Sharp Lp-decay rates for perturbations of degenerate ...Introduction Interpolation inequalities...

IntroductionInterpolation inequalities

Decay ratesEnergy estimates

Sharp Lp-decay rates for perturbations ofdegenerate scalar viscous shocks:the return of the energy method

Ramón G. Plaza

Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de México

IX Americas Conference on Differential EquationsTrujillo, Perú

January 13, 2012

Ramón G. Plaza Sharp decay rates for sonic shocks

Degenerate (or sonic) shocksHistory (abridged)PreliminariesMain result

Introduction

Interpolation inequalities

Decay rates

Energy estimates

Degenerate viscous shock wavesScalar viscous conservation law:

ut + f (u)x = (b(u)ux)x,

where (x, t) ∈ R× [0,+∞), f , b ∈ C2, b > 0.

Viscous shock profile:

u(x, t) = u(x− st),

(b(u)u′)′ = f (u)′ − su′,

u(ξ)→ u±, as ξ → ±∞.′ = d/dξ, ξ = x− st.

Degenerate viscous shock wavesScalar viscous conservation law:

ut + f (u)x = (b(u)ux)x,

where (x, t) ∈ R× [0,+∞), f , b ∈ C2, b > 0.

Viscous shock profile:

u(x, t) = u(x− st),

(b(u)u′)′ = f (u)′ − su′,

u(ξ)→ u±, as ξ → ±∞.′ = d/dξ, ξ = x− st.

Generalized shock front: (u+, u−, s) ∈ R3, u− 6= u+.

Rankine-Hugoniot condition:

−s(u+ − u−) + f (u+)− f (u−) = 0,

Generalized entropy condition:

−s(u− u−) + f (u)− f (u−) < 0, ∀u ∈ (u+, u−).

Degenerate shocks

f changes convexity in (u+, u−).

Characteristic (sonic) shock speed:

s = f ′(u+), or, s = f ′(u−).

Toy model for sonic gas waves (e.g. Chapman-Jouguet combustion).Case boundary bet. undercompressive and Lax shock.

History (abridged)

I MEI (1995). One inflection point for f .I MATSUMURA-NISHIHARA (1994). Asymmetric, non-bounded

on the sonic side, weight function.I NISHIKAWA (1998). Multi-d.I MEI (1999). Exponentially decaying perturbations.

All results: not sharp decay rates in L∞.

History (abridged)

Pointwise Green’s function bounds methods

Under spectral stability assumption, resolvent kernel bounds;ponitwise bounds for the Green function.

I Non degenerate case: ZUMBRUN-HOWARD (1998): parabolicviscosity. MASCIA-ZUMBRUN (2002-2008), relaxation and realviscosity. ZUMBRUN (2000-2008) Multi-d.

I HOWARD (2002a), (2002b). Degenerate viscous shocks: scalarcase, first order degeneracy.

I HOWARD (2004). Degenerate shocks, 2× 2 systems.

Sharper decay rates in Lp, slower decaying data. Drawback:degeneracy of order one only. Very technical estimates.

For zero-mass perturbations∫

u− u0 = 0, u0 decaying as (1 + |x|)−r,there holds (HOWARD (2002a))

‖u− u‖Lp ≤ C(1 + t)1−r/2 log t, 1 < r ≤ 3, p > 1.

Preliminaries

Normalization:x→ x− st, f → f − su− (f (u±)− su±).

Assumptions:

f , b ∈ C2, b > 0, (regularity and positive diffusion), (A1)

f (u−) = f (u+) = 0, (Rankine-Hugoniot condition), (A2)

f (u) < 0 ∀ u ∈ (u+, u−), (generalized entropy condition). (A3)

Preliminaries

Normalization:x→ x− st, f → f − su− (f (u±)− su±).

Assumptions:

f , b ∈ C2, b > 0, (regularity and positive diffusion), (A1)

f (u−) = f (u+) = 0, (Rankine-Hugoniot condition), (A2)

f (u) < 0 ∀ u ∈ (u+, u−), (generalized entropy condition). (A3)

Existence and structure of profiles:

Under assumptions (A), let

θ := min{k ∈ Z+ :dkfduk (u+) 6= 0} ≥ 1,

degree of degeneracy of the shock. Then there exists a profile u,u(±∞) = u±, unique up to translations, monotone decreasing,ux < 0, decaying as

|∂jx(u(x)− u−)| ≤ Ce−c|x|,

|∂jx(u(x)− u+)| ≤ C|x|−1/θ,

j = 0, 1, 2, uniform C > 0, all x ∈ R. HOWARD (2002a).

Perturbation equations

Zero-mass perturbations.∫R

(u0(x)− u(x− δ)) dx = 0.

W.l.o.g. δ = 0. Integrated perturbation:

u(x, t) = vx(x, t) + u(x)

Perturbation equations

vt = b(u)vxx − a(x)vx + F,

v(x, 0) = v0(x) =

−∞(u0(y)− u(y)) dy

a(x) = f ′(u)− b(u)x,

F := −(f (vx + u)− f (u)− f ′(u)vx)+

+ (b(vx + u)− b(u)− b′(u)vx)(ux + vxx)+

+ b′(u)vxvxx = b′(u)vxvxx +O(v2x),

The Matsumura-Nishihara weight function

η(x) := η(u(x)),

η(u) :=(u− u+)(u− u−)

f (u)> 0, u ∈ (u+, u−).

η ∼ |u− u+|−θ, as x→ +∞,η ∼ C > 0, as x→ −∞,

The Matsumura-Nishihara weight function

η(x) := η(u(x)),

η(u) :=(u− u+)(u− u−)

f (u)> 0, u ∈ (u+, u−).

η ∼ |u− u+|−θ, as x→ +∞,η ∼ C > 0, as x→ −∞,

Properties:

η ∼ 〈x〉+ :=

{(1 + x2)1/2, x ≥ 0,1, x < 0.

η ≥ C−1 > 0, |ηx| ≤ Cη,

Right sign for energy estimates:

Φ(x) = ((b(u)η)x + a(x)η)x

= ux(d2/du2)(f (u)η(u))|u=u = −2|ux| < 0

Properties:

η ∼ 〈x〉+ :=

{(1 + x2)1/2, x ≥ 0,1, x < 0.

η ≥ C−1 > 0, |ηx| ≤ Cη,

Right sign for energy estimates:

Φ(x) = ((b(u)η)x + a(x)η)x

= ux(d2/du2)(f (u)η(u))|u=u = −2|ux| < 0

Main result

I P, J. Math Anal. Appl. 382 (2011).

Theorem. Under assumptions, with u+ < u−. Suppose zero masscondition and that

−∞(u0(x)− u(x)) dx ∈ Zη,p,

Zη,p = L1η ∩ L2

η ∩ Lpη ∩W2,p, for some 2 ≤ p < +∞.

There exists ε > 0 such that if ‖v0‖Zη,p < ε, then the Cauchy problemfor perturbation equation with initial condition u(0) = u0 has aunique global solution u− u ∈ C([0,+∞]; W1,p) satisfying

‖u− u‖Lp ≤ CME0t−1/2(1 + t)−12 (1−1/p), 2 ≤ p < +∞,

‖u− u‖L∞ ≤ CME0t−1/2−1/p(1 + t)−12 (1−1/p),

for all 0 < t < +∞, E0 = ‖v0‖L1η

+ ‖v0‖2L2η

+ ‖v0‖pLpη, with uniform

M > 0.

Features:

I Better decay rates, no log t factor.I All orders of degeneracyI Elementary proof.

Interpolation inequalities in Lpη

Example: L2

Introduction

Decay rates

Energy estimates

Example: L2

Interpolation inequalities in weighted Lp spaces

Inequalities 1:For 2 ≤ p < +∞, all u ∈ W2,p:

‖u‖p∞ ≤ C‖ux‖Lp‖u‖p−1

‖ux‖Lp ≤ C‖(|ux|p/2)x‖2/(p+2)

L2 ‖u‖2/(p+2)Lp ,

some constants C = C(p) > 0.

Example: L2

Inequality 2:For 2 ≤ p < +∞, all u ∈ W2,p ∩ Lp

η ∩ L1η, ux ∈ W2,p ∩ Lp

‖u‖p(p+1)/(p−1)

≤ C‖u‖2p/(p−1)

‖(|u|p/2)x‖2L2η,

some C = C(p) > 0. η = Matsumura-Nishihara function.

Reference (without η): ESCOBEDO, ZUAZUA (1991).

Example: L2

Inequality 2:For 2 ≤ p < +∞, all u ∈ W2,p ∩ Lp

η ∩ L1η, ux ∈ W2,p ∩ Lp

‖u‖p(p+1)/(p−1)

≤ C‖u‖2p/(p−1)

‖(|u|p/2)x‖2L2η,

some C = C(p) > 0. η = Matsumura-Nishihara function.

Reference (without η): ESCOBEDO, ZUAZUA (1991).

Example: L2

Observation: By Hölder,

‖(|u|p/2)x‖2L2η

∫Rη|u|p−2u2

x dx ≤ p2

4‖ux‖2

Lpη‖u‖p−2

Lpη< +∞.

if u, ux ∈ W2,p ∩ Lpη.

Example: L2

Example: in L2

Use ‖u‖2∞ ≤ 2‖u‖L2‖ux‖L2 :

‖u‖8L2 =

u2 dx)4≤ ‖u‖4

∞‖u‖4L1 ≤ 4‖u‖2

L2‖ux‖2L2‖u‖4

⇒ ‖u‖6L2 ≤ C‖ux‖2

L2‖u‖4L1 .

Ineq. with p = 2.

Example: Decay rates in L2

Lp decay rates

Introduction

Decay rates

Energy estimates

Lp decay rates

Decay rates

Fact: If ρ ≥ 0, C1 in t ≥ 0,

dρdt≤ −Cρβ,

C > 0, β > 1, with ρ(0) = ρ0 > 0. Then

ρ ≤ ξ, a.e. in t,

= −Cξβ, ξ(0) = ρ0.

Lp decay rates

Hypotheses: For 0 < t < T ≤ +∞, some C0,C1 > 0,

ddt‖u(t)‖2

L2 ≤ −C1‖ux‖2L2 ,

‖u(t)‖L1 ≤ C0.

‖u(t)‖L2 ≤ C(1 + t)−1/4,

C = O(C0 + ‖u(0)‖L2)

Lp decay rates

Proof:

ddt‖u‖2

L2 ≤ −C1‖ux‖2L2 ≤ −

C‖u‖6

L2‖u‖−4L1 ≤ −C2‖u‖6

C2 =C1

ξ =(2C2t + ‖u(0)‖−4

)−1/2,

solution to dξ/dt = −C2ξ3, ξ(0) = ‖u(0)‖2

L2 , β = 3.

Lp decay rates

ξ(t)−2 ≥ C3(1 + t), C3 = min{2C2, ‖u(0)‖−4L2 }.

⇒ ‖u(t)‖2L2 ≤ C−1/2

3 (1 + t)−1/2 ≤ C(C0 + ‖u(0)‖L2)2(1 + t)−1/2.

Lp decay rates

Lp decay ratesProposition. Let 2 ≤ p < +∞, and u a solution such that

ddt‖u(t)‖p

Lpη≤ −C1‖(|u|p/2)x(t)‖2

‖u(t)‖L1η≤ C0,

for all 0 < t < T ≤ +∞, and uniform C1,C0 > 0. Then, there existsC > 0 such that

‖u(t)‖Lpη≤ C(1 + t)−

12 (1−1/p),

for all 0 < t < T. Moreover,

C = O(C0 + ‖u(0)‖Lpη).

Local existenceBasic estimateL1η estimate

Higher order estimates

Introduction

Decay rates

Energy estimates

Local existence

Zη,p := W2,p ∩ Lpη ∩ L1

η ∩ L2η,

Xη,p(0,T) := {v ∈ C([0,T]; Zη,p), vx ∈ L2([0,T]; Zη,p)}, 0 < T ≤ +∞.

Proposition. For any ε0 > 0 there exists a positive constant T0depending on ε0 such that if v0 ∈ Zη,p and ‖v0‖Zη,p ≤ ε0, then theCauchy problem for v has a unique solution v ∈ Xη,p(0,T0) satisfying‖v(t)‖Zη,p < 2ε0 for each 0 ≤ t ≤ T0.

Local existence

Zη,p := W2,p ∩ Lpη ∩ L1

η ∩ L2η,

Xη,p(0,T) := {v ∈ C([0,T]; Zη,p), vx ∈ L2([0,T]; Zη,p)}, 0 < T ≤ +∞.

Proposition. For any ε0 > 0 there exists a positive constant T0depending on ε0 such that if v0 ∈ Zη,p and ‖v0‖Zη,p ≤ ε0, then theCauchy problem for v has a unique solution v ∈ Xη,p(0,T0) satisfying‖v(t)‖Zη,p < 2ε0 for each 0 ≤ t ≤ T0.

Basic energy estimate

For v(t) ∈ Xη,p(0,T), T > 0,

1p‖v(t)‖p

+4(p− 1)

0‖b(u)1/2(|v(τ)|p/2)x‖2

∫R|ux||v(τ)|p dx dτ =

=1p‖v(0)‖p

∫RηFv(τ)|v(τ)|p−2 dx dτ

R(t) := supτ∈[0,t]

‖v(τ)‖Zη,p ,

Consequences:1. If R(t) < ε1 for t ∈ [0,T], for 0 < ε1 � 1,

‖v(t)‖pLpη

+ C∫ t

0‖(|v(τ)|p/2)x‖2

dτ ≤ ‖v(0)‖pLpη.

R(t) := supτ∈[0,t]

‖v(τ)‖Zη,p ,

Consequences:1. If R(t) < ε1 for t ∈ [0,T], for 0 < ε1 � 1,

‖v(t)‖pLpη

+ C∫ t

0‖(|v(τ)|p/2)x‖2

dτ ≤ ‖v(0)‖pLpη.

2. When p = 2, R(t) < ε1,

‖v(t)‖2L2η

+ C∫ t

0‖vx(τ)‖2

dτ ≤ ‖v(0)‖2L2η

3. If R(t) < ε1,

ddt‖v(t)‖p

+ C1‖(|v|p/2)x(t)‖2L2η≤ 0.

L1η estimate

If R(t) < ε1, then

‖v(t)‖L1η≤ C(‖v(0)‖L1

η+ ‖v(0)‖2

L2η),

Consequence: Decay rate, with C0 = ‖v(0)‖L1η

+ ‖v(0)‖2L2η:

For v ∈ Xη,p(0,T), if R(t) < ε1 then

‖v(t)‖pLpη≤ CEp

0(1 + t)−(p−1)/2,

E0 := ‖v(0)‖L1η

+ ‖v(0)‖2L2η

+ ‖v(0)‖pLpη.

L1η estimate

If R(t) < ε1, then

‖v(t)‖L1η≤ C(‖v(0)‖L1

η+ ‖v(0)‖2

L2η),

Consequence: Decay rate, with C0 = ‖v(0)‖L1η

+ ‖v(0)‖2L2η:

For v ∈ Xη,p(0,T), if R(t) < ε1 then

‖v(t)‖pLpη≤ CEp

0(1 + t)−(p−1)/2,

E0 := ‖v(0)‖L1η

+ ‖v(0)‖2L2η

+ ‖v(0)‖pLpη.

Estimate for vx:Suppose v ∈ Xη,p(0,T), 2 ≤ p < +∞, solution. For ε2 > 0 suff.small if R(t) < ε2 then

tα(1 + t)β‖vx(t)‖pLp + C2

0τα(1 + τ)β‖(|vx|p/2)x(τ)‖2

dτ ≤

CEp0tα−p/2(1 + t)β−

12 (p−1),

for 0 ≤ t ≤ T , with α > p/2, β > 12(p− 1). Moreover,

‖vx(t)‖pLp ≤ CEp

0t−p/2(1 + t)−12 (p−1).

Estimate for vxx:Suppose v ∈ Xη,p(0,T), 2 ≤ p < +∞, solution. For 0 < ε3 ≤ ε2 suff.small if R(t) < ε3 then

tγ(1 + t)δ‖vxx(t)‖pLp + C3

0τγ(1 + τ)δ‖(|vxx|p/2)x(τ)‖2

≤ CEp0tγ−p(1 + t)δ−

12 (p−1),

0 ≤ t ≤ T , with γ > p, δ > 12(p− 1). Moreover,

‖vxx(t)‖pLp ≤ CEp

0t−p(1 + t)−12 (p−1).

Global existenceTheorem. Suppose v0 ∈ Zη,p, 2 ≤ p < +∞. Then there exists ε > 0s. t. if ‖v0‖Zη,p < ε, then the Cauchy problem for v has a uniqueglobal solution v ∈ Xη,p(0,+∞) with

‖v(t)‖L1 ≤ ME0,

‖v(t)‖Lp ≤ ME0(1 + t)−12 (1−1/p),

‖v(t)‖L2 ≤ ME0(1 + t)−14 ,

‖vx(t)‖Lp ≤ ME0t−12 (1 + t)−

12 (1−1/p),

‖vxx(t)‖Lp ≤ ME0t−1(1 + t)−12 (1−1/p),

for all 0 < t < +∞, uniform M > 0, with

E0 = ‖v0‖L1η

+ ‖v0‖2L2η

+ ‖v0‖pLpη

.Ramón G. Plaza Sharp decay rates for sonic shocks

Thank you!

Sharp Lp-decay rates for perturbations of degenerate ...Introduction Interpolation inequalities...

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