On a low Mach number limit for Supernovae Donatella Donatelli joint - - PowerPoint PPT Presentation

On a low Mach number limit for Supernovae

Donatella Donatelli joint work with E. Feireisl

(arXiv:1604.05860, [math.AP]) Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit` a degli Studi dell’Aquila 67100 L’Aquila, Italy donatella.donatelli@univaq.it

Motivation

A broad range of interesting phenomena in science and engineering

• ccur in a low Mach number regime, in which the fluid velocity is

much less than the speed of sound Ma = fluid velocity sound speed

2

What are supernovae?

A supernova is the explosion of a star. It is the largest explosion that takes place in space. A supernova happens where there is a change in the core, or center, of a star. For example if we have two stars that orbit the same point (binary star systems) and one of the stars, a carbon-oxygen white dwarf, steals matter from its companion star. Eventually, the white dwarf accumulates too much matter. Having too much matter causes the star to explode, resulting in a supernova.

3

What are supernovae?

But supernovas are difficult to see in our own Milky Way galaxy because dust blocks our view. In 1604, Johannes Kepler discovered the last observed supernova in the Milky

• Way. NASA’s Chandra telescope discovered the remains of a

more recent supernova. A supernova burns for only a short period of time, but it can tell scientists a lot about the universe. When the star explodes, it shoots elements and debris into

• space. Many of the elements we find here on Earth are made

in the core of stars.

4

Cassiopeia A

Cassiopeia A is a supernova remnant in the constellation Cassiopeia. The supernova

• ccurred approximately 11,000 light-years away within the Milky Way.

Discovery date: 1947.

5

Supernova 1987A

Credits: Nasa - Hubble Space Telescope

6

Why construct a mathematical model?

to track the final steps that leads from a white dwarf to a supernova ⇓ to know more about how the supernova ignite

7

Range of timescale

100 years convection that precedes ignition last few hours convection becomes more vigorous as the heat release intensifies and convection can no longer carry away the heat. 1 second duration of the explosion

• In the last minutes of the convective phase, velocity reach ∼ 1% of

the sound speed

• low Mach number regime

8

Our Goal

Understand the separation of scales that occurs in a low Mach number regime Physically, one can think of the solution to a low Mach number model as supporting infinitely fast acoustic equilibration rather than finite-velocity acoustic wave propagation Mathematically, this is manifest in the addition of a constraint on the velocity field to the system of evolution equations.

9

The mathematical model

Compressible viscous fluid ̺ = ̺(x, t) density u = u(x, t) velocity vector Θ = Θ(x, t) temperature Xk = Xk(x, t) mass fraction of the species ˙ ωk = ˙ ωk(x, t) production rates Viscous stress tensor S(∇xu) =

• ∇xu + ∇t

xu − 2

3divxuI

• + λdivxuI, λ ≥ 0,

10

The mathematical model

Mass conservation ∂t̺ + div(̺u) = 0 Momentum balance ∂t(̺u) + div(̺u ⊗ u) + ∇xp = µ div S(∇xu) + ̺f Energy balance ∂t(̺E) + div(̺uE + pu) = −

• k

̺qk ˙ ωk Combustion equation ∂t(̺Xk) + div(̺uXk) = ̺ ˙ ωk E = e + |u|2/2, total energy p = p(̺, Θ) pressure

11

Nondimensional equations

Tref =reference time Lref =reference lenght Uref =reference velocity ρref =reference density pref =reference pressure

12

Nondimensional equations

Tref =reference time Lref =reference lenght Uref =reference velocity ρref =reference density pref =reference pressure Nondimensional variables =X′ = X Xref Sr =

Lref TrefUref

Ma =

Uref

√

pref/ρref

Re = ρrefUrefLref

Trefµref

Fr =

Uref

√

Lreffref

12

Nondimensional equations

Tref =reference time Lref =reference lenght Uref =reference velocity ρref =reference density pref =reference pressure Nondimensional variables =X′ = X Xref Sr =

Lref TrefUref

Ma =

Uref

√

pref/ρref

Re = ρrefUrefLref

Trefµref

Fr =

Uref

√

Lreffref

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇xp = div S(∇xu) + ̺f

12

Nondimensional equations

Tref =reference time Lref =reference lenght Uref =reference velocity ρref =reference density pref =reference pressure Nondimensional variables =X′ = X Xref Sr =

Lref TrefUref

Ma =

Uref

√

pref/ρref

Re = ρrefUrefLref

Trefµref

Fr =

Uref

√

Lreffref

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f

12

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f

13

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f Sr = Lref TrefUref = 1 ( the characteristic time scale Tref equals the convection time scale Lref/Uref)

13

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f Sr = Lref TrefUref = 1 ( the characteristic time scale Tref equals the convection time scale Lref/Uref) Ma = ε

13

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f Sr = Lref TrefUref = 1 ( the characteristic time scale Tref equals the convection time scale Lref/Uref) Ma = ε strong stratification Fr ∼ Ma ∼ ε

13

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f Sr = Lref TrefUref = 1 ( the characteristic time scale Tref equals the convection time scale Lref/Uref) Ma = ε strong stratification Fr ∼ Ma ∼ ε high Reynolds number Re ∼ ε−α

13

Nondimensional equations

Sr∂t̺ + div(̺u) = 0 Sr∂t(̺u) + div(̺u ⊗ u) + 1 Ma2 ∇p = 1 Re div S(∇xu) + 1 Fr2 ̺f Sr = Lref TrefUref = 1 ( the characteristic time scale Tref equals the convection time scale Lref/Uref) Ma = ε strong stratification Fr ∼ Ma ∼ ε high Reynolds number Re ∼ ε−α ∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + 1 ε2 ∇p = εα div S(∇xu) + 1 ε2 ̺f

13

Low Mach number limit

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + 1 ε2 ∇p = εα div S(∇xu) + 1 ε2 ̺f

14

Low Mach number limit

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + 1 ε2 ∇p = εα div S(∇xu) + 1 ε2 ̺f

• ∂tV + V · ∇xV + ∇xΠ = −̺0f

where ̺0 is the unique solution of the static equation ∇xp(̺0) = ̺0f and.... div(̺0V) = 0

14

Low Mach number limit

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + 1 ε2 ∇p = εα div S(∇xu) + 1 ε2 ̺f

• ∂tV + V · ∇xV + ∇xΠ = −̺0f

where ̺0 is the unique solution of the static equation ∇xp(̺0) = ̺0f and.... div V = −∇x̺0 ̺0 V constrain on the velocity field

14

Main problem: fast oscillating acoustic waves

Linearized system Fourier Transform ρt + ux = 0 ut + 1 ε2 ρx = 0 ˆ ρt ˆ ut

• +

iξ iξ ε2 ˆ ρ ˆ u

• = 0

15

Main problem: fast oscillating acoustic waves

Linearized system Fourier Transform ρt + ux = 0 ut + 1 ε2 ρx = 0 ˆ ρt ˆ ut

• +

iξ iξ ε2 ˆ ρ ˆ u

• = 0

Solutions ˆ ρ(ξ, t) ˆ u(ξ, t)

• =

ˆ ρ+(ξ) ˆ u+(ξ)

• eiθ(ξ)t +

ˆ ρ−(ξ) ˆ u−(ξ)

• e−iθ(ξ)t

θ(ξ) = ξ ε

15

Main problem: fast oscillating acoustic waves

Where do we see these waves? u = H[u]

• soleinodal part

+ H⊥[u]

gradient part

div H[u] = 0 H⊥[u] = ∇Ψ

16

Main problem: fast oscillating acoustic waves

Where do we see these waves? u = H[u]

• soleinodal part

+ H⊥[u]

gradient part

div H[u] = 0 H⊥[u] = ∇Ψ ρt + div u = 0 ut + 1 ε2 ∇r = 0 ∂tρ + ∆Ψ = 0 ∂t∇Ψ + 1 ε2 ∇ρ = 0

16

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇xp = µ div S(∇xu) + ̺f ∂t(̺E) + div(̺uE + pu) = −

• k

̺qk ˙ ωk ∂t(̺Xk) + div(̺uXk) = ̺ ˙ ωk E = e + |u|2/2, total energy p = p(̺, Θ) pressure e = e(Θ, ρ) internal energy p = p(̺, Θ) = pion + prad,

17

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u)+∇xp = µ div S(∇xu) + ̺f ∂t(̺E) + div(̺uE + pu) = −

✟✟✟✟ ✟ ❍❍❍❍ ❍

• k

̺qk ˙ ωk

✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤

∂t(̺Xk) + div(̺uXk) = ̺ ˙ ωk E = e + |u|2/2, total energy p = p(̺, Θ) pressure e = e(Θ, ρ) internal energy p = p(̺, Θ) = pion + prad,

17

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u)+∇xp = µ div S(∇xu) + ̺f ∂t(̺E) + div(̺uE + pu) = −

✟✟✟✟ ✟ ❍❍❍❍ ❍

• k

̺qk ˙ ωk

✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤

∂t(̺Xk) + div(̺uXk) = ̺ ˙ ωk E = e + |u|2/2, total energy p = p(̺, Θ) pressure e = e(Θ, ρ) = cvΘ internal energy p = p(̺, Θ) = pion + prad,

17

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u)+∇xp = µ div S(∇xu) + ̺f ∂t(̺E) + div(̺uE + pu) = −

✟✟✟✟ ✟ ❍❍❍❍ ❍

• k

̺qk ˙ ωk

✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤

∂t(̺Xk) + div(̺uXk) = ̺ ˙ ωk E = e + |u|2/2, total energy p = p(̺, Θ) pressure e = e(Θ, ρ) = cvΘ internal energy p = p(̺, Θ) = pion + prad = (̺Θ)γ, γ > 3/2

17

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇x(̺Θ)γ = µ div S(∇xu) + ̺f ∂t(̺Θ) + div(̺Θu) = 0

18

The model (a simplified)

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇x(̺Θ)γ = µ div S(∇xu) + ̺f ∂t(̺Θ) + div(̺Θu) = 0 f = ∇xF F ∼ standard Gravitational Potential F ∈ C∞(R3), F(x) > 0 for all x ∈ R3 F 1 |x| ≤ F(x) ≤ F 1 |x| for all |x| > R, |x|2|∇xF(x)| + |x|3|∇2

xF(x)| ≤ c for all x ∈ R3.

18

The simplified model.........

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇x(̺Θ)γ = µ div S(∇xu) + ̺∇xF ∂t(̺Θ) + div(̺Θu) = 0

19

The simplified model.........

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + ∇x(̺Θ)γ = µ div S(∇xu) + ̺∇xF ∂t(̺Θ) + div(̺Θu) = 0 far-field conditions ̺ → ̺, u → 0, Θ → 1 as |x| → ∞

19

.........adimensional variables

∂t̺ + div(̺u) = 0 ∂t(̺u) + div(̺u ⊗ u) + 1 ε2 ∇x(̺Θ)γ = εα div S(∇xu) + 1 ε2 ̺∇xF ∂t(̺Θ) + div(̺Θu) = 0 far-field conditions ̺ → ̺, u → 0, Θ → 1 as |x| → ∞ Sr = 1 Ma = ε Fr = ε, Re ∼ ε−α, 0 < α < 4 3

20

singular limit as ε → 0...

.......ill-prepared initial data ̺(0, ·) = ̺0+ε̺(1)

0,ε,

̺(1)

0,εL1∩L∞(R3) ≤ c,

̺(1)

0,ε → ̺(1)

in L1(R3) u(0, ·) = u0,ε, u0,εL2∩L∞(R3) ≤ c, u0,ε → v0 in L2(R3), Θ(0, ε) = 1+ε2Θ(2)

0,ε,

Θ(2)

0,εL1∩L∞(R3) ≤ c,

Θ(2)

0,ε → Θ(2)

in L1(R3).

21

Existence of global dissipative weak solutions solutions for NS

( Maltese, Novotny, et al. arXiv:1603.08965, [math.AP], 2016.)

22

Existence of global dissipative weak solutions solutions for NS

( Maltese, Novotny, et al. arXiv:1603.08965, [math.AP], 2016.) Equation of continuity holds in the sense of distributions (renormalized equation also satisfied) Momentum equation holds in the sense of distributions

• R3 ̺G(Θ)ϕ

dx t=τ2

t=τ1

= τ2

τ1

• R3 [̺G(Θ)∂tϕ + ̺G(Θ)u · ∇xϕ]dx dt

for any 0 ≤ τ1 ≤ τ2 ≤ T and for any ϕ ∈ C1

c ([0, T] × R3) and any

G ∈ C(R).

22

Existence of global dissipative weak solutions solutions for NS

( Maltese, Novotny, et al. arXiv:1603.08965, [math.AP], 2016.) Equation of continuity holds in the sense of distributions (renormalized equation also satisfied) Momentum equation holds in the sense of distributions

• R3 ̺G(Θ)ϕ

dx t=τ2

t=τ1

= τ2

τ1

• R3 [̺G(Θ)∂tϕ + ̺G(Θ)u · ∇xϕ]dx dt

for any 0 ≤ τ1 ≤ τ2 ≤ T and for any ϕ ∈ C1

c ([0, T] × R3) and any

G ∈ C(R). dissipative solutions are characterized by the energy inequality

• R3

1 2̺|u|2 + H(̺Θ) − H′(̺0)(̺ − ̺0) − H(̺0) ε2

• dx

t=τ

t=0

+ εα τ

• R3 S(∇xu) : ∇xu

dx dt ≤ 0, for a.e. τ ∈ (0, T) where H(Z) = 1 γ − 1Zγ.

22

Main result

Let {̺ε, uε, Θε}ε>0 be a family of dissipative solutions of NS, with ill prepared initial data, then, as ε → 0

23

Main result

Let {̺ε, uε, Θε}ε>0 be a family of dissipative solutions of NS, with ill prepared initial data, then, as ε → 0 sup

t∈[0,T]

̺ε(t, ·)−̺0(L2+Lγ)(R3) → 0 sup

t∈[0,T]

Θε(t, ·) − T (t, ·)L2(R3) → 0 T

• ̺ε

̺0 uε − V

• 2

L2(K)

dt → 0 for any compact K ∈ R3

23

Main result

Let {̺ε, uε, Θε}ε>0 be a family of dissipative solutions of NS, with ill prepared initial data, then, as ε → 0 sup

t∈[0,T]

̺ε(t, ·)−̺0(L2+Lγ)(R3) → 0 sup

t∈[0,T]

Θε(t, ·) − T (t, ·)L2(R3) → 0 T

• ̺ε

̺0 uε − V

• 2

L2(K)

dt → 0 for any compact K ∈ R3 where [V, Π, T ] in (0, T) × R3 is a strong solution of div(̺0V) = 0 ∂tV+V·∇xV+∇xΠ = −T ∇xF ∂t(T ) + div(T V) = 0

23

Main result

Let {̺ε, uε, Θε}ε>0 be a family of dissipative solutions of NS, with ill prepared initial data, then, as ε → 0 sup

t∈[0,T]

̺ε(t, ·)−̺0(L2+Lγ)(R3) → 0 sup

t∈[0,T]

Θε(t, ·) − T (t, ·)L2(R3) → 0 T

• ̺ε

̺0 uε − V

• 2

L2(K)

dt → 0 for any compact K ∈ R3 where [V, Π, T ] in (0, T) × R3 is a strong solution of div(̺0V) = 0 ∂tV+V·∇xV+∇xΠ = −T ∇xF ∂t(T ) + div(T V) = 0 and the background density profile ̺0 is the solution of ∇x̺γ

0 = ̺0∇xF in R3,

̺0 → ̺ > 0 as |x| → ∞.

23

Strategy

Uniform bounds Analysis of the acoustic waves Convergence to the target system

24

Strategy

Uniform bounds Analysis of the acoustic waves Convergence to the target system Construction of dispersive estimates

24

Strategy

Uniform bounds Analysis of the acoustic waves Convergence to the target system Construction of dispersive estimates Relative entropy (energy)

24

Relative entropy (energy)

E

• ̺, Θ, u
• r, U
• =
• R3

1 2̺|u − U|2 + H(̺Θ) − H′(r)(̺Θ − r) − H(r) ε2

• dx

for any pair of “test functions” (r − ̺) ∈ C∞

c ([0, T] × R3), r > 0, U ∈ C∞ c ([0, T] × R3; R3).

and where H(Z) = 1 γ − 1Zγ

25

Relative energy inequality

Any dissipative solution to the primitive system satisfies

• E
• ̺ε, Θε, uε
• r, U

t=τ

t=0+εα

τ

• R3S(∇x(uε−U)) : ∇x(uε−U)

dx dt ≤ τ

• R3 ̺ε (∂tU + uε · ∇xU)·(U − uε)+εαS(∇xU) : ∇x(U−uε)

dx dt + 1 ε2 τ

• R3
• (r − ̺εΘε)∂tH′(r) + ∇xH′(r) · (rU − ̺εΘεuε)
• dx dt

− 1 ε2 τ

• R3
• div U
• (̺εΘε)γ − rγ

+ ̺ε∇xF · (U − uε)

• dx dt

= R (̺ε, Θε, uε, r, U)

26

Relative energy inequality

Any dissipative solution to the primitive system satisfies

• E
• ̺ε, Θε, uε
• r, U

t=τ

t=0+εα

τ

• R3S(∇x(uε−U)) : ∇x(uε−U)

dx dt ≤ τ

• R3 ̺ε (∂tU + uε · ∇xU)·(U − uε)+εαS(∇xU) : ∇x(U−uε)

dx dt + 1 ε2 τ

• R3
• (r − ̺εΘε)∂tH′(r) + ∇xH′(r) · (rU − ̺εΘεuε)
• dx dt

− 1 ε2 τ

• R3
• div U
• (̺εΘε)γ − rγ

+ ̺ε∇xF · (U − uε)

• dx dt

= R (̺ε, Θε, uε, r, U) if we choose as test functions r = ̺0, U = 0 we get.....

26

Uniform bounds

ess sup

t∈[0,T]

√̺εuεL2(R3;R3) ≤ c = c(data), εα/2

• ∇xuε + ∇t

xuε − 2

3divxuεI

• L2((0,T)×R3)

+div uεL2((0,T)×R3)

• ≤ c,

ess sup

t∈[0,T]

• Θε − 1

ε2

• L1(R3)

+

• Θε − 1

ε2

• L∞(R3)
• ≤ c

ess sup

t∈[0,T]

• ̺ε − ̺0

ε

• ess
• L2(R3)

+

• ̺εΘε − ̺0

ε

• ess
• L2(R3)
• ≤ c

ess sup

t∈[0,T]

• R3 ([1]res + |[̺ε]res|γ + |[̺εΘε]res|γ)dx ≤ ε2c.

27

Uniform bounds

ess sup

t∈[0,T]

√̺εuεL2(R3;R3) ≤ c = c(data), εα/2

• ∇xuε + ∇t

xuε − 2

3divxuεI

• L2((0,T)×R3)

+div uεL2((0,T)×R3)

• ≤ c,

ess sup

t∈[0,T]

• Θε − 1

ε2

• L1(R3)

+

• Θε − 1

ε2

• L∞(R3)
• ≤ c

ess sup

t∈[0,T]

• ̺ε − ̺0

ε

• ess
• L2(R3)

+

• ̺εΘε − ̺0

ε

• ess
• L2(R3)
• ≤ c

ess sup

t∈[0,T]

• R3 ([1]res + |[̺ε]res|γ + |[̺εΘε]res|γ)dx ≤ ε2c.

h = [h]ess + [h]res [h]ess = χ(̺εΘε)h, [h]res = (1 − χ(̺εΘε))h χ ∈ C∞

c (0, ∞), χ ≥ 0, χ(Y ) = 1 if 1 2(min x∈R3 ̺0(x)) ≤ Y ≤ 2(max x∈R3 ̺0(x)).

27

Local in space pressure estimate

T

• K

([̺εΘε]res)γ+β dx dt ≤ c(K)ε2+ω β, ω > 0 and any compact set K ⊂ R3.

28

Local in space pressure estimate

T

• K

([̺εΘε]res)γ+β dx dt ≤ c(K)ε2+ω β, ω > 0 and any compact set K ⊂ R3. It is obtained by using as test functions in the momentum equation ϕ(t, x) = φ(x)∇x∆−1

x [b (̺εΘε)]res , φ ∈ C∞ c (R3), φ ≥ 0

b ∈ C∞[0, ∞), b ≥ 0, b(Y ) = aY β, a ≥ 0, 0 ≤ β < γ for Y >> 1 and by estimating 1 ε2 T

• R3 φ
• p(̺εΘε) − p(̺0)
• [b (̺εΘε)]res dx dt =

7

• j=1

Ij,ε Here we effectively use the restriction 0 < α < 4 3

28

Main result

Let {̺ε, uε, Θε}ε>0 be a family of dissipative solutions of NS, with ill prepared initial data, then, as ε → 0 sup

t∈[0,T]

̺ε(t, ·)−̺0(L2+Lγ)(R3) → 0 sup

t∈[0,T]

Θε(t, ·) − T (t, ·)L2(R3) → 0 T

• ̺ε

̺0 uε − V

• 2

L2(K)

dt → 0 for any compact K ∈ R3 where [V, Π, T ] in (0, T) × R3 is a strong solution of div(̺0V) = 0 ∂tV+V·∇xV+∇xΠ = −T ∇xF ∂t(T ) + div(T V) = 0 and the background density profile ̺0 is the solution of ∇x̺γ

0 = ̺0∇xF in R3,

̺0 → ̺ > 0 as |x| → ∞.

29

Convergence- step 1

E

• ̺ε, Θε, uε
• r, U
• =
• R3

1 2̺ε|uε − U|2+ H(̺εΘε)−H′(r)(̺εΘε − r)−H(r) ε2

• dx

30

Convergence- step 1

E

• ̺ε, Θε, uε
• r, U
• =
• R3

1 2̺ε|uε − U|2+ H(̺εΘε)−H′(r)(̺εΘε − r)−H(r) ε2

• dx

We take r = rε = ̺0 + εsε, U = Uε = V + ∇xΦε, where sε, Φε solve the acoustic wave equation

30

A variant of the Helmoltz decomposition

v = H̺0[v] + H⊥[v], H⊥[v] = ∇xΦ, where divx (̺0∇xΦ) = divx(̺0v), in R3 and, div(̺0H̺0[v]) = 0

31

Acoustic waves

∂t̺ + div(̺u) = 0 ∂t(̺u) + 1 ε2 (∇x(̺Θ)γ − ̺∇xF) = − div(̺u ⊗ u) + εα div S(∇xu) ̺(0, ·) = ̺0 + ε(̺(1)

0,ε − ̺(1) 0 ) + ε̺(1) 0 ,

̺(1)

0,ε → ̺(1)

in L1(R3) u(0, ·) = u0,ε − v0 + H̺0[v0] + v0 − H̺0[v0] u0,ε → v0 in L2(R3), = u0,ε − v0 + V0 + ∇xΦ0,

32

Acoustic waves

∂t̺ + div(̺u) = 0 ∂t(̺u) + 1 ε2 (∇x(̺Θ)γ − ̺∇xF) = − div(̺u ⊗ u) + εα div S(∇xu) ̺(0, ·) = ̺0 + ε(̺(1)

0,ε − ̺(1) 0 ) + ε̺(1) 0 ,

̺(1)

0,ε → ̺(1)

in L1(R3) u(0, ·) = u0,ε − v0 + H̺0[v0] + v0 − H̺0[v0] u0,ε → v0 in L2(R3), = u0,ε − v0 + V0 + ∇xΦ0, sε = ̺ε − ̺0 ε , ∇Φε = H⊥[uε]

32

Acoustic system

ε∂tsε + divx [̺0∇xΦε] = R.h.s, ε̺0∂t∇xΦε + ̺0∇x p′(̺0) ̺0 sε

• = R.h.s,

sε(0, ·) = ̺(1)

0 , ∇xΦε(0, ·) = ∇xΦ0.

33

Convergence- step 2

E

• ̺ε, Θε, uε
• r, U
• =
• R3

1 2̺ε|uε − U|2+ H(̺εΘε)−H′(r)(̺εΘε − r)−H(r) ε2

• dx

We take r = rε = ̺0 + εsε, U = Uε = V + ∇xΦε, where sε, Φε solve the acoustic wave equation

34

Convergence- step 2

E

• ̺ε, Θε, uε
• r, U
• =
• R3

1 2̺ε|uε − U|2+ H(̺εΘε)−H′(r)(̺εΘε − r)−H(r) ε2

• dx

We take r = rε = ̺0 + εsε, U = Uε = V + ∇xΦε, where sε, Φε solve the acoustic wave equation

• E
• ̺ε, Θε, uε
• rε, Uε

t=τ

t=0+εα

τ

• R3 S(∇x(uε−Uε)) : ∇x(uε−Uε)

dx dt ≤ R (̺ε, Θε, uε, rε, Uε)

34

R (̺ε, Θε, uε, rε, Uε) ≤ τ

• R3 ̺ε (uε − Uε) · ∇xUε · (Uε − uε)dx dt

+ τ

• R3 ̺ε
• V · ∇2

xΦε + ∇xΦε · ∇xUε

• · (Uε − uε)dx dt

+ τ

• R3 ̺ε (∂tV + V · ∇xV) · (Uε − uε)dx dt

+ τ

• R3εαS(∇xUε) : ∇x(Uε − uε)

dx dt − 1 ε2 τ

• R3 divxUε
• p(̺εΘε) − p′(rε)(̺εΘε − rε) − p(rε)
• dx dt

+ 1 ε2 τ

• R3 ̺εΘε∇x
• H′(rε) − H′′(̺0)(rε − ̺0) − H′(̺0)
• · (Uε − uε)

dx dt + 1 ε τ

• R3(rε − ̺εΘε)H′′(rε)divx(sεUε)

dx dt + τ

• R3
• ̺ε(1 − Θε)∂t∇xΦε·(Uε − uε) + ̺ε

ε2 (1 − Θε) ∇xH′(̺0) · (uε − Uε)

• 35

R (̺ε, Θε, uε, rε, Uε) ≤ τ

• R3 ̺ε
• uε − Uε
• · ∇x Uε ·
• Uε − uε
• dx dt

+ τ

• R3 ̺ε
• V · ∇2

xΦε + ∇xΦε · ∇x Uε

• ·
• Uε − uε
• dx dt

+ τ

• R3 ̺ε (∂tV + V · ∇xV) ·
• Uε − uε
• dx dt

+ τ

• R3εαS(∇xUε) : ∇x( Uε − uε)

dx dt − 1 ε2 τ

• R3 divx Uε
• p(̺εΘε) − p′(rε)(̺εΘε − rε) − p(rε)
• dx dt

+ 1 ε2 τ

• R3̺εΘε∇x
• H′(rε) − H′′(̺0)( rε − ̺0) − H′(̺0)
• · ( Uε − uε)

dx d + 1 ε τ

• R3( rε − ̺εΘε)H′′(rε)divx(sεUε)

dx dt + τ

• R3
• ̺ε(1 − Θε)∂t∇xΦε·( Uε − uε)+ ̺ε

ε2 (1 − Θε) ∇xH′(̺0)·(uε − Uε )

36

Convergence- step 3

If we have Dispersive estimates sup

t∈[0,T]

• Φε(t, ·)W m,2(R3) + sε(t, ·)W m,2(R3)
• ≤ cE(ε, m)
• Φ0,εL2(R3) + s0,εL2(R3)
• T
• Φε(t, ·)W m,∞(R3) + sε(t, ·)W m,∞(R3)
• dt

≤ cD(ε, m)

• ∇xΦ0,εL2(R3;R3) + s0,εL2(R3)
• ,

for any m > 0, cD(ε, m) → 0 as ε → 0.

37

Convergence- final step

• then. . .
• E
• ̺ε, Θε, uε
• rε, Uε

t=τ

t=0 ≤ c

τ E

• ̺ε, Θε, uε
• rε, Uε
• dt+o(ε),

....... and by using Gronwall’s lemma we complete the proof.

38

Convergence- final step

• then. . .
• E
• ̺ε, Θε, uε
• rε, Uε

t=τ

t=0 ≤ c

τ E

• ̺ε, Θε, uε
• rε, Uε
• dt+o(ε),

....... and by using Gronwall’s lemma we complete the proof. E

• ̺ε, Θε, uε
• rε, Uε
• =
• R3

1 2̺ε|uε−Uε|2+H(̺εΘε)−H′(r)(̺εΘε−rε)−H(rε) ε2

• 38

Acoustic waves

ε∂tsε + divx [̺0∇xΦε] = 0, ε̺0∂t∇xΦε + ̺0∇x p′(̺0) ̺0 sε

• = 0,

39

Acoustic waves

ε∂tsε + divx [̺0∇xΦε] = 0, ε̺0∂t∇xΦε + ̺0∇x p′(̺0) ̺0 sε

• = 0,

∂ttΦε − 1 ε2 p′(̺0) ̺0 div(̺0∇xΦε) = 0

39

Acoustic waves

ε∂tsε + divx [̺0∇xΦε] = 0, ε̺0∂t∇xΦε + ̺0∇x p′(̺0) ̺0 sε

• = 0,

∂ttΦε − 1 ε2 p′(̺0) ̺0 div(̺0∇xΦε) = 0 Φε(t) = 1 2 exp

• i
• A̺0

t ε Φ0,ε − i

• A̺0

p′(̺0) ̺0 s0,ε

• + 1

2 exp

• −i
• A̺0

t ε Φ0,ε + i

• A̺0

p′(̺0) ̺0 s0,ε

• ,

where A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv)

39

The Acoustic “operator”

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv)

40

The Acoustic “operator”

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv) A̺0[v] = −divx

• p′(̺0)∇xv
• + ̺0∇x

p′(̺0) ̺0

• · ∇xv

= − div(A(x)∇xv) + B(x) · ∇xv where ∇x̺γ

0 = ̺0∇xF

in R3, ̺0 → ̺ > 0 as |x| → ∞. ⇓ ̺0 shares the asymptotic properties of the long range potential F

40

Our Goal

. . . to derive frequency localized Strichartz estimates for the

• perator A̺0 in the form

∞

−∞

• G(A̺0) exp
• ±i
• A̺0t
• [h]
• p

Lq(R3) dt ≤ c(G)hp L2(R3)

for any any G ∈ C∞

c (0, ∞) provided

1 p + 3 q = 1 2

41

Decomposition method of Smith and Sogge

We consider U = G(A̺0) exp

• ±i
• A̺0t
• [h]

the solution of the problem ∂2

ttU − divx (A(x)∇xU) + B(x) · ∇xU = 0,

U(0, ·) = G(A̺0)[h], ∂tU(0, ·) = ±i

• A̺0G(−A̺0)[h],

42

Decomposition method of Smith and Sogge

We consider U = G(A̺0) exp

• ±i
• A̺0t
• [h]

the solution of the problem ∂2

ttU − divx (A(x)∇xU) + B(x) · ∇xU = 0,

U(0, ·) = G(A̺0)[h], ∂tU(0, ·) = ±i

• A̺0G(−A̺0)[h],

χ ∈ C∞

c (R), χ(z) =

1 for |z| < M + 1 0 for |z| > M + 2, We decompose U = χ(|x|)U + (1 − χ(|x|))U = W + V, W = χ(|x|)U, V = (1 − χ(|x|))U

42

Our approach

for W = χ(|x|)U we show frequency and space localized decay estimates. These are estimates on solutions of the wave equation generated by A̺0 localized in both the physical and frequency space. for V = (1 − χ(|x|))U we show frequency localized estimates by combining the local estimates with a global result of Metcalfe and Tataru (Math. Ann., 353(4):1183–1237, 2012)

43

Frequency and space localized decay estimates

W = χ(|x|)G(A̺0) exp

• ±i
• A̺0t
• [h]

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv)

44

An Abstract result of Kato

(Reed and Simon, Methods of Modern Mathematical Physics IV, Theorem XIII.25)

Let A be a closed densely defined linear operator and H a self adjoint densely defined linear operator in a Hilbert space X. For λ / ∈ R, let RH[λ] = (H − λId)−1 denote the resolvent of H. Suppose that Γ = sup

λ/ ∈R, v∈D(A∗), vX=1

A ◦ RH[λ] ◦ A∗[v]X < ∞. Then sup

w∈X, wX=1

π 2 ∞

−∞

A exp(−itH)[w]2

X dt ≤ Γ2.

X = L2(Ω), H =

• −A̺0, A[v] = ϕG(A̺0)[v], v ∈ X,

A ◦ RH[λ] ◦ A∗ = ϕG(A̺0) 1

• −A̺0 − λG(A̺0)ϕ

where G ∈ C∞

0 (0, ∞), ϕ ∈ C∞ 0 (Ω) are given functions.

clicha qui

45

Frequency and space localized decay estimates

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv)

46

Frequency and space localized decay estimates

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv) the point spectrum of A̺0 is empty, (see DeBi`

evre and Pravica,

• Comm. Partial Differential Equations, 17(1-2):69–97, 1992, Theorem 2.1(a)]);

46

Frequency and space localized decay estimates

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv) the point spectrum of A̺0 is empty, (see DeBi`

evre and Pravica,

• Comm. Partial Differential Equations, 17(1-2):69–97, 1992, Theorem 2.1(a)]);

the operator A̺0 satisfies the Limiting absorption principle (DeBi`

evre and Pravica, J. Funct. Anal., 98(2):404–436, 1991, Theorem 1.10).

The cut-off resolvent operator (1 + |x|2)− s

2 ◦ (−A̺0 − µ ± iδ)−1 ◦ (1 + |x|2)− s 2

can be extended as bounded linear operator on L2, for δ → 0 and λ belonging to compact set of (0, ∞).

46

Frequency and space localized decay estimates

A̺0 : v → −p′(̺0) ̺0 divx (̺0∇xv) By applying an abstract theorem on linear operators by Kato, clicca quiwe get the frequency localized decay estimates ∞

−∞

• ϕG(A̺0) exp
• ±i
• A̺0t
• [h]
• 2

L2(R3) dt ≤ c(ϕ, G)h2 L2(R3)

for any ϕ ∈ C∞

c (R3) and any G ∈ C∞ c (0, ∞)

47

Estimates for V

the function V = (1 − χ(|x|))U satisfies ∂2

ttV − divx

• ˜

A(x)∇xV

• + ˜

B(x) · ∇xV = G, G = ˜ A∇xχ · ∇xU + divx( ˜ A∇xχU) − ˜ B · ∇xχU V (0, ·) = (1 − χ)U0, ∂tV (0, ·) = (1 − χ)U1.

48

Estimates for V

the function V = (1 − χ(|x|))U satisfies ∂2

ttV − divx

• ˜

A(x)∇xV

• + ˜

B(x) · ∇xV = G, G = ˜ A∇xχ · ∇xU + divx( ˜ A∇xχU) − ˜ B · ∇xχU V (0, ·) = (1 − χ)U0, ∂tV (0, ·) = (1 − χ)U1. ˜ A(x) =

• p′(̺0)

if |x| > L p′(¯ ̺) if |x| < L ˜ B(x) =

• ̺0∇x
• p′(̺0)

̺0

• if |x| > L

if |x| < L

48

Strichartz estimates by Metcalfe and Tataru

( Math. Ann., 353 (4):1183–1237, 2012.)

∂2

ttV −divx

• ˜

A(x)∇xV

• + ˜

B(x)·∇xV = Z, V (0, ·) = V0, ∂tV (0, ·) = V1 V Lp(R;Lq(R3)) ≤

• ∇xV0L2(R3;R3) + V1L2(R3) + ZLr(R;Ls(R3))
• ,

whenever 1 p + 3 q = 1 2 = 1 r + 3 s − 2.

49

Strichartz estimates by Metcalfe and Tataru

( Math. Ann., 353 (4):1183–1237, 2012.)

∂2

ttV −divx

• ˜

A(x)∇xV

• + ˜

B(x)·∇xV = Z, V (0, ·) = V0, ∂tV (0, ·) = V1 V Lp(R;Lq(R3)) ≤

• ∇xV0L2(R3;R3) + V1L2(R3) + ZLr(R;Ls(R3))
• ,

whenever 1 p + 3 q = 1 2 = 1 r + 3 s − 2. The coefficients ˜ A(x), ˜ B(x) must be “asymptotically flat” in the sense that

• j∈Z

sup

x∈Aj

• |x|2|∇2

x ˜

A(x)| + |x|| ˜ A(x)| + | ˜ A(x) − A|

• ≤

δ, A > 0,

• j∈Z

sup

x∈Aj

• |x|2|∇x ˜

B(x)| + |x|| ˜ B(x)|

• ≤

δ, where Aj = {2j ≤ |x| ≤ 2j+1} are the dyadic regions covering R3, and δ > 0 is sufficiently small.

49

∂2

ttV − divx

• ˜

A(x)∇xV

• + ˜

B(x) · ∇xV = G

50

∂2

ttV − divx

• ˜

A(x)∇xV

• + ˜

B(x) · ∇xV = G ∇x ˜ A, ˜ B = 0 for all |x| < L, |∇x ˜ A(x)| + | ˜ B(x)| ≤ c|∇xF| ≤ c |x|2 for |x| ≥ L. ∇2

x ˜

A, ∇x ˜ B = 0 for all |x| < L, |∇2

x ˜

A(x)| + |∇x ˜ B(x)| ≤ c

• |∇xF(x)|2 + |∇2

xF(x)|2

≤ c 1 |x|3 + 1 |x|4

• for |x| ≥ L.

= ⇒ the coefficients are “asymptotically flat”, we can apply Metcalfe and Tataru.

50

Finally.....

. . . we got frequency localized Strichartz estimates for the operator A̺0 in the form ∞

−∞

• G(A̺0) exp
• ±i
• A̺0t
• [h]
• p

Lq(R3) dt ≤ c(G)hp L2(R3)

for any any G ∈ C∞

c (0, ∞) provided

1 p + 3 q = 1 2

51

Finally.....

. . . we got frequency localized Strichartz estimates for the operator A̺0 in the form ∞

−∞

• G(A̺0) exp
• ±i
• A̺0

t ε

• [h]
• p

Lq(R3)

dt ≤ εc(G)hp

L2(R3)

for any any G ∈ C∞

c (0, ∞) provided

1 p + 3 q = 1 2

51

Finally.....

. . . we got frequency localized Strichartz estimates for the operator A̺0 in the form ∞

−∞

• G(A̺0) exp
• ±i
• A̺0

t ε

• [h]
• p

Lq(R3)

dt ≤ εc(G)hp

L2(R3)

for any any G ∈ C∞

c (0, ∞) provided

1 p + 3 q = 1 2 Φε(t) = 1 2 exp

• i
• A̺0

t ε Φ0,ε − i

• A̺0

p′(̺0) ̺0 s0,ε

• + 1

2 exp

• −i
• A̺0

t ε Φ0,ε + i

• A̺0

p′(̺0) ̺0 s0,ε

• ,

51

Dispersive estimate for Φε

Φε(t) = 1 2 exp

• i
• A̺0

t ε Φ0,ε − i

• A̺0

p′(̺0) ̺0 s0,ε

• + 1

2 exp

• −i
• A̺0

t ε Φ0,ε + i

• A̺0

p′(̺0) ̺0 s0,ε

• ,

T

• Φε(t, ·)W m,∞(R3)
• dt ≤ cD(ε, m)
• ∇xΦ0,εL2(R3;R3)
• 52

Final remarks

☞ To get the estimate of the acoustic operator it is important the far field behavior of ̺ε, ̺ε → ¯ ̺ = 0 (̺0 → ¯ ̺ ), as |x| → 0

53

Final remarks

☞ To get the estimate of the acoustic operator it is important the far field behavior of ̺ε, ̺ε → ¯ ̺ = 0 (̺0 → ¯ ̺ ), as |x| → 0 ☞ If ̺0 → 0, as |x| → 0, then we don’t have any information on the dispersive behavior of A̺0[v] := p′(̺0) ̺0 divx (̺0∇xv)

53

Final remarks

☞ To get the estimate of the acoustic operator it is important the far field behavior of ̺ε, ̺ε → ¯ ̺ = 0 (̺0 → ¯ ̺ ), as |x| → 0 ☞ If ̺0 → 0, as |x| → 0, then we don’t have any information on the dispersive behavior of A̺0[v] := p′(̺0) ̺0 divx (̺0∇xv) ☞ It could be interesting to study a more “physical model” (including combustion and more physical constitutive equation for the pressure law)

53