Solving the initial value problem for Euler and Burgers equations by convex optimization

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Solving the initial value problem for Euler and Burgers equations by convex optimization

Yann Brenier

CNRS, DMA, Ecole Normale Supérieure, 45 rue d’Ulm, FR-75005 Paris in association with CNRS-INRIA "MOKAPLAN" team

COLOCVIUL FRANCO-ROMAN DE MATEMATICI APLICATE Bordeaux, IMB, 27-31/08/2018

Yann Brenier (CNRS, DMA-ENS) IVP for Euler by convex optimization Bordeaux, 29/08/2018 1 / 20

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OUR GOAL

We want to solve the initial value problem for a large class of equations including the Euler equations of fluid mechanics through a concave maximization problem, similar to the ones considered in Control Theory (or, for instance, Mean Field Games).

Reference: Y.B. ArXiv Oct. 2017, to appear in CMP

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OUR GOAL

We want to solve the initial value problem for a large class of equations including the Euler equations of fluid mechanics through a concave maximization problem, similar to the ones considered in Control Theory (or, for instance, Mean Field Games).

Reference: Y.B. ArXiv Oct. 2017, to appear in CMP

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Euler 1755/57: The first PDEs ever written (at least in modern style)!

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The class of "entropic conservation laws"

∂_tU +∇ ·(F(U)) = 0, U = U(t,x) ∈ R^m, x ∈ R^d enjoying an additional conservation law

∂_t(E(U)) +∇ ·(Q(U)) =0,

for all smooth solutions U, where E is strictly convex.

A typical example: the equations of an isothermal gas (Euler, 1755)

∂tρ+∇ ·q=0, ∂tq+∇ ·(q⊗q

ρ ) +∇ρ=0, E= |q|²

2ρ +ρlogρ, ρ >0, q∈R^d. Typically, such systems are locally well-posed, with generic formation of shock waves.

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∂_t(E(U)) +∇ ·(Q(U)) =0,

∂tρ+∇ ·q=0, ∂tq+∇ ·(q⊗q

ρ ) +∇ρ=0, E= |q|²

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∂_t(E(U)) +∇ ·(Q(U)) =0,

∂tρ+∇ ·q=0, ∂tq+∇ ·(q⊗q

ρ ) +∇ρ=0, E= |q|²

2ρ +ρlogρ, ρ >0, q∈R^d.

Typically, such systems are locally well-posed, with generic formation of shock waves.

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∂_t(E(U)) +∇ ·(Q(U)) =0,

∂tρ+∇ ·q=0, ∂tq+∇ ·(q⊗q

ρ ) +∇ρ=0, E= |q|²

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0 0.05 0.1 0.15 0.2 0.25

0 0.2 0.4 0.6 0.8 1

’fort.10’

Inviscid Burgers equation :∂_tu+∂_x(u²/2) =0,u=u(t,x),x ∈R/Z,t ≥0.

Formation of two shock waves. (Vertical axis:t ∈[0,1/4], horizontal axis:x ∈T.)

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The least square approach?

GivenU₀ on T^d = R^d/Z^d and T > 0, if F(U) is linear inU, the least square method obviously leads to a (degenerate) convex problem

U(t=0,·)=Uinf ₀ Z

[0,T]×T^d

|∂_tU +∇ ·(F(U))|²

but this is no longer true for nonlinear systems.

We need a different idea!

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[0,T]×T^d

|∂_tU +∇ ·(F(U))|² but this is no longer true for nonlinear systems.

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[0,T]×T^d

|∂_tU +∇ ·(F(U))|² but this is no longer true for nonlinear systems.

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An entropy minimization approach

We want to minimize the entropy among all weak solutions U of the initial value problem:

infU

Z

[0,T]×T^d

E(U), U = U(t,x) ∈ R^m subject to

Z

[0,T]×T^d

∂_tA ·U +∇A ·F(U) + Z

T^d

A(0,·)·U₀ = 0

for all smoothA=A(t,x)∈R^m withA(T,·) =0. A priori, by "convex integration" à la De Lellis-Székelyhidi, there are many non entropy-preserving weak solutions attached toU₀.

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infU

Z

[0,T]×T^d

Z

[0,T]×T^d

∂_tA ·U +∇A ·F(U) + Z

T^d

A(0,·)·U₀ = 0

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infU

Z

[0,T]×T^d

Z

[0,T]×T^d

∂_tA ·U +∇A ·F(U) + Z

T^d

A(0,·)·U₀ = 0

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infU

Z

[0,T]×T^d

Z

[0,T]×T^d

∂_tA ·U +∇A·F(U) + Z

T^d

A(0,·)·U₀ = 0

for all smoothA=A(t,x)∈R^mwithA(T,·) =0.

A priori, by "convex integration" à la De Lellis-Székelyhidi, there are many non entropy-preserving weak solutions attached toU₀.

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infU

Z

[0,T]×T^d

Z

[0,T]×T^d

∂_tA ·U +∇A·F(U) + Z

T^d

A(0,·)·U₀ = 0

for all smoothA=A(t,x)∈R^mwithA(T,·) =0. A priori, by "convex integration" à la De Lellis-Székelyhidi, there are many non entropy-preserving weak solutions attached toU₀.

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The resulting saddle-point problem

infU sup

A

Z

[0,T]×T^d

E(U)−∂_tA·U − ∇A·F(U)

− Z

T^d

A(0,·)·U₀

whereA = A(t,x) ∈ R^m is smooth with A(T,·) =0. HereU₀ is the initial condition and T the final time.

N.B. The supremum inA exactly encodes thatU is a weak solution with initial condition U₀, all test functions A acting like Lagrange multipliers.

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The resulting saddle-point problem infU sup

A

Z

[0,T]×T^d

E(U)−∂_tA·U − ∇A ·F(U)

− Z

T^d

A(0,·)·U₀

whereA = A(t,x) ∈ R^m is smooth with A(T,·) =0.

HereU0 is the initial condition and T the final time.

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The resulting saddle-point problem infU sup

A

Z

[0,T]×T^d

E(U)−∂_tA·U − ∇A ·F(U)

− Z

T^d

A(0,·)·U₀

whereA = A(t,x) ∈ R^m is smooth with A(T,·) =0.

HereU0 is the initial condition and T the final time.

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Reversing infimum and supremum...

leads to a concave maximization problem inA, namely sup

A

infU

Z

[0,T]×T^d

E(U)−∂_tA·U− ∇A·F(U)− Z

T^d

A(0,·)·U₀

= sup

A

Z

[0,T]×T^d

K(∂_tA,∇A)− Z

T^d

A(0,·)·U₀.

K(E,B) = inf

U∈R^m

E(U)−E ·U−B ·F(U).

Notice that K is automatically concave. However they might be a duality gap with possibly inf sup > sup inf.

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A

infU

Z

[0,T]×T^d

E(U)−∂_tA·U− ∇A·F(U)− Z

T^d

A(0,·)·U₀

= sup

A

Z

[0,T]×T^d

K(∂_tA,∇A)− Z

T^d

A(0,·)·U₀.

K(E,B) = inf

U∈R^m

E(U)−E ·U−B ·F(U).

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A

infU

Z

[0,T]×T^d

E(U)−∂_tA·U− ∇A·F(U)− Z

T^d

A(0,·)·U₀

= sup

A

Z

[0,T]×T^d

K(∂_tA,∇A)− Z

T^d

A(0,·)·U0.

K(E,B) = inf

U∈R^m

E(U)−E ·U −B ·F(U).

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A

infU

Z

[0,T]×T^d

E(U)−∂_tA·U− ∇A·F(U)− Z

T^d

A(0,·)·U₀

= sup

A

Z

[0,T]×T^d

K(∂_tA,∇A)− Z

T^d

A(0,·)·U0.

K(E,B) = inf

U∈R^m

E(U)−E ·U −B ·F(U).

Notice that K is automatically concave.

However they might be a duality gap with possibly inf sup > sup inf.

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A

infU

Z

[0,T]×T^d

E(U)−∂_tA·U− ∇A·F(U)− Z

T^d

A(0,·)·U₀

= sup

A

Z

[0,T]×T^d

K(∂_tA,∇A)− Z

T^d

A(0,·)·U0.

K(E,B) = inf

U∈R^m

E(U)−E ·U −B ·F(U).

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The elementary example of the Burgers equation

Then, the maximization problem in A simply reads sup

A

Z

[0,T]×T

− (∂_tA)² 2(1−∂_xA) −

Z

T

A(0,·)u₀.

with A = A(t,x) ∈ R subject toA(T,·) = 0, ∂_xA ≤ 1. Introducing ρ = 1−∂_xA ≥ 0,q = ∂_tA ∈ R, we get sup{

Z

[0,T]×T

−q²

2ρ + qu₀, s.t. ∂_tρ+∂_xq = 0, ρ(T,·) = 1} looking very similar to an optimal transport problem!

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A

Z

[0,T]×T

− (∂_tA)² 2(1−∂_xA) −

Z

T

A(0,·)u₀.

with A = A(t,x) ∈ R subject toA(T,·) = 0, ∂_xA ≤ 1.

Introducing ρ = 1−∂_xA ≥ 0,q = ∂_tA ∈ R, we get sup{

Z

[0,T]×T

−q²

2ρ + qu₀, s.t. ∂_tρ+∂_xq = 0, ρ(T,·) = 1} looking very similar to an optimal transport problem!

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A

Z

[0,T]×T

− (∂_tA)² 2(1−∂_xA) −

Z

T

A(0,·)u₀.

Introducing ρ = 1−∂_xA ≥0, q = ∂_tA ∈ R, we get sup{

Z

[0,T]×T

−q²

2ρ + qu₀, s.t. ∂_tρ+∂_xq = 0, ρ(T,·) = 1}

looking very similar to an optimal transport problem!

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A

Z

[0,T]×T

− (∂_tA)² 2(1−∂_xA) −

Z

T

A(0,·)u₀.

Introducing ρ = 1−∂_xA ≥0, q = ∂_tA ∈ R, we get sup{

Z

[0,T]×T

−q²

2ρ + qu₀, s.t. ∂_tρ+∂_xq = 0, ρ(T,·) = 1}

looking very similar to an optimal transport problem!

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Example 2: the isothermal Euler equations

∂_tρ+ ∇ ·q = 0, ∂_tq +∇ ·(q⊗q

ρ ) +∇ρ = 0. We maximize in u = u(t,x) ∈ R, Q = Q(t,x) ∈ R^d, M = M(t,x) =M^t(t,x) ∈ R^d×d, M ≥ 0,

Z

[0,T]×D

−exp(u)exp(1

2Q·M⁻¹ ·Q)− Z

D

σ₀ρ₀ −w₀ ·q₀,

u = ∂_tσ +∂ⁱw_i, Q_i = ∂_tw_i +∂_iσ, M_ij = δ_ij −∂_iw_j −∂_jw_i, whereσ and w must vanish at t = T.

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∂_tρ+ ∇ ·q = 0, ∂_tq +∇ ·(q⊗q

ρ ) +∇ρ = 0.

We maximize in u = u(t,x) ∈ R, Q = Q(t,x) ∈ R^d, M = M(t,x) =M^t(t,x) ∈ R^d×d, M ≥ 0,

Z

[0,T]×D

−exp(u)exp(1

2Q·M⁻¹ ·Q)− Z

D

σ₀ρ₀ −w₀ ·q₀,

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∂_tρ+ ∇ ·q = 0, ∂_tq +∇ ·(q⊗q

ρ ) +∇ρ = 0.

Z

[0,T]×D

−exp(u)exp(1

2Q·M⁻¹ ·Q)− Z

D

σ₀ρ₀ −w₀ ·q₀,

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∂_tρ+ ∇ ·q = 0, ∂_tq +∇ ·(q⊗q

ρ ) +∇ρ = 0.

Z

[0,T]×D

−exp(u)exp(1

2Q·M⁻¹ ·Q)− Z

D

σ₀ρ₀ −w₀ ·q₀,

u = ∂_tσ +∂ⁱw_i, Q_i = ∂_tw_i +∂_iσ, M_ij = δ_ij −∂_iw_j −∂_jw_i, whereσ and w must vanish att = T.

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Example 3:

the Euler equations of incompressible fluids

In this limit case, the resulting problem reads sup

(M,Q)

− Z

[0,T]×D

q₀ ·Q + 1

2 Q ·M⁻¹ ·Q,

where, again, Q is a vector field and M = M^t ≥ 0 is a field of semi-definite symmetric matrices, subject to M_ij(T,·) =δ_ij, ∂_tM_ij = ∂_jQ_i +∂_iQ_j +2∂_i∂_j(−4)⁻¹∂_kQ^k.

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Example 3:

the Euler equations of incompressible fluids

In this limit case, the resulting problem reads sup

(M,Q)

− Z

[0,T]×D

q₀ ·Q + 1

2 Q ·M⁻¹ ·Q,

where, again, Q is a vector field and M = M^t ≥ 0 is a field of semi-definite symmetric matrices, subject to M_ij(T,·) =δ_ij, ∂_tM_ij = ∂_jQ_i +∂_iQ_j +2∂_i∂_j(−4)⁻¹∂_kQ^k.

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Main results (ArXiv Oct. 2017, to appear in CMP)

Theorem 1: If U is a smooth solution to the Cauchy problem and T is not too large(*), then U can be recovered from the concave maximization problem which admits A(t,x) = (t −T)E⁰(U(t,x)) as solution. Theorem 2: For the Burgers equation, all entropy solutions can be recovered, for arbitrarily large T. (*) more precisely if