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High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws Stphane Junca Labo JAD, Universit de Nice Sophia-Antipolis & Coffee Team INRIA June 25 2012 Stphane Junca (Nice) HYPERBOLIC PDEs 2012

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SLIDE 1

High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws

Stéphane Junca

Labo JAD, Université de Nice Sophia-Antipolis & Coffee Team INRIA

June 25 2012

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 1 / 33

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SLIDE 2

Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 2 / 33

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SLIDE 3

Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 3 / 33

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I) Lions, Perthame, Tadmor 1994

∂u ∂t + ∇ · F(u) = u(0, X) = u0(X) X ∈ I Rd, u : [0, +∞) × I Rd → I R, F : I R → I Rd Smoothing effect for NONLINEAR flux M > 0, ∃ s = s(F, M) > 0 sup

X

|u0(X)| ≤ M ⇒ u ∈ W s,1

loc (]0, +∞[×I

Rd) ∩ W s,1

loc (I

Rd), ∀t > 0

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 4 / 33

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II) Lions, Perthame, Tadmor 1994

NONLINEAR flux velocity : a(v) = F ′(v) : ∃α > 0, ∃C > 0, sup

τ 2+|ξ|2=1

measure{|v| ≤ M, |τ + ξ · a(v)| ≤ δ} ≤ Cδα Sobolev exponent : ∀ s < α 2 + α Improvement : Tadmor, Tao, 2007 : ∀ s < α 1 + 2α

Conjecture : Lions, Perthame, Tadmor 1994

ssup=αsup

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 5 / 33

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SLIDE 6

III) Lions, Perthame, Tadmor 1994

suniform UNIFORM Sobolev bound and boundary layers Let M > 0, ε > 0, A > 0, s < s(F, M), ∃ C = C(ε, s, u0∞, A) u0L∞ ≤ M uW s,1([ε,A]×[−A,A]d)+ supt>ε u(t, .)W s,1([−A,A]d) ≤ C Proof : Kinetic formulation & averaging lemmas ∂tf + a(v).∇xf = ∂vm f(t, x, v) , v ∈ I R, m(t, x, v) ≥ 0

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 6 / 33

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Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 7 / 33

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SLIDE 8

One dimensional case : d = 1

Linear case : a′ ≡ 0 ∂u ∂t + c ∂u ∂x = 0 No smoothing effect : u(t, x) = u0(x − ct) Propagations of high oscillations with large amplitude : u0

ε(x) = u0

x ε

  • ⇒

uε(t, x) = u0 x − ct ε

  • ,

where u0 is periodic.

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 8 / 33

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The strongest Nonlinear case : strictly convex flux

with u0(x + 1) ≡ u0(x) ∈ L∞(I R) and mean u0 = 1 u0(θ)dθ ∂u ∂t + ∂f(u) ∂x , ∀ v ∈ I R, a′(v) = 0, f(u) = u2 2 Smoothing effect, O. Oleinik ; P . D. Lax 50’ On (0, 1) Total variation u(t, .) ≤ C t . ⇒ Decay of the amplitude, |u(t, x) − u0| ≤ C t ⇒ No propagations of high oscillations with large amplitude : u0

ε(x) = u0

x ε

  • ⇒

|uε(t, x) − u0| ≤ εC t Proof : y = x ε , s = t ε

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 9 / 33

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Propagations of high oscillations with small amplitude

u0

ε(x) = εu0

x εq

  • uε(t, x) ≃

       εu0 x

εq

  • if q < 1

linear propagation εU

  • t, x

ε

  • if q = 1

nonlinear propagation εu0 if q > 1 nonlinear smoothing effect Critical exponent, q = 1 = α,

  • R. Diperna, A. Majda, C.M.P

., 1985, ∂U ∂t + ∂ ∂θ U2 2 = 0, U(0, θ) = u0(θ) uε(t, x) = εU

  • t, x

ε

  • .

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 10 / 33

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Degenerate Nonlinear case

p ≥ 2, a(v) = vp , αsup = 1 p < 1 ∂u ∂t + ∂ ∂x u1+p 1 + p

  • = 0

u0

ε(x) = εu0

x εq

  • ⇒

uε(t, x) ≃      εu0 x

εq

  • if q < p

εU

  • t, x

εq

  • if q = p

εu0 if q > p Profile equation for critical exponent, q = p, ∂U ∂t + ∂ ∂θ U1+p 1 + p = 0, U(0, θ) = u0(θ) uε(t, x) = εU

  • t, x

εp

  • Stéphane Junca (Nice)

HYPERBOLIC PDE’s 2012 June 25 2012 11 / 33

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Burgers 2D is not non-linear

∂tu + ∂xu2 + ∂yu2 = u(0, x, y) = u0 x − y ε2012

  • Panov : ξ = (1, −1), F = (u2, u2),

ξ · F(u) ≡ 0 Lions-Perthame-Tadmor : ξ · F ′(v) = ξ · a(v) ≡ 0, αsup = 0 Enguist-E : ξ · F ′′(v) ≡ 0, stationary solutions without smoothing effect u(t, x, y) = u0(x − y)

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 12 / 33

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2D genuine nonlinear example

∂tu + ∂x u2 2 + ∂y u3 3 = u(0, x, y) = 0 + εu0 φ0(x, y) ε2

  • Genuine nonlinear flux : a(u) = (u, u2)

a′(u) a′′(u)

  • =

1 2u 2

  • φ0(x, y) = x cancellations

φ0(x, y) = y propagations : (0, 1) = ∇φ0 ⊥ a(u = 0) = (1, 0)

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 13 / 33

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High frequency waves

∂u ∂t + ∇x.F(u) = u(0, x) = u + εU0 φ0(x) εq

  • φ0(x)

= v · x

1

Propagation : uε(t, x) = u + εU (t, ε−qφ(t, x)) + . . .

2

Cancellation uε(t, x) = u + εU + . . . Chen,J, Rascle (JDE 06), multiphase : u(0, x) = u + εU0 (ε−q1φ1(x), · · · , ε−qdφd(x))

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 14 / 33

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Propagation of smooth high oscillations

Theorem

q > 1 an integer, F ∈ C∞(I R, I Rd), U0 ∈ C1(I R/Z Z , I R), v = (0, · · · , 0), dka duk (u) v = 0, k = 1, · · · , q − 1 (1) then ∃ T0 > 0 such that, for all ε ∈]0, 1], uε smooth on [0, T0] × I R : uε(t, x) = u + εU

  • t, φ(t, x)

εq

  • + O(ε2) in C1([0, T0] × I

Rd), ∂U ∂t + b∂Uq+1 ∂θ = 0, U(0, θ) = U0(θ). b =

1 (q+1)!

  • a(q)(u) v
  • ,

φ(t, x) = v · (x − t a(u)).

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 15 / 33

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Proof : WKB expansions

uε(t, x) = u + ε Uε

  • t, φ(t, x)

εq

  • ,

Uε(0, θ) = U0(θ) Taylor expansions : v = ∇φ ∂tUε

  • t, φ(t, x)

εq

  • =

∂tUε − ε−q(a(u) · v)∂θUε divxF(uε) =

q

  • k=0

1 εq−(k+1) ∂θUk+1

ε

(k + 1)!a(k)(u) · v + εq+2divxGε

q(Uε)

= ε1−q(a(u) · v)∂θUε + εb∂θUq+1

ε

+ ε2∂θgε

q(Uε),

Simplification and characteristics 0 = ∂tuε + divxF(uε) = ε

  • ∂tUε + b∂θUq+1

ε

+ ε∂θgε

q(Uε)

  • .

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 16 / 33

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Uniform Sobolev bounds

sequence bounded in L∞ for 0 < ε ≤ 1 uε(t, x) = u + εU

  • t, φ(t, x)

εq

  • + . . .

and if d

dθU0 = 0 a.e. there exists C > 0, 0 ≤ t ≤ T0

1 C ≤ uε(t, .)W s,1

loc (I

Rd) ≤ C,

s = 1

q

For s > 1

q the Sobolev norm W s,1 loc (I

Rd) blows up.

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 17 / 33

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Sobolev estimates

0 < s < 1 , d“s′′ dx“s′′

  • εV

x εq

  • ∼

ε εsq = ε1−sq Upper bound : interpolation between L1 and W 1,1

  • rder ε in L1

loc

  • rder ε1−q in W 1,1

loc

  • rder ε(1−s)1+s(1−q) in W s,1

loc

Lower bound : intrinsic semi-norm W s,p(I Rd) |V(x) − V(y)|p |x − y|d+sp dxdy

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 18 / 33

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If the orthogonality condition : dka duk (u) v = 0, k = 1, · · · , q − 1, is violated then

Cancellation of high oscillations, Chen, J, Rascle, 06’

Let F belongs to Cq+1 and U0 ∈ L∞(I R/Z Z , I R), If for some 0 < j < q dja duj (u) v = then uε(t, x) = u + εU0 + o(ε) in L1

loc(]0, +∞[×I

Rd). proof : compactness with kinetic formulation

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 19 / 33

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Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 20 / 33

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Nonlinear flux

Lax : d=1, f ′′ = a′ = 0 ⇐ ⇒ genuine nonlinear Tartar : d=1, not a linear function on any interval Engquist-E : independence of functions F′′

1(v), · · · , F′′ d(v)

Lions-Perthame-Tadmor : related to a(v) = F′(v) : αsup Panov : ∀ξ = 0, ξ · F is non constant on any interval genuine nonlinearity condition for smooth multi-D flux F : det(a′(v), a′′(v), · · · , a(d)(v)) = 0 everywhere

Chen, J., Rascle 2006, Crippa ; Otto ; Westdickenberg 2008.

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 21 / 33

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Nonlinear Flux

Definition

F ∈ C∞(I R, I Rd), I = [−M, M], dF[u] = inf{k ≥ 1, rank{a′(u), · · · , a(k)(u)} = d} ≥ d, = inf{k ≥ 1, rank{F′′(u), · · · , F(k+1)(u)} = d}, dF = sup

|u|≤M

dF[u] ∈ {d, d + 1, · · · } ∪ {+∞}.

1

nonlinear flux F on [−M, M] if dF < +∞

2

genuine nonlinear if dF = d

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 22 / 33

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Sharp measurement of the flux non-linearity

Theorem

F ∈ C∞([−M, M], I Rd), the Lions-Perthame-Tadmor measurement of the flux non-linearity : αsup is given by αsup = 1 dF ≤ 1 d . If αsup > 0 there exists u ∈ [−M, M] such that dF = dF[u].

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 23 / 33

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Sketch of the proof

1

F(u) = c u1+p (1 + p)!, a(v) = c vp p! , c = 0 dF = p, u = 0 (a′(0), · · · , a(p−1)(0), a(p)(0)) = (0, · · · , 0, c) α = 1 p measure{|v| ≤ M, |a(v)| ≤ δ} = (2|c/p!|−α) × δα

2

u → dF[u] is upper semi-continuous

3

Uniform control of measure{|v| ≤ M, |A(v; τ, ξ)| ≤ δ} where A(v; τ, ξ) = τ + ξ · a(v), τ 2 + |ξ|2 = 1.

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 24 / 33

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Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 25 / 33

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Maximal uniform Sobolev bound

Theorem

IF : nonlinear flux F ∈ C∞([−M, M], I Rd). and : αsup > 0 : the sharp measurement of the flux non-linearity. THEN ∃ u ∈ [−M, M], ∃ T0 > 0, ∃ (uε

0)0<ε<1 such that

uε

0 − uL∞(I Rd) < ε,

and the sequence of entropy solutions (uε)0<ε<1 for all s ≤ αsup, the sequence (uε)0<ε<1 is uniformly bounded in W s,1

loc ([0, T0] × I

Rd) ∩ C0([0, T0], W s,1

loc (I

Rd)), for all s > αsup, the sequence (uε)0<ε<1 is unbounded in W s,1

loc ([0, T0] × I

Rd) and in C0([0, T0], W s,1

loc (I

Rd)).

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 26 / 33

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Construction of the supercritical oscillating sequences

1

1 αsup = dF = dF[u]

2

Oscillating initial data near u : with U′

0 = 0 a.e.

uε(0, x) = u + εU0 φ0(x) εq

  • 3

The best phase φ0 : (0, · · · , 0) = v = ∇φ0 ⊥ {a′(u), · · · , a(q−1)(u)} q = dF ∇φ0 · a(q)(u) =

4

Sobolev uniform optimal bounds for (uε)0<ε≤1 s = 1 q = 1 dF = αsup

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 27 / 33

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Highlight of the conjecture

Optimal oscillating example ∃(uε)0<ε<1 uniformly bounded in the best Sobolev spaces conjectured by Lions-Perthame-Tadmor s = αsup for s > ssup there is no uniform bound in W s,1 for solutions with initial data bounded in L∞ Upper bound for the maximal Sobolev exponent with it uniform control of the Sobolev norm ssup uniform ≤ αsup

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 28 / 33

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Sommaire

1

Conjecture : Lions, Perthame, Tadmor 1994

2

High frequency waves dimension d = 1 d>1 Sobolev estimates

3

Characterization of Nonlinear Flux

4

Bound of the uniform maximal smoothing effect

5

Recent Works

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 29 / 33

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SLIDE 30

Recent Works

1

Optimal smoothing effect s = αsup for nonlinear degenerate convex flux in one dimension :d = 1. Smoothing effect with Christian Bourdarias & Marguerite Gisclon with fractional BV spaces : BV s ≃ W s,1/s ssup = αsup for d = 1 & a(v) ր Examples for optimality with Pierre Castelli

2

d ≥ 1 bound for the smoothing effect with Pierre Castelli ∀ε > 0, ∃ |u0(x)| ≤ M such that for t > 0 u(t, .) ∈ W αsup−ε,1 and u(t, .) / ∈ W αsup+ε,1 i.e. : ssup ≤ αsup

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 30 / 33

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Optimality for the 1D case

1

f(u) = u3, u0(x) = x sin 1 x

  • Kuo-Shung Cheng.

The space BV is not enough for hyperbolic conservation laws. J.D.E. 1983.

2

For power-law flux and piecewise constant solutions : Camillo De Lellis ; Michael, Westdickenberg. On the optimality of velocity averaging lemmas.

  • Ann. Inst. H. Poincaré Anal. Non Linéaire, 2003.

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 31 / 33

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Optimality for all smooth flux with continuous solutions

joint work with Pierre Castelli

1

space dimension d = 1 For all smooth flux and continuous solutions : u0(x) = xβ cos 1 xγ

  • Pierre Castelli,

Master’s Thesis 2012 and current work.

2

d > 1 Like High frequency waves : construction using the new characterisation of smooth nonlinear flux : slides 20-24

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 32 / 33

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THANK YOU FOR YOUR ATTENTION

Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 33 / 33