), ε = 0.07 ξ(t), ε = 0.05 ξ(t), ε = 0.04 ξ(t), ε = 0.02 0.2 −0.4008 −0.4020 −0.4029 −0.4040 −0.4059 1 −0.3314 −0.3345 −0.3360 −0.3374 −0.3389 10 −0.3070 −0.3263 −0.3304 −0.3320 −0.3326 103 −0.0103 −0.1600 −0.2562 −0.3181 −0.3325 104 ∼ −10−12 −0.0084 −0.1115 −0.2531 −0.3320 0.5 ∗ 106 ∼ −10−12 ∼ −10−11 ∼ −10−10 −0.0379 −0.3099

ε = 0.10

ε = 0.07

ε = 0.05

ε = 0.04

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