Weak and Measure-Valued Solutions of the Incompressible Euler - PowerPoint PPT Presentation

Weak and Measure-Valued Solutions for Euler 1 / 20 Weak and Measure-Valued Solutions of the Incompressible Euler Equations Emil Wiedemann (joint work with L´ aszl´ o Sz´ ekelyhidi Jr.) Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 2 / 20 Outline 1 Weak Solutions 2 Measure-Valued Solutions Young Measures Measure-Valued Solutions for Euler Admissibility 3 The Relationship between Weak and Measure-Valued Solutions The Result Ingredients of the Proof 4 Outlook Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 2 / 20 Outline 1 Weak Solutions 2 Measure-Valued Solutions Young Measures Measure-Valued Solutions for Euler Admissibility 3 The Relationship between Weak and Measure-Valued Solutions The Result Ingredients of the Proof 4 Outlook Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 2 / 20 Outline 1 Weak Solutions 2 Measure-Valued Solutions Young Measures Measure-Valued Solutions for Euler Admissibility 3 The Relationship between Weak and Measure-Valued Solutions The Result Ingredients of the Proof 4 Outlook Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 2 / 20 Outline 1 Weak Solutions 2 Measure-Valued Solutions Young Measures Measure-Valued Solutions for Euler Admissibility 3 The Relationship between Weak and Measure-Valued Solutions The Result Ingredients of the Proof 4 Outlook Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 3 / 20 Incompressible Euler Equations The Cauchy problem for the incompressible Euler equations of inviscid fluid motion reads ∂ t v + div( v ⊗ v ) + ∇ p = 0 div v = 0 v ( · , 0) = v 0 where v : R d × R + → R d and p : R d × R + → R are sought for and v 0 : R d → R d is a given initial velocity field with div v 0 = 0. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 4 / 20 Weak Formulation loc ( R d × R + ; R d ) is a weak solution with initial data We say that v ∈ L 2 v 0 ∈ L 2 ( R d ) if � ∞ � � R d ( v · ∂ t φ + v ⊗ v : ∇ φ ) dxdt + R d v 0 ( x ) φ ( x , 0) dx = 0 0 c ( R d × [0 , ∞ ); R d ) with div φ = 0 and for every φ ∈ C ∞ � R d v ( x , t ) · ∇ ψ ( x ) dx = 0 for a.e. t ∈ R + and every ψ ∈ C ∞ c ( R d ). Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 5 / 20 Motivation for Measure-Valued Solutions Vanishing viscosity method: • Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 • Send ǫ → 0 • Show that the corresponding solutions v ǫ converge to v , and that v is a weak solution of Euler. This approach fails! The problem is that we can not pass to the limit in the nonlinearity due to conceivable oscillation and concentration effects. Measure-valued solutions (mvs) are designed to capture complex oscillation and concentration phenomena and thus to overcome this problem. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 5 / 20 Motivation for Measure-Valued Solutions Vanishing viscosity method: • Solve the Cauchy problem for Navier-Stokes with viscosity ǫ > 0 • Send ǫ → 0 • Show that the corresponding solutions v ǫ converge to v , and that v is a weak solution of Euler. This approach fails! The problem is that we can not pass to the limit in the nonlinearity due to conceivable oscillation and concentration effects. Measure-valued solutions (mvs) are designed to capture complex oscillation and concentration phenomena and thus to overcome this problem. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 6 / 20 Young Measures A (generalised) Young measure on R d with parameters in R m is a triple ( ν x , λ, ν ∞ x ), where • ν x ∈ P ( R d ) for a.e. x ∈ R m (oscillation measure) • λ ∈ M + ( R m ) (concentration measure) ∈ P ( S d − 1 ) for λ -a.e. x ∈ R m (concentration-angle measure) • ν ∞ x Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 6 / 20 Young Measures A (generalised) Young measure on R d with parameters in R m is a triple ( ν x , λ, ν ∞ x ), where • ν x ∈ P ( R d ) for a.e. x ∈ R m (oscillation measure) • λ ∈ M + ( R m ) (concentration measure) ∈ P ( S d − 1 ) for λ -a.e. x ∈ R m (concentration-angle measure) • ν ∞ x Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 6 / 20 Young Measures A (generalised) Young measure on R d with parameters in R m is a triple ( ν x , λ, ν ∞ x ), where • ν x ∈ P ( R d ) for a.e. x ∈ R m (oscillation measure) • λ ∈ M + ( R m ) (concentration measure) ∈ P ( S d − 1 ) for λ -a.e. x ∈ R m (concentration-angle measure) • ν ∞ x Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 6 / 20 Young Measures A (generalised) Young measure on R d with parameters in R m is a triple ( ν x , λ, ν ∞ x ), where • ν x ∈ P ( R d ) for a.e. x ∈ R m (oscillation measure) • λ ∈ M + ( R m ) (concentration measure) ∈ P ( S d − 1 ) for λ -a.e. x ∈ R m (concentration-angle measure) • ν ∞ x Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 7 / 20 Generation Let ( v n ) be a sequence of maps R m → R d which is bounded in L 2 ( R m ). We say that ( v n ) generates the Young measure ( ν x , λ, ν ∞ x ) if �� � �� � ∗ S d − 1 f ∞ ( θ ) d ν ∞ f ( v n ) dx ⇀ R d f ( z ) d ν x ( z ) dx + x ( θ ) λ in the sense of measures for every suitable f : R d → R . Here, f ( s θ ) f ∞ ( θ ) = lim s 2 s →∞ is the recession function of f . Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 8 / 20 The Fundamental Theorem Fundamental Theorem of Young Measures (DiPerna-Majda ’87, Alibert-Bouchitt´ e ’97) If ( v n ) is a bounded sequence in L 2 ( R m ; R d ) , then there exists a subsequence which generates some Young measure ( ν x , λ, ν ∞ x ) , i.e. �� � �� � ∗ S d − 1 f ∞ ( θ ) d ν ∞ f ( v n ) dx R d f ( z ) d ν x ( z ) dx + x ( θ ) ⇀ λ. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 9 / 20 Basic Examples ( m = d = 1) Example 1. (Oscillation) � if x ∈ [ k , k + 1 +1 2 ) , v ( x ) = if x ∈ [ k + 1 − 1 2 , k + 1) and v n ( x ) = v ( nx ) . Then clearly ⇀ 1 2 f (+1) + 1 � f ( v n ) ∗ 2 f ( − 1) = f ( z ) d ν ( z ) R with ν = 1 2 δ +1 + 1 2 δ − 1 . Moreover λ = 0. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 10 / 20 Example 2. (Concentration) v n ( x ) = √ n χ [ − 1 2 n ] . 2 n ; 1 Then ν x = δ 0 for a.e. x , λ = δ 0 , ν ∞ 0 = δ +1 . Example 3. (Concentration in various directions) v n ( x ) = √ n � � χ [ − 1 2 n ;0 ] − χ [ 0; 1 . 2 n ] 0 = 1 2 δ +1 + 1 Then ν x = δ 0 for a.e. x , λ = δ 0 , ν ∞ 2 δ − 1 . Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 10 / 20 Example 2. (Concentration) v n ( x ) = √ n χ [ − 1 2 n ] . 2 n ; 1 Then ν x = δ 0 for a.e. x , λ = δ 0 , ν ∞ 0 = δ +1 . Example 3. (Concentration in various directions) v n ( x ) = √ n � � χ [ − 1 2 n ;0 ] − χ [ 0; 1 . 2 n ] 0 = 1 2 δ +1 + 1 Then ν x = δ 0 for a.e. x , λ = δ 0 , ν ∞ 2 δ − 1 . Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 11 / 20 Example 4. (Diffuse concentration) n − 1 v n ( x ) = √ n � χ � � . k n ; k n + 1 n 2 k =0 Then ν x = δ 0 for a.e. x , λ = χ [0;1] dx , ν ∞ = δ +1 for a.e. x ∈ [0; 1]. x Example 5. (Strong convergence) If v n → v strongly in L 2 , then the v n generate the Young measure ν x = δ v ( x ) , λ = 0. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 11 / 20 Example 4. (Diffuse concentration) n − 1 v n ( x ) = √ n � χ � � . k n ; k n + 1 n 2 k =0 Then ν x = δ 0 for a.e. x , λ = χ [0;1] dx , ν ∞ = δ +1 for a.e. x ∈ [0; 1]. x Example 5. (Strong convergence) If v n → v strongly in L 2 , then the v n generate the Young measure ν x = δ v ( x ) , λ = 0. Emil Wiedemann Universit¨ at Bonn

Weak and Measure-Valued Solutions for Euler 12 / 20 Measure-Valued Solutions for Euler Let now, as before, ( v ǫ ) be a sequence of weak (Hopf-Leray) solutions for Navier-Stokes with ǫ → 0 and v ǫ ( t = 0) = v 0 . Since � R d | v ǫ ( x , t ) | 2 dx < ∞ , sup sup ǫ> 0 t ≥ 0 we can apply the Fundamental Theorem of Young measures to obtain ( ν x , t , λ, ν ∞ x , t ) such that � � ν x , t , z ⊗ z � + � ν ∞ � ∂ t � ν x , t , z � + div x , t , θ ⊗ θ � λ + ∇ p ( x , t ) = 0 div � ν x , t , z � = 0 � in the sense of distributions. Here, we wrote � ν, f ( z ) � := f ( z ) d ν ( z ) and similarly for ν ∞ . Emil Wiedemann Universit¨ at Bonn