non standard solutions in compressible gas dynamics · introduction main result the strategies...

Introduction Main result The strategies Recent achievements Conclusions

Non–standard solutions incompressible gas dynamics

Elisabetta Chiodaroli

EPFLLausanne

Heraklion, September 19th, 2013

Elisabetta Chiodaroli EPFL Lausanne

Non–standard solutions in compressible gas dynamics 1/26

Plan of the talk1 Introduction

The compressible isentropic Euler system of gas dynamicsIll–posedness results

2 Main resultLipschitz initial dataEntropy rate admissibility criterion

3 The strategiesRiemann problemConvex integrationEntropy rate

4 Recent achievementsHeat conducting gasIll-posedness results

5 ConclusionsElisabetta Chiodaroli EPFL Lausanne

The compressible isentropic Euler system of gas dynamics

The Euler system [Euler, 1757]: a paradigm

Compressible Euler system of isentropic gas dynamics in Euleriancoordinates in Rn, n ≥ 2

∂tρ+ divx(ρv) = 0∂t(ρv) + divx (ρv ⊗ v) +∇x [p(ρ)] = 0ρ(·, 0) = ρ0

v(·, 0) = v0 .

Unknowns:

ρ(x , t): density of the gas

v(x , t): velocity of the gas

The pressure p is a given function of ρ s.t. p′ > 0 (hyperbolicity).Typical example: p(ρ) = kργ with k > 0, γ > 1.

v(·, 0) = v0 .

Unknowns:

v(·, 0) = v0 .

Unknowns:

The Euler system: basic definitions I

Definition

A weak solution of (1) on Rn × [0,∞) is a pair of boundedfunctions (ρ, v) such that:

∫ ∫[ρ∂tψ + ρv · ∇xψ] +

∫ρ0(x)ψ(x , 0)dx = 0∫ ∫

[ρv · ∂tφ+ ρv ⊗ v : ∇xφ+ p(ρ) divx φ] +

∫ρ0(x)v 0(x) · φ(x , 0)dx = 0.

for all C∞ functions ψ, φ compactly supported in Rn × [0,∞)

Well–posedness issue

Weak solutions are non-unique

Problems

How to develop a well-posedness theory? In which functionalspace?How to go beyond singularities but restoring uniqueness? How toselect unique weak solutions? =⇒ entropy inequalities?

Well–posedness issue

Weak solutions are non-unique

Problems

How to develop a well-posedness theory? In which functionalspace?How to go beyond singularities but restoring uniqueness? How toselect unique weak solutions? =⇒ entropy inequalities?

The Euler system: basic definitions II

Possible admissibility criteria for singling out unique weaksolutions: ENTROPY INEQUALITIES

Definition

A bounded weak solution (ρ, v) of (1) is an admissible or entropysolution if

(ρε(ρ) +

2ρ |v |2

)+ divx

[(ρε(ρ) +

2ρ |v |2 + p(ρ)

]≤ 0

in the sense of distributions. The internal energy ε is giventhrough p(ρ) = ρ2ε′(ρ).

The Euler system: basic definitions II

Possible admissibility criteria for singling out unique weaksolutions: ENTROPY INEQUALITIES

Definition

A bounded weak solution (ρ, v) of (1) is an admissible or entropysolution if

(ρε(ρ) +

2ρ |v |2

)+ divx

[(ρε(ρ) +

2ρ |v |2 + p(ρ)

]≤ 0

in the sense of distributions. The internal energy ε is giventhrough p(ρ) = ρ2ε′(ρ).

Ill–posedness results

The Euler system: a first striking ill-posedness result

Theorem (De Lellis, Szekelyhidi, 2010)

n ≥ 2. For any pressure law p, there are bounded initial data(ρ0, v0) with ρ0 ≥ c > 0 with infinitely many bounded admissibleweak solutions (ρ, v) of (1) with ρ ≥ c > 0.

Remarks:

proof based on previous work on the incompressible Eulerequations

initial data a fortiori sufficiently irregular due to weak-stronguniqueness: as long as a classical solution exists, any boundedadmissible solution must coincide with it

ill-posedness of entropy solutions in L∞

Remarks:

Non-uniqueness with arbitrary density

Theorem (E.C., 2011)

n ≥ 2. For any pressure law p, for any sufficiently regular initialdensity ρ0 ≥ c > 0 there are bounded initial data v0 allowing forinfinitely many bounded admissible weak solutions (ρ, v) of (1)with ρ ≥ c > 0.

Comments:

result proven in space–periodic setting

method as in [De Lellis, Szekelyhidi, 2010] for incompressibleEuler but partially adapted to compressible world

non-uniqueness due to irregularity of the velocity field

entropy inequality does not select unique weak solution

Comments:

Possible developments

So far, one could still argue that non-uniqueness is due to theirregularity of the initial velocity, rather than to the irregularity ofthe solutions. What happens, for instance, in case of smoothinitial data after the first blow-up time?

Lipschitz initial data

Ill-posedness with Lipschitz initial data

Theorem (E.C., De Lellis, C., Kreml, O., 2013)

Let p(ρ) = ρ2. There exist Lipschitz initial data (ρ0, v0) forwhich there are infinitely many bounded admissible weak solutions(ρ, v) of Euler system (1) on R2 × [0,∞) with inf ρ > 0.

!! ATTENTION !! These solutions are all locally Lipschitz on afinite interval where they all coincide with the unique classicalsolution: non–uniqueness arises after the first blow–up time.→ → → → → → → → → → WEAK–STRONG UNIQUENESS

Ill-posedness with Lipschitz initial data

!! ATTENTION !! These solutions are all locally Lipschitz on afinite interval where they all coincide with the unique classicalsolution: non–uniqueness arises after the first blow–up time.→ → → → → → → → → → WEAK–STRONG UNIQUENESS

Ill-posedness with Lipschitz data: comments

Inspired by a work of Szekelyhidi on incompressible Euler withvortex sheet initial data.

The proof:1 is not completely in the “compressible world”, but exploits

several specific properties of compressible Euler.2 builds directly upon convex integration method of De Lellis &

Szekelyhidi on incompressible Euler equations

QUESTIONS1: Irregularity of the solutions for the ill-posedness property?2: Description of the set of Lipschitz initial data?3: Further admissibility conditions to rule out non-standard sols?

Entropy rate admissibility criterion

The criterion ([Dafermos, 1973])

We define the local total entropy at time t ∈ [0,∞)

H(ρ,v)(t) =

(ρε(ρ) +

2ρ |v |2

Definition

A weak solution (ρ, v) of (1) on Rn × [0,∞) is “entropy rate”admissible if there is no other solution (ρ, v) with the property thatfor some τ ∈ [0,∞), (ρ, v)(x , t) = (ρ, v)(x , t) on R2 × [0, τ ] and

H ′(ρ,v)(τ) < H ′(ρ,v)(τ).

The criterion ([Dafermos, 1973])

We define the local total entropy at time t ∈ [0,∞)

H(ρ,v)(t) =

(ρε(ρ) +

2ρ |v |2

Definition

A weak solution (ρ, v) of (1) on Rn × [0,∞) is “entropy rate”admissible if there is no other solution (ρ, v) with the property thatfor some τ ∈ [0,∞), (ρ, v)(x , t) = (ρ, v)(x , t) on R2 × [0, τ ] and

H ′(ρ,v)(τ) < H ′(ρ,v)(τ).

Good news

Let p(ρ) = ρ2. The non-standard solutions originating fromLipschitz initial data and constructed with convex integrationmethods are not entropy rate admissible.

HOPE: The entropy rate admissibility criterion could single outunique weak solutions.

Good news

Let p(ρ) = ρ2. The non-standard solutions originating fromLipschitz initial data and constructed with convex integrationmethods are not entropy rate admissible.

HOPE: The entropy rate admissibility criterion could single outunique weak solutions.

Lipschitz initial data: ill-posedness

Theorem (E.C., De Lellis, Kreml, 2013)

Key idea: We build our solutions from solutions to a Riemannproblem, i.e. a Cauchy problem with initial data of the specialform:

(ρ0(x), v0(x)) :=

(ρ−, v−) if x2 < 0

(ρ+, v+) if x2 > 0,(2)

where ρ±, v± are constants and x = (x1, x2) ∈ R2

Lipschitz initial data: ill-posedness

Theorem (E.C., De Lellis, Kreml, 2013)

Key idea: We build our solutions from solutions to a Riemannproblem, i.e. a Cauchy problem with initial data of the specialform:

(ρ0(x), v0(x)) :=

(ρ−, v−) if x2 < 0

(ρ+, v+) if x2 > 0,(2)

where ρ±, v± are constants and x = (x1, x2) ∈ R2

Riemann problem

(ρ0(x), v0(x)) :=

(ρ−, v−) if x2 < 0

(ρ+, v+) if x2 > 0,

Step 1: Find (ρ±, v±) such that there is a unique locallyLipschitz self–similar solution backwards in time (compressionwave)

Step 2: With these data (ρ±, v±) find infinitely manyadmissible weak solutions forward in time

Riemann problem

(ρ0(x), v0(x)) :=

(ρ−, v−) if x2 < 0

(ρ+, v+) if x2 > 0,

Step 1: Find (ρ±, v±) such that there is a unique locallyLipschitz self–similar solution backwards in time (compressionwave)

Step 2: With these data (ρ±, v±) find infinitely manyadmissible weak solutions forward in time

Riemann problem

Step 1

We look for solutions independent of x1. Observe that if (ρ, v) is asolution then also

(ρ(x2, t), v(x2, t)) := (ρ(−x2,−t), v(−x2,−t))

is. Moreover, if (ρ, v) is locally Lipschitz and hence satisfies theadmissibility condition with equality, so does (ρ, v).

Therefore we can look for RAREFACTION WAVE forward intime simply by switching (ρ−, v−) and (ρ+, v+). Such solution willhave the form

(ρ, v)(x2, t) = (R,W )(x2

), −∞ < x2 <∞, 0 < t <∞

Riemann problem

Step 1 → Lemma

We have

Let 0 < ρ− < ρ+, v− = (− 1ρ+, 2√

2(√ρ+ −

√ρ−)) and

v+ = (− 1ρ+, 0). Then there is a pair

(ρ, v) ∈W 1,∞loc ∩ L∞(R2 × (−∞, 0),R+ × R2) such that

(i) ρ+ ≥ ρ ≥ ρ− > 0;

(ii) The pair solves the Euler system with p(ρ) = ρ2 in theclassical sense (pointwise a.e. and distributionally);

(iii) for t ↑ 0 the pair (ρ(·, t), v(·, t)) converges pointwise a.e. to(ρ0, v0) as in (3);

(iv) (ρ(·, t), v(·, t)) ∈W 1,∞ for every t < 0.

Riemann problem

Partial summary

We have found Riemann initial data (ρ−, v−), (ρ+, v+) which”produce” on time (−∞, 0) a locally Lipschitz self–similarcompression wave which is a unique classical (and thereforeadmissible) solution to the Euler equations

What remains is to find infinitely many solutions forward intime with the Riemann initial data as in the previous Lemma

We heavily use the tools developed by De Lellis - Szekelyhidi

Key point of the theory is the notion of a subsolution

Convex integration

Subsolution

x2ν+ν−

Subsolution (ρ, v) piecewise constant.

(ρ, v) = (ρ−, v−)1P− + (ρ1, v1)1P1 + (ρ+, v+)1P+

Convex integration

From subsolution to solutions

The upshot is the following

Proposition

Let (ρ±, v±) be such that there exists at least one admissiblesubsolution of the Euler equations with initial data (3). Then thereare infinitely many bounded admissible solutions (ρ, v) to (1)-(3)(forward in time).

Proof: Use convex integration on P1 as developed by De Lellisand Szekelyhidi

Convex integration

From subsolution to solutions

The upshot is the following

Proposition

Let (ρ±, v±) be such that there exists at least one admissiblesubsolution of the Euler equations with initial data (3). Then thereare infinitely many bounded admissible solutions (ρ, v) to (1)-(3)(forward in time).

Proof: Use convex integration on P1 as developed by De Lellisand Szekelyhidi

Entropy rate

Non-standard solutions are not entropy rate admissible

The Riemann data

(ρ0(x), v0(x)) :=

(ρ−, v−) if x2 < 0

(ρ+, v+) if x2 > 0,(3)

with ρ±, v± constants allowing for infinitely many non-standardsolutions (ρ, v)(x1, x2, t) forward in time, admit also a forward intime self-similar solution (ρS , vS)(x2, t) depending only onone-space variable. The result is that

H ′(ρS ,vS ) < H ′(ρ,v).

Heat conducting gas

Full Euler-Fourier system

Full Euler-Fourier system in R3,

∂tρ+ divx(ρv) = 0∂t(ρv) + divx (ρv ⊗ v) +∇x [p(ρ, θ)] = 0∂t(ρe(ρ, θ)) + divx(ρe(ρ, θ)v) + divxq = −p(ρ, θ)divxv .

(4)Unknowns:

θ(x , t): temperature of the gas

Heat conducting gas

(4)Unknowns:

Heat conducting gas

(4)Unknowns:

Heat conducting gas

(4)Unknowns:

Ill-posedness results

Ill-posedness

In case of- PERFECT MONOATOMIC GAS: p(ρ) = ρθ, e(ρ, θ) = 3

2θ,- standard FOURIER LAW: q = −∇xθ

Theorem (E.C., Feireisl, E., Kreml, O., 2013)

For any sufficiently regular initial density, initial temperature andinitial velocity there are infinitely many global weak solutions(ρ, v , θ) of (4).

NOTE: These solutions satisfy also the associated entropyequation (they comply with the Second law of thermodynamics),but they violate the First law of Thermodynamics.

Ill-posedness

Ill-posedness of dissipative solutions

To eliminate non-physical solutions → DISSIPATIVESOLUTIONS → TOTAL ENERGY CONSERVATION:

E (t) =

2|v |2 + e(ρ, θ)

)(t, ·)dx = E (0)

Let T > 0. For any sufficiently regular initial density and initialtemperature there exists a bounded initial velocity such thatthere are infinitely many dissipative solutions (ρ, v , θ) of (4) in(0,T )× R3.

NOTE: Initial velocity has to be irregular due to WEAK-STRONGUNIQUENESS.

E (t) =

2|v |2 + e(ρ, θ)

)(t, ·)dx = E (0)

E (t) =

2|v |2 + e(ρ, θ)

)(t, ·)dx = E (0)

Conclusions

For compressible Euler:Ill-posedness of entropy solutions → non-standardsolutions:

1 in any dimension → for any regular initial density and suitableconstructed initial velocities

2 in 2D and quadratic pressure → even for Lipschitz initial data

Entropy rate admissibility criterion seems to rule outnon-standard solutions

Future perspectives:

study non-isentropic casedescribe set of initial data allowing for ill-posedness ofentropy solutions.understand the general effectiveness of Entropy rateadmissibility criterion.

Conclusions

Thank you for yourattention!

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