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TRANSCRIPT
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
Introduction Main result The strategies Recent achievements Conclusions
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
Non–standard solutions in compressible gas dynamics 2/26
Introduction Main result The strategies Recent achievements Conclusions
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 .
(1)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 3/26
Introduction Main result The strategies Recent achievements Conclusions
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 .
(1)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 3/26
Introduction Main result The strategies Recent achievements Conclusions
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 .
(1)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 3/26
Introduction Main result The strategies Recent achievements Conclusions
The compressible isentropic Euler system of gas dynamics
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,∞)
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 4/26
Introduction Main result The strategies Recent achievements Conclusions
The compressible isentropic Euler system of gas dynamics
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 5/26
Introduction Main result The strategies Recent achievements Conclusions
The compressible isentropic Euler system of gas dynamics
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 5/26
Introduction Main result The strategies Recent achievements Conclusions
The compressible isentropic Euler system of gas dynamics
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
∂t
(ρε(ρ) +
1
2ρ |v |2
)+ divx
[(ρε(ρ) +
1
2ρ |v |2 + p(ρ)
)v
]≤ 0
in the sense of distributions. The internal energy ε is giventhrough p(ρ) = ρ2ε′(ρ).
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 6/26
Introduction Main result The strategies Recent achievements Conclusions
The compressible isentropic Euler system of gas dynamics
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
∂t
(ρε(ρ) +
1
2ρ |v |2
)+ divx
[(ρε(ρ) +
1
2ρ |v |2 + p(ρ)
)v
]≤ 0
in the sense of distributions. The internal energy ε is giventhrough p(ρ) = ρ2ε′(ρ).
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 6/26
Introduction Main result The strategies Recent achievements Conclusions
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∞
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 7/26
Introduction Main result The strategies Recent achievements Conclusions
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∞
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 7/26
Introduction Main result The strategies Recent achievements Conclusions
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∞
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 7/26
Introduction Main result The strategies Recent achievements Conclusions
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∞
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 7/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 8/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 8/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 8/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 8/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 8/26
Introduction Main result The strategies Recent achievements Conclusions
Ill–posedness results
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 9/26
Introduction Main result The strategies Recent achievements Conclusions
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 10/26
Introduction Main result The strategies Recent achievements Conclusions
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 10/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 11/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 11/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 11/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 11/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data
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?
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 11/26
Introduction Main result The strategies Recent achievements Conclusions
Entropy rate admissibility criterion
The criterion ([Dafermos, 1973])
We define the local total entropy at time t ∈ [0,∞)
H(ρ,v)(t) =
∫K
(ρε(ρ) +
1
2ρ |v |2
)dx
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).
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 12/26
Introduction Main result The strategies Recent achievements Conclusions
Entropy rate admissibility criterion
The criterion ([Dafermos, 1973])
We define the local total entropy at time t ∈ [0,∞)
H(ρ,v)(t) =
∫K
(ρε(ρ) +
1
2ρ |v |2
)dx
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).
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 12/26
Introduction Main result The strategies Recent achievements Conclusions
Entropy rate admissibility criterion
Good news
Theorem (E.C., De Lellis, C., Kreml, O., 2013)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 13/26
Introduction Main result The strategies Recent achievements Conclusions
Entropy rate admissibility criterion
Good news
Theorem (E.C., De Lellis, C., Kreml, O., 2013)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 13/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data: ill-posedness
Theorem (E.C., De Lellis, Kreml, 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.
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 14/26
Introduction Main result The strategies Recent achievements Conclusions
Lipschitz initial data: ill-posedness
Theorem (E.C., De Lellis, Kreml, 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.
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 14/26
Introduction Main result The strategies Recent achievements Conclusions
Riemann problem
Plan
(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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 15/26
Introduction Main result The strategies Recent achievements Conclusions
Riemann problem
Plan
(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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 15/26
Introduction Main result The strategies Recent achievements Conclusions
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
t
), −∞ < x2 <∞, 0 < t <∞
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 16/26
Introduction Main result The strategies Recent achievements Conclusions
Riemann problem
Step 1 → Lemma
We have
Lemma
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 17/26
Introduction Main result The strategies Recent achievements Conclusions
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 18/26
Introduction Main result The strategies Recent achievements Conclusions
Convex integration
Subsolution
x2ν+ν−
1
P1
P+
P−
t
Subsolution (ρ, v) piecewise constant.
(ρ, v) = (ρ−, v−)1P− + (ρ1, v1)1P1 + (ρ+, v+)1P+
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 19/26
Introduction Main result The strategies Recent achievements Conclusions
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 20/26
Introduction Main result The strategies Recent achievements Conclusions
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
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 20/26
Introduction Main result The strategies Recent achievements Conclusions
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).
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 21/26
Introduction Main result The strategies Recent achievements Conclusions
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:
ρ(x , t): density of the gas
v(x , t): velocity of the gas
θ(x , t): temperature of the gas
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 22/26
Introduction Main result The strategies Recent achievements Conclusions
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:
ρ(x , t): density of the gas
v(x , t): velocity of the gas
θ(x , t): temperature of the gas
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 22/26
Introduction Main result The strategies Recent achievements Conclusions
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:
ρ(x , t): density of the gas
v(x , t): velocity of the gas
θ(x , t): temperature of the gas
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 22/26
Introduction Main result The strategies Recent achievements Conclusions
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:
ρ(x , t): density of the gas
v(x , t): velocity of the gas
θ(x , t): temperature of the gas
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 22/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 23/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 23/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 23/26
Introduction Main result The strategies Recent achievements Conclusions
Ill-posedness results
Ill-posedness of dissipative solutions
To eliminate non-physical solutions → DISSIPATIVESOLUTIONS → TOTAL ENERGY CONSERVATION:
E (t) =
∫R3
ρ
(1
2|v |2 + e(ρ, θ)
)(t, ·)dx = E (0)
Theorem (E.C., Feireisl, E., Kreml, O., 2013)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 24/26
Introduction Main result The strategies Recent achievements Conclusions
Ill-posedness results
Ill-posedness of dissipative solutions
To eliminate non-physical solutions → DISSIPATIVESOLUTIONS → TOTAL ENERGY CONSERVATION:
E (t) =
∫R3
ρ
(1
2|v |2 + e(ρ, θ)
)(t, ·)dx = E (0)
Theorem (E.C., Feireisl, E., Kreml, O., 2013)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 24/26
Introduction Main result The strategies Recent achievements Conclusions
Ill-posedness results
Ill-posedness of dissipative solutions
To eliminate non-physical solutions → DISSIPATIVESOLUTIONS → TOTAL ENERGY CONSERVATION:
E (t) =
∫R3
ρ
(1
2|v |2 + e(ρ, θ)
)(t, ·)dx = E (0)
Theorem (E.C., Feireisl, E., Kreml, O., 2013)
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 24/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 25/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 25/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 25/26
Introduction Main result The strategies Recent achievements Conclusions
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.
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 25/26
Introduction Main result The strategies Recent achievements Conclusions
Thank you for yourattention!
Elisabetta Chiodaroli EPFL Lausanne
Non–standard solutions in compressible gas dynamics 26/26