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 ∂ t v + div( v ⊗ v ) + ∇ p = 0 div v = 0 v ( · , 0) = v 0 , where v : R d × R + → R d is the velocity and p : R d × R + → R the pressure. Here v 0 : R d → R d is a given initial divergence-free velocity field. Agnieszka ´ Swierczewska Weak and measure-valued solutions

Weak Formulation loc ( R d × R + ; R d ) is a weak solution with initial We say that v ∈ L 2 data 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 ). 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 � R d | v ( x , t ) | 2 dx ≤ 1 � R d | v 0 ( x ) | 2 dx . 2 2 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 v ǫ 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 � R d | v ( x , t ) | 2 dx ≤ 1 � R d | v 0 ( x ) | 2 dx . 2 2 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 v ǫ 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 � R d | v ( x , t ) | 2 dx ≤ 1 � R d | v 0 ( x ) | 2 dx . 2 2 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 v ǫ 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 � R d | v ( x , t ) | 2 dx ≤ 1 � R d | v 0 ( x ) | 2 dx . 2 2 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 v ǫ 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 � R d | v ( x , t ) | 2 dx ≤ 1 � R d | v 0 ( x ) | 2 dx . 2 2 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 v ǫ 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 v ǫ ⇀ v it does not follow that v ǫ ⊗ v ǫ ⇀ v ⊗ v . Measure-valued solutions (mvs) are designed to capture complex oscillation 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 v ǫ ⇀ v it does not follow that v ǫ ⊗ v ǫ ⇀ v ⊗ v . Measure-valued solutions (mvs) are designed to capture complex oscillation and concentration phenomena and thus to overcome this problem. Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures A (generalized) Young measure on R d with parameters in R d × R + is a triple ( ν x , t , m , ν ∞ x , t ), where ν x , t ∈ P ( R d ) for a.e. ( x , t ) ∈ R d × R + (oscillation measure) m ∈ M + ( R d × R + ) (concentration measure) x , t ∈ P ( S d − 1 ) for m -a.e. ( x , t ) ∈ R d × R + ν ∞ (concentration-angle measure) Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures A (generalized) Young measure on R d with parameters in R d × R + is a triple ( ν x , t , m , ν ∞ x , t ), where ν x , t ∈ P ( R d ) for a.e. ( x , t ) ∈ R d × R + (oscillation measure) m ∈ M + ( R d × R + ) (concentration measure) x , t ∈ P ( S d − 1 ) for m -a.e. ( x , t ) ∈ R d × R + ν ∞ (concentration-angle measure) Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures A (generalized) Young measure on R d with parameters in R d × R + is a triple ( ν x , t , m , ν ∞ x , t ), where ν x , t ∈ P ( R d ) for a.e. ( x , t ) ∈ R d × R + (oscillation measure) m ∈ M + ( R d × R + ) (concentration measure) x , t ∈ P ( S d − 1 ) for m -a.e. ( x , t ) ∈ R d × R + ν ∞ (concentration-angle measure) Agnieszka ´ Swierczewska Weak and measure-valued solutions

Young Measures A (generalized) Young measure on R d with parameters in R d × R + is a triple ( ν x , t , m , ν ∞ x , t ), where ν x , t ∈ P ( R d ) for a.e. ( x , t ) ∈ R d × R + (oscillation measure) m ∈ M + ( R d × R + ) (concentration measure) x , t ∈ P ( S d − 1 ) for m -a.e. ( x , t ) ∈ R d × R + ν ∞ (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 L 2 ( R 2 × 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) ∂ t v + div U + ∇ q = 0 div v = 0 v ( · , 0) = v 0 , with a nonlinear (pointwise) constraint U = v ⊗ v − 1 q = p + 1 n | v | 2 Id , n | v | 2 Id . 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 2 λ max ( v ⊗ v − U ) where λ max 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 v 0 and the energy ¯ e is a pair ( v , U ) which solves the linear system ∂ t v + div U + ∇ q = 0 div v = 0 for some q and such that v ( · , 0) = v 0 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 � � 1 � 2 | v | 2 − ¯ I ( v ) = inf e dx t R n on X is non-positive I ( v ) = 0 iff v is a weak solution to Euler system with initial 2 | v | 2 = ¯ data v 0 and energy density 1 e . If ¯ e is sufficiently large, then the set X is non-empty and has infinite cardinality. X – the closure of X wrt weak L 2 − topology. Indeed, assuming that subsolutions are bounded, the set X is bounded in L 2 and the weak L 2 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 L 2 − 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 v 0 ∈ L 2 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 L 2 − dense subset of initial data Weak-strong uniqueness Agnieszka ´ Swierczewska Weak and measure-valued solutions