On weak and measure-valued solutions to compressible Euler and - - PowerPoint PPT Presentation

On weak and measure-valued solutions to compressible Euler and similar systems

Agnieszka ´ Swierczewska–Gwiazda joint work with Eduard Feireisl (Czech Academy of Sciences), Piotr Gwiazda and Emil Wiedemann (University of Bonn)

Institute of Applied Mathematics and Mechanics University of Warsaw Porquerolles, 14th September 2015

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Excursion to Incompressible Euler Equations

The Cauchy problem for the incompressible Euler equations of inviscid fluid motion reads ∂tv + div(v ⊗ v) + ∇p = 0 div v = 0 v(·, 0) = v0, where v : Rd × R+ → Rd is the velocity and p : Rd × R+ → R the

• pressure. Here v0 : Rd → Rd is a given initial divergence-free

velocity field.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Weak Formulation

We say that v ∈ L2

loc(Rd × R+; Rd) is a weak solution with initial

data v0 ∈ L2(Rd) if ∞

• Rd(v · ∂tφ + v ⊗ v : ∇φ)dxdt +
• Rd v0(x)φ(x, 0)dx = 0

for every φ ∈ C ∞

c (Rd × [0, ∞); Rd) with div φ = 0 and

• Rd v(x, t) · ∇ψ(x)dx = 0

for a.e. t ∈ R+ and every ψ ∈ C ∞

c (Rd).

Agnieszka ´ Swierczewska Weak and measure-valued solutions

The Vanishing Viscosity Problem

We want to consider admissible weak solutions, i.e. solutions satisfying E(t) := 1 2

• Rd |v(x, t)|2dx ≤ 1

2

• Rd |v0(x)|2dx .

What is the idea to show existence? Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 Send ǫ → 0 Show that the corresponding solutions vǫ converge to v, and that v is a weak solution of Euler.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

The Vanishing Viscosity Problem

We want to consider admissible weak solutions, i.e. solutions satisfying E(t) := 1 2

• Rd |v(x, t)|2dx ≤ 1

2

• Rd |v0(x)|2dx .

What is the idea to show existence? Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 Send ǫ → 0 Show that the corresponding solutions vǫ converge to v, and that v is a weak solution of Euler.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

The Vanishing Viscosity Problem

We want to consider admissible weak solutions, i.e. solutions satisfying E(t) := 1 2

• Rd |v(x, t)|2dx ≤ 1

2

• Rd |v0(x)|2dx .

What is the idea to show existence? Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 Send ǫ → 0 Show that the corresponding solutions vǫ converge to v, and that v is a weak solution of Euler.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

The Vanishing Viscosity Problem

We want to consider admissible weak solutions, i.e. solutions satisfying E(t) := 1 2

• Rd |v(x, t)|2dx ≤ 1

2

• Rd |v0(x)|2dx .

What is the idea to show existence? Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 Send ǫ → 0 Show that the corresponding solutions vǫ converge to v, and that v is a weak solution of Euler.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

The Vanishing Viscosity Problem

We want to consider admissible weak solutions, i.e. solutions satisfying E(t) := 1 2

• Rd |v(x, t)|2dx ≤ 1

2

• Rd |v0(x)|2dx .

What is the idea to show existence? Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 Send ǫ → 0 Show that the corresponding solutions vǫ converge to v, and that v is a weak solution of Euler.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Oscillations and Concentrations

What happens? The problem is that we can not pass to the limit in the nonlinearity due to conceivable oscillation and concentration effects. More precisely, from vǫ ⇀ v it does not follow that vǫ ⊗ vǫ ⇀ v ⊗ v. Measure-valued solutions (mvs) are designed to capture complex

• scillation and concentration phenomena and thus to overcome

this problem.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Oscillations and Concentrations

What happens? The problem is that we can not pass to the limit in the nonlinearity due to conceivable oscillation and concentration effects. More precisely, from vǫ ⇀ v it does not follow that vǫ ⊗ vǫ ⇀ v ⊗ v. Measure-valued solutions (mvs) are designed to capture complex

• scillation and concentration phenomena and thus to overcome

this problem.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures

A (generalized) Young measure on Rd with parameters in Rd × R+ is a triple (νx,t, m, ν∞

x,t), where

νx,t ∈ P(Rd) for a.e. (x, t) ∈ Rd × R+ (oscillation measure) m ∈ M+(Rd × R+) (concentration measure) ν∞

x,t ∈ P(Sd−1) for m-a.e. (x, t) ∈ Rd × R+

(concentration-angle measure)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures

A (generalized) Young measure on Rd with parameters in Rd × R+ is a triple (νx,t, m, ν∞

x,t), where

νx,t ∈ P(Rd) for a.e. (x, t) ∈ Rd × R+ (oscillation measure) m ∈ M+(Rd × R+) (concentration measure) ν∞

x,t ∈ P(Sd−1) for m-a.e. (x, t) ∈ Rd × R+

(concentration-angle measure)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures

A (generalized) Young measure on Rd with parameters in Rd × R+ is a triple (νx,t, m, ν∞

x,t), where

νx,t ∈ P(Rd) for a.e. (x, t) ∈ Rd × R+ (oscillation measure) m ∈ M+(Rd × R+) (concentration measure) ν∞

x,t ∈ P(Sd−1) for m-a.e. (x, t) ∈ Rd × R+

(concentration-angle measure)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures

A (generalized) Young measure on Rd with parameters in Rd × R+ is a triple (νx,t, m, ν∞

x,t), where

νx,t ∈ P(Rd) for a.e. (x, t) ∈ Rd × R+ (oscillation measure) m ∈ M+(Rd × R+) (concentration measure) ν∞

x,t ∈ P(Sd−1) for m-a.e. (x, t) ∈ Rd × R+

(concentration-angle measure)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Weak solutions

Overview of the recent results Scheffer ’93, Shnirelman ’97 constructed examples of weak solutions in L2(R2 × R) compactly supported in space and time De Lellis and Sz´ ekelyhidi 2010 showed that weak solutions need not be unique

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Different formulation of the Euler system

Consider a linear system (highly underdetermined) ∂tv + div U + ∇q = 0 div v = 0 v(·, 0) = v0, with a nonlinear (pointwise) constraint U = v ⊗ v − 1 n|v|2Id, q = p + 1 n|v|2Id. so that U is trace-free symmetric matrix.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Energy

Kinetic energy density for an Euler system is given by e(v) = 1 2|v|2. Given v, U satisfying the above problem for some pressure q one introduces the generalized energy density e(v, U) = n 2λmax(v ⊗ v − U) where λmax is the largest eigenvalue.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Introducing the set of subsolutions

If ¯ e is a given energy density, a subsolution corresponding to initial data v0 and the energy ¯ e is a pair (v, U) which solves the linear system ∂tv + div U + ∇q = 0 div v = 0 for some q and such that v(·, 0) = v0 and e(v(x, t), U(x, t)) ≤ ¯ e for almost all x, t

Agnieszka ´ Swierczewska Weak and measure-valued solutions

X - the set of velocity fields which can be complemented by some U to become a subsolution. The functional I(v) = inf

t

• Rn

1 2|v|2 − ¯ e

• dx
• n X is non-positive

I(v) = 0 iff v is a weak solution to Euler system with initial data v0 and energy density 1

2|v|2 = ¯

• e. If ¯

e is sufficiently large, then the set X is non-empty and has infinite cardinality. X – the closure of X wrt weak L2−topology. Indeed, assuming that subsolutions are bounded, the set X is bounded in L2 and the weak L2 topology is metrizable on X and X becomes a complete metric space.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Baire category argument

I is a lower semi-continuous functional with respect to the weak L2−topology hence by virtue of Baire category argument the set of points of continuity of I on X is residual (its complement in X is of first Baire category) and hence has infinite cardinality. Finally one shows that I[v] = 0 whenever v is a point of continuity of I

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Short summary of these results (incompressible Euler)

De Lellis, Sz´ ekelyhidi 2010 showed that weak solutions need not to be unique Wiedemann 2011: for any initial data v0 ∈ L2 there exists infinitely many weak solutions Sz´ ekelyhidi, Wiedemann 2012: if we require the energy to be non-increasing, then a global existence and non-uniqueness result is known for an L2−dense subset of initial data Weak-strong uniqueness

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Original 1-D Savage-Hutter Model ’89

Find the height h : R × R+ − → R the velocity v : R × R+ − → R satisfying the system of conservation laws ∂th + ∂x(hv) = ∂t(hv) + ∂x

• hv2 + βh2

? hg, (SH)

• x

v(t,x) h(t,x)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Here β(x), g(x, v) are defined by β(x) = k cos ξ(x), g(x, v) = sinξ(x) − sign0(v)cos ξ(x) where ξ(x)− inclination angle of bottom topography at point x. The function g is not continuous at v = 0. Therefore extension to a multivalued sign(v) : sign(v) =      −1 for v < 0 [−1, 1] for v = 0 1 for v > 0

Agnieszka ´ Swierczewska Weak and measure-valued solutions

• x

Agnieszka ´ Swierczewska Weak and measure-valued solutions

• x

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Physical motivation (Gwiazda ’02, Hadeler & Kuttler ’03, Bouchut & Westdickenberg ’04)

Consider static problem (v = 0) with ξ = 0 ⇒ Governing equation: khx ∈ [−1, 1] ⇒ h is Lipschitz with Lipschitz constant ≤

1 k

Physically relevant solution!

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Two-dimensional case

Agnieszka ´ Swierczewska Weak and measure-valued solutions

2d System

∂th + div(hu) = 0, ∂t(hu) + div(hu ⊗ u) + ∇(ah2) = h

• −γ u

|u| + f

• ,

where Ω =

• [0, 1]|{0,1}

2 . initial conditions h(0, ·) = h0, u(0, ·) = u0. The term

u |u| has to be understood as a multi-valued mapping,

which for non-zero velocities takes the value

u |u|, whereas for u = 0

takes the values in the whole closed unit ball.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Weak solutions

A weak solution is a triple [h, u, Bu], whereas Bu is the selection from the multi-valued graph satisfying T

• Ω

(h∂tϕ + hu · ∇ϕ) dx dt = −

• Ω

h0ϕ(0, ·) dx dt for any ϕ ∈ C 1([0, T) × Ω); T

• Ω
• hu · ∂tΦ + hu ⊗ u : ∇Φ + ah2 div Φ
• dx dt

= T

• Ω

h (γB − f) · Φ dx dt −

• Ω

h0u0 · Φ(0, ·) dx for any Φ ∈ C 1([0, T) × Ω; R2), where B = Bu(t, x) =     

u(t,x) |u(t,x)| if u(t, x) = 0,

∈ B1(0) if u(t, x) = 0,

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Existence of weak solutions

Theorem (Feireisl, Gwiazda, ´ S-G 2015) Let T > 0 and the initial data h0, u0 be given. Suppose that f ∈ C 1([0, T] × Ω; R2). Then the problem (S-H) admits infinitely many weak solutions in (0, T) × Ω. The weak solutions belong to the class h, ∂th, ∇h ∈ C 1([0, T] × Ω), u ∈ Cweak([0, T]; L2(Ω; R2)) ∩ L∞((0, T) × Ω; R2). div u ∈ C([0, T] × Ω), B ∈ L∞((0, T) × Ω; R2)

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Application of the method of convex integration to S-H system

∂th + div(hu) = 0, ∂t(hu) + div(hu ⊗ u) + ∇(ah2) = h

• −γ u

|u| + f

• ,

h0 ∈ C 2(Ω), u0 ∈ C 2(Ω; R2), h0 > 0 in Ω. using the standard Helmholtz decomposition, we may write h0u0 = v0+V0+∇Ψ0, div v0 = 0,

• Ω

Ψ0 dx = 0,

• Ω

v0 dx = 0, We look for solutions in the form hu = v + V + ∇Ψ, where div v = 0,

• Ω

Ψ(t, ·) = 0,

• Ω

v(t, ·) = 0, V = V(t) ∈ R2.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Continuity equation

∂th + ∆xΨ = 0 in (0, T) × Ω, h(0, ·) = h0, Ψ(0, ·) = Ψ0. We can choose h = h(t, x) ∈ C 2([0, T] × Ω) such that h(0, ·) = h0, ∂th(0, ·) = −∆xΨ0, h(t, ·) > 0,

• Ω

h(t, ·) dx =

• Ω

h0 dx for all t ∈ [0, T], and compute −∆xΨ(t, ·) = ∂th(t, ·),

• Ω

Ψ(t, ·) dx = 0.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Consequently, the original problem reduces to finding the functions v, V satisfying (weakly) ∂tv+∂tV+div (v + V + ∇Ψ) ⊗ (v + V + ∇Ψ) h +

• ah2 + ∂tΨ
• I
• = h
• −γ v + V + ∇Ψ

|v + V + ∇Ψ| + f

• ,

div v = 0,

• Ω

v(t, ·) dx = 0, v(0, ·) = v0, V(0) = V0.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Kinetic energy

We denote E = 1 2 |v + V + ∇Ψ|2 h the kinetic energy density associated with the Savage-Hutter system. Analogously, we rewrite the system in the form ∂tv+∂tV+div (v + V + ∇Ψ) ⊗ (v + V + ∇Ψ) h − 1 2 |v + V + ∇Ψ|2 h I

• +∇
• E − Λ + ah2 + ∂tΨ
• = −γ

h 2E 1/2 (v + V + ∇Ψ) + hf, where Λ = Λ(t) is a spatially homogeneous function to be determined below. Finally, for E = Λ − ah2 − ∂tΨ, the above equation reduces.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Kinetic energy

We denote E = 1 2 |v + V + ∇Ψ|2 h the kinetic energy density associated with the Savage-Hutter system. Analogously, we rewrite the system in the form ∂tv+∂tV+div (v + V + ∇Ψ) ⊗ (v + V + ∇Ψ) h − 1 2 |v + V + ∇Ψ|2 h I

• = −γ

h 2E 1/2 (v + V + ∇Ψ) + hf, where Λ = Λ(t) is a spatially homogeneous function to be determined below. Finally, for E = Λ − ah2 − ∂tΨ, the above equation reduces.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Determine V as the unique solution of the ODE ∂tV −

• 1

|Ω|

• Ω

γ h 2E 1/2 dx

• V

= 1 |Ω|

• Ω
• γ

h 2E 1/2 (v + ∇Ψ) + hf

• dx, V(0) = V0.

Finally, we find a tensor M = M[v] such that M(t, x) ∈ R2×2

sym,0, for any t, x, and

div M = −γ h 2E 1/2 (v + V[v] + ∇Ψ) + 1 |Ω|

• Ω

γ h 2E 1/2 (v + V[v] + ∇Ψ) dx +hf − 1 |Ω|

• Ω

hf dx,

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Finally we write S-H system in a concise form

∂tv+div (v + V[v] + ∇Ψ) ⊗ (v + V[v] + ∇Ψ) h − 1 2 |v + V[v] + ∇Ψ|2 h I − M M M[v]

• = 0

divv = 0

and proceed with the method of convex integration.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Subsolutions

X0 - the set of velocity fields such that ∂tw + div F = 0 in (0, T) × Ω for some F λmax (w + V[w] + ∇Ψ) ⊗ (w + V[w] + ∇Ψ) h − F − M[w]

• < E−δ

in (0, T) × Ω for some δ > 0, where E is the kinetic energy introduced before.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Key steps

1 Recall that

E = Λ − ah2 − ∂tΨ. The first observation is that the set X0 is non-empty provided Λ(t) ≥ Λ0 > 0 in [0, T]

2 Take the closure X 0 of the set of subsolutions X0. 3 Show that X0 has infinite cardinality.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

1 The first observation is that the set X0 is non-empty provided

Λ(t) ≥ Λ0 > 0 in [0, T] and Λ0 is large enough. Here “large enough” means in terms

• f the initial data, f, and the time T.

λmax (v0 + V[v0] + ∇Ψ) ⊗ (v0 + V[v0] + ∇Ψ) h − M[v0]

• < E−δ

= Λ − ah2 − ∂tΨ − δ.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Infinitely many solutions

We introduce the functional I[v] = T

• Ω

1 2 |v + V[v] + ∇Ψ|2 h − E

• dx dt

I : X 0 → (−∞, 0] is a lower semi-continuous functional with respect to the topology of the space Cweak([0, T]; L2(Ω; R2)). Consequently, by virtue of Baires category argument, the set of points of continuity of I in X 0 has infinite cardinality.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Lemma Let U ⊂ R × RN, N = 2, 3 be a bounded open set. Suppose that g ∈ C(U; RN), W ∈ C(U; RN×N

sym,0), e, r ∈ C(U), r > 0, e ≤ e in U

are given such that N 2 λmax g ⊗ g r − W

• < e in U.

Then there exist sequences wn ∈ C ∞

c (U; RN), Gn ∈ C ∞ c (U; RN×N sym,0), n = 0, 1, . . .

such that wn → 0 weakly in L2(U; RN) and ∂twn + div Wn = 0, div wn = 0 in RN, N 2 λmax (g + wn) ⊗ (g + wn) r − (W + Gn)

• < e in U,

Agnieszka ´ Swierczewska Weak and measure-valued solutions

admissible solutions

The constructed solutions satisfy 1 2h|u|2 = E = Λ − ah2 − ∂tΨ for a.a. (t, x) ∈ (0, T) × Ω. In particular, as Λ has been chosen large, the total energy Etot of the flow, Etot(t) =

• Ω

1 2h|u|2 + ah2

• (t, ·) dx

may (and does in “most” cases) experience a jump at the initial time, lim inf

t→0+ Etot(t) >

• Ω

1 2h0|u0|2 + ah2

• dx.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

admissible solutions

Definition We say that [h, u, Bu] is a admissible weak solution to the Savage-Hutter system if in addition the energy inequality holds for a.a. τ ∈ (0, T). Etot(τ) ≡

• Ω

1 2h|u|2 + ah2

• (τ, ·) dx +

τ

• Ω

hγBu · u dx dt ≤

• Ω

1 2h0|u0|2 + ah2

• dx +

τ

• Ω

hf · u dx dt,

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Weak-strong uniqueness

Theorem Let [h, u, Bu] be a admissible weak solution of the Savage-Hutter system in (0, T) × Ω. Let [H, U, BU], H > 0 be a globally Lipschitz (strong) solution of the same problem, with h0 = H(0, ·), u0 = U(0, ·). Then h = H, u = U a.e. in (0, T) × Ω.

Agnieszka ´ Swierczewska Weak and measure-valued solutions

admissible solutions

Theorem Under the hypotheses of previous theorem, let T > 0, and h0 ∈ C 2(Ω), h0 > 0, be given. Then there exists u0 ∈ L∞(Ω; R2) such that the Savage-Hutter system admits infinitely many admissible weak solutions in (0, T) starting from the initial data [h0; u0].

Agnieszka ´ Swierczewska Weak and measure-valued solutions

References for the results for S-H model, 2-d case

1 existence of measure-valued solutions: Gwiazda 2005,

Asymptotic Analysis

2 weak-strong uniqueness for measure valued solutions:

Gwiazda, ´ S-G., Wiedemann, 2015, to appear in Nonlinearity

3 existence of weak solutions: Feireisl, Gwiazda, ´

S-G., 2015, to appear in Comm. PDE

Agnieszka ´ Swierczewska Weak and measure-valued solutions

Thank you for your attention

Agnieszka ´ Swierczewska Weak and measure-valued solutions