Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples Ex.1: The potential energy functional � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.2: The entropy functional Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.2: The entropy functional � L 2 ( ρ ) := R n ρ ( x ) log( ρ ( x )) d x Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.2: The entropy functional � � L 2 ( x , ρ, ∇ ρ ) = ρ log( ρ ) , L 2 ( ρ ) := R n ρ ( x ) log( ρ ( x )) d x , δ L 2 δρ = ∂ ρ L 2 ( ρ ) = log( ρ ) + 1 , Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.2: The entropy functional � � L 2 ( x , ρ, ∇ ρ ) = ρ log( ρ ) , L 2 ( ρ ) := R n ρ ( x ) log( ρ ( x )) d x , δ L 2 δρ = ∂ ρ L 2 ( ρ ) = log( ρ ) + 1 , ∂ t ρ − div ( ρ ∇ (log( ρ ) + 1)) = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 The entropy functional � The heat equation � � L 2 ( x , ρ, ∇ ρ ) = ρ log( ρ ) , L 2 ( ρ ) := R n ρ ( x ) log( ρ ( x )) d x , δ L 2 δρ = ∂ ρ L 2 ( ρ ) = log( ρ ) + 1 , ∂ t ρ − ∆ ρ = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.3: The internal energy functional Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.3: The internal energy functional 1 � m − 1 ρ m ( x ) d x , L 3 ( ρ ) := m � = 1 R n Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.3: The internal energy functional � 1 m − 1 ρ m , � 1 L 3 ( x , ρ, ∇ ρ ) = m − 1 ρ m ( x ) d x , L 3 ( ρ ) := δ L 3 m − 1 ρ m − 1 , m δρ = ∂ ρ L 3 ( ρ ) = R n Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.3: The internal energy functional � 1 m − 1 ρ m , � 1 L 3 ( x , ρ, ∇ ρ ) = m − 1 ρ m ( x ) d x , L 3 ( ρ ) := δ L 3 m − 1 ρ m − 1 , m δρ = ∂ ρ L 3 ( ρ ) = R n ∂ t ρ − div ( ρ ∇ ρ m ) = 0 ρ Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 The internal energy functional � The porous media equation � 1 m − 1 ρ m , � 1 L 3 ( x , ρ, ∇ ρ ) = m − 1 ρ m ( x ) d x , L 3 ( ρ ) := δ L 3 m − 1 ρ m − 1 , m δρ = ∂ ρ L 3 ( ρ ) = R n ∂ t ρ − ∆ ρ m = 0 Otto ’01 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.4: The (Entropy+ Potential) energy functional Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.4: The (Entropy+ Potential) energy functional � L 4 ( ρ ) := R n ( ρ ( x ) log( ρ ( x )) + ρ ( x ) V ( x )) d x , Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.4: The (Entropy+ Potential) energy functional � � L 4 ( x , ρ, ∇ ρ ) = ρ log( ρ ) + ρ V ( x ) , L 4 ( ρ ) := R n ( ρ log( ρ )+ ρ V ) , δ L 4 δρ = ∂ ρ L 4 ( x , ρ ) = log( ρ ) + 1 + V ( x ) , Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Ex.4: The (Entropy+ Potential) energy functional � � L 4 ( x , ρ, ∇ ρ ) = ρ log( ρ ) + ρ V ( x ) , L 4 ( ρ ) := R n ( ρ log( ρ )+ ρ V ) , δ L 4 δρ = ∂ ρ L 4 ( x , ρ ) = log( ρ ) + 1 + V ( x ) , ∂ t ρ − div ( ρ ∇ (log( ρ ) + 1 + V )) = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Examples The potential energy functional � The linear transport equation � L 1 ( x , ρ, ∇ ρ ) = L 1 ( x , ρ ) = ρ V ( x ) , � L 1 ( ρ ) := R n V ( x ) ρ ( x ) d x , δ L 1 δρ = ∂ ρ L 1 ( x , ρ ) = V ( x ) , ∂ t ρ − div ( ρ ∇ V ) = 0 Entropy+Potential � The Fokker-Planck equation � � L 4 ( x , ρ, ∇ ρ ) = ρ log( ρ ) + ρ V ( x ) , L 4 ( ρ ) := R n ( ρ log( ρ )+ ρ V ) , δ L 4 δρ = ∂ ρ L 4 ( x , ρ ) = log( ρ ) + 1 + V ( x ) , ∂ t ρ − ∆ ρ − div ( ρ ∇ V ) = 0 Jordan-Kinderlehrer-Otto ’97 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples Ex.5: The Dirichlet integral Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples Ex.5: The Dirichlet integral L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x 2 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples Ex.5: The Dirichlet integral � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + div ( ρ ∇ ∆ ρ ) = 0 Otto ’98 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + ( ρ ∇ ∆ ρ ) = 0 Otto ’98 Ex.6: The Fisher information Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + ( ρ ∇ ∆ ρ ) = 0 Otto ’98 Ex.6: The Fisher information |∇ ρ ( x ) | 2 L 6 ( ρ ) := 1 d x = 1 � � R n |∇ log( ρ ( x )) | 2 ρ ( x ) d x 2 ρ ( x ) 2 R n Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + ( ρ ∇ ∆ ρ ) = 0 Otto ’98 Ex.6: The Fisher information L 6 ( x , ρ, ∇ ρ ) = |∇ log( ρ ) | 2 ρ, � L 6 ( ρ ) := 1 � |∇ log( ρ ) | 2 ρ δρ = − 2 ∆ √ ρ δ L 6 2 √ ρ Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + ( ρ ∇ ∆ ρ ) = 0 Otto ’98 Ex.6: The Fisher information L 6 ( x , ρ, ∇ ρ ) = |∇ log( ρ ) | 2 ρ, � L 6 ( ρ ) := 1 � |∇ log( ρ ) | 2 ρ δρ = − 2 ∆ √ ρ δ L 6 2 √ ρ � ∆ √ ρ � �� ∂ t ρ + 2 div ρ ∇ = 0 Gianazza-Savar´ √ ρ e-Toscani 2006 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Fourth order examples The Dirichlet integral � The thin film equation � L 5 ( x , ρ, ∇ ρ ) = L 5 ( ρ ) = 1 2 |∇ ρ | 2 , L 5 ( ρ ) := 1 � R n |∇ ρ ( x ) | 2 d x , δ L 5 2 δρ = − ∆ ρ, ∂ t ρ + ( ρ ∇ ∆ ρ ) = 0 Otto ’98 The Fisher information � Quantum drift diffusion equation L 6 ( x , ρ, ∇ ρ ) = |∇ log( ρ ) | 2 ρ, � L 6 ( ρ ) := 1 � |∇ log( ρ ) | 2 ρ δρ = − 2 ∆ √ ρ δ L 6 2 √ ρ � ∆ √ ρ � �� ∂ t ρ + 2 div ρ ∇ = 0 Gianazza-Savar´ √ ρ e-Toscani 2006 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. New insight • This gradient flow approach has brought several developments in: ◮ approximation algorithms ◮ asymptotic behaviour of solutions (new contraction and energy estimates) ([Otto’01]: the porous medium equation) ◮ applications to functional inequalities (Logarithmic Sobolev inequalities ↔ trends to equilibrium a class of diffusive PDEs) ..... [ Agueh, Brenier, Carlen, Carrillo, Dolbeault, Gangbo, Ghoussoub, McCann, Otto, Vazquez, Villani.. ] Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Wasserstein spaces ◮ the space of Borel probability measures on R n with finite second moment � � � µ probability measures on R n : R n | x | 2 d µ ( x ) < + ∞ P 2 ( R n ) = Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Wasserstein spaces ◮ the space of Borel probability measures on R n with finite second moment � � � µ probability measures on R n : R n | x | 2 d µ ( x ) < + ∞ P 2 ( R n ) = ◮ Given µ 1 , µ 2 ∈ P 2 ( R n ), a transport plan between µ 1 and µ 2 is a measure µ ∈ P 2 ( R n × R n ) with marginals µ 1 and µ 2 , i.e. π 1 ♯ µ = µ 1 , π 2 ♯ µ = µ 2 Γ( µ 1 , µ 2 ) is the set of all transport plans between µ 1 and µ 2 . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Wasserstein spaces ◮ the space of Borel probability measures on R n with finite second moment � � � µ probability measures on R n : R n | x | 2 d µ ( x ) < + ∞ P 2 ( R n ) = ◮ Given µ 1 , µ 2 ∈ P 2 ( R n ), a transport plan between µ 1 and µ 2 is a measure µ ∈ P 2 ( R n × R n ) with marginals µ 1 and µ 2 , i.e. π 1 ♯ µ = µ 1 , π 2 ♯ µ = µ 2 Γ( µ 1 , µ 2 ) is the set of all transport plans between µ 1 and µ 2 . ◮ The (squared) Wasserstein distance between µ 1 and µ 2 is �� � R n × R n | x − y | 2 d µ ( x , y ) : µ ∈ Γ( µ 1 , µ 2 ) W 2 2 ( µ 1 , µ 2 ) := min . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Wasserstein spaces Given p ≥ 1 ◮ the space of Borel probability measures on R n with finite p th-moment � � � µ probability measures on R n : R n | x | p d µ ( x ) < + ∞ P p ( R n ) = ◮ Given µ 1 , µ 2 ∈ P p ( R n ), a transport plan between µ 1 and µ 2 is a measure µ ∈ P p ( R n × R n ) with marginals µ 1 and µ 2 , i.e. π 1 ♯ µ = µ 1 , π 2 ♯ µ = µ 2 Γ( µ 1 , µ 2 ) is the set of all transport plans between µ 1 and µ 2 . ◮ The ( p th-power of the) p - Wasserstein distance between µ 1 and µ 2 is �� � R n × R n | x − y | p d µ ( x , y ) : µ ∈ Γ( µ 1 , µ 2 ) W p p ( µ 1 , µ 2 ) := min . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Wasserstein spaces Given p ≥ 1 ◮ the space of Borel probability measures on R n with finite p th-moment � � � µ probability measures on R n : R n | x | p d µ ( x ) < + ∞ P p ( R n ) = ◮ Given µ 1 , µ 2 ∈ P p ( R n ), a transport plan between µ 1 and µ 2 is a measure µ ∈ P p ( R n × R n ) with marginals µ 1 and µ 2 , i.e. π 1 ♯ µ = µ 1 , π 2 ♯ µ = µ 2 Γ( µ 1 , µ 2 ) is the set of all transport plans between µ 1 and µ 2 . ◮ The ( p th-power of the) p - Wasserstein distance between µ 1 and µ 2 is �� � R n × R n | x − y | p d µ ( x , y ) : µ ∈ Γ( µ 1 , µ 2 ) W p p ( µ 1 , µ 2 ) := min . ◮ the Wasserstein distance is tightly related with the Monge-Kantorovich optimal mass transportation problem. Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Towards metric spaces ◮ the metric space ( P p ( R n ) , W p ) is not a Riemannian manifold . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Towards metric spaces ◮ the metric space ( P p ( R n ) , W p ) is not a Riemannian manifold . (In [ Jordan-Kinderlehrer-Otto ’97] Fokker-Planck equation interpreted as a gradient flow by switching to the steepest descent, discrete time formulation ).... Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Towards metric spaces ◮ the metric space ( P p ( R n ) , W p ) is not a Riemannian manifold . (In [ Jordan-Kinderlehrer-Otto ’97] Fokker-Planck equation interpreted as a gradient flow by switching to the steepest descent, discrete time formulation ).... ◮ However, Otto develops formal Riemannian calculus in Wasserstein spaces to provide heuristical proofs of qualitative properties (eg., asymptotic behaviour) of Wasserstein gradient flows Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Towards metric spaces ◮ the metric space ( P p ( R n ) , W p ) is not a Riemannian manifold . (In [ Jordan-Kinderlehrer-Otto ’97] Fokker-Planck equation interpreted as a gradient flow by switching to the steepest descent, discrete time formulation ).... ◮ However, Otto develops formal Riemannian calculus in Wasserstein spaces to provide heuristical proofs of qualitative properties (eg., asymptotic behaviour) of Wasserstein gradient flows ◮ rigorous proofs through technical arguments, based on the “classical” theory and regularization procedures, and depending on the specific case.. Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Towards metric spaces ◮ the metric space ( P p ( R n ) , W p ) is not a Riemannian manifold . (In [ Jordan-Kinderlehrer-Otto ’97] Fokker-Planck equation interpreted as a gradient flow by switching to the steepest descent, discrete time formulation ).... ◮ However, Otto develops formal Riemannian calculus in Wasserstein spaces to provide heuristical proofs of qualitative properties (eg., asymptotic behaviour) of Wasserstein gradient flows ◮ rigorous proofs through technical arguments, based on the “classical” theory and regularization procedures, and depending on the specific case.. Metric spaces are a suitable framework for rigorously interpreting diffusion PDE as gradient flows in the Wasserstein spaces in the full generality . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces In [Gradient flows in metric and in the Wasserstein spaces Ambrosio, Gigli, Savar´ e ’05] : Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces In [Gradient flows in metric and in the Wasserstein spaces Ambrosio, Gigli, Savar´ e ’05] : • refined existence, approximation, uniqueness, long-time behaviour results for general Gradient Flows in Metric Spaces Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces In [Gradient flows in metric and in the Wasserstein spaces Ambrosio, Gigli, Savar´ e ’05] : • refined existence, approximation, uniqueness, long-time behaviour results for general Gradient Flows in Metric Spaces Approach based on the theory of Minimizing Movements & Curves of Maximal Slope [ De Giorgi, Marino, Tosques, Degiovanni, Ambro- sio.. ’80 ∼ ’90] Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces In [Gradient flows in metric and in the Wasserstein spaces Ambrosio, Gigli, Savar´ e ’05] : • refined existence, approximation, uniqueness, long-time behaviour results for general Gradient Flows in Metric Spaces • The applications of these results to gradient flows in Wasserstein spaces are made rigorous through development of a “differential/metric calcu- lus” in Wasserstein spaces: ◮ notion of tangent space and of (sub)differential of a functional on P p ( R n ) ◮ calculus rules ◮ link between the weak formulation of evolution PDEs and their formulation as a gradient flow in P p ( R n ) Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces In [Gradient flows in metric and in the Wasserstein spaces Ambrosio, Gigli, Savar´ e ’05] : • refined existence, approximation, uniqueness, long-time behaviour results for general Gradient Flows in Metric Spaces • In [ R., Savar´ e, Segatti, Stefanelli ’06]: complement the Ambro- sio, Gigli, Savar´ e ’s results on the long-time behaviour of Curves of Maximal Slope Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Data: ◮ A complete metric space ( X , d ), ◮ a proper functional φ : X → ( −∞ , + ∞ ] Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Data: ◮ A complete metric space ( X , d ), ◮ a proper functional φ : X → ( −∞ , + ∞ ] Problem: How to formulate the gradient flow equation “ u ′ ( t ) = −∇ φ ( u ( t ))” , t ∈ (0 , T ) Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Data: ◮ A complete metric space ( X , d ), ◮ a proper functional φ : X → ( −∞ , + ∞ ] Problem: How to formulate the gradient flow equation “ u ′ ( t ) = −∇ φ ( u ( t ))” , t ∈ (0 , T ) in absence of a natural linear or differentiable structure on X ? Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Data: ◮ A complete metric space ( X , d ), ◮ a proper functional φ : X → ( −∞ , + ∞ ] Problem: How to formulate the gradient flow equation “ u ′ ( t ) = −∇ φ ( u ( t ))” , t ∈ (0 , T ) in absence of a natural linear or differentiable structure on X ? To get some insight, let us go back to the euclidean case... Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 So we get the equivalent formulation: Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 So we get the equivalent formulation: dt φ ( u ( t )) = − 1 d 2 | u ′ ( t ) | 2 − 1 2 |∇ φ ( u ( t )) | 2 This involves Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 So we get the equivalent formulation: dt φ ( u ( t )) = − 1 d 2 | u ′ ( t ) | 2 − 1 2 |∇ φ ( u ( t )) | 2 This involves the modulus of derivatives, rather than derivatives, Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 So we get the equivalent formulation: dt φ ( u ( t )) = − 1 d 2 | u ′ ( t ) | 2 − 1 2 |∇ φ ( u ( t )) | 2 This involves the modulus of derivatives, rather than derivatives, hence it can make sense in the setting of a metric space! Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Gradient flows in metric spaces: heuristics Given a proper (differentiable) function φ : R n → ( −∞ , + ∞ ] u ′ ( t ) = −∇ φ ( u ( t )) ⇔ | u ′ ( t ) + ∇ φ ( u ( t )) | 2 = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 � u ′ ( t ) , ∇ φ ( u ( t )) � = 0 ⇔ | u ′ ( t ) | 2 + |∇ φ ( u ( t )) | 2 + 2 d dt φ ( u ( t )) = 0 So we get the equivalent formulation: dt φ ( u ( t )) = − 1 d 2 | u ′ ( t ) | 2 − 1 2 |∇ φ ( u ( t )) | 2 This involves the modulus of derivatives, rather than derivatives, hence it can make sense in the setting of a metric space! We introduce suitable “surrogates” of (the modulus of) derivatives . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Metric derivatives • Setting: A complete metric space ( X , d ) Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Metric derivatives • Setting: A complete metric space ( X , d ) Metric derivative & geodesics Given an absolutely continuous curve u : (0 , T ) → X ( u ∈ AC (0 , T ; X )), its metric derivative is defined by d ( u ( t ) , u ( t + h )) | u ′ | ( t ) := lim for a.e. t ∈ (0 , T ) , | h | h → 0 ( � u ′ ( t ) � � | u ′ | ( t )), Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Metric derivatives • Setting: A complete metric space ( X , d ) Metric derivative & geodesics Given an absolutely continuous curve u : (0 , T ) → X ( u ∈ AC (0 , T ; X )), its metric derivative is defined by d ( u ( t ) , u ( t + h )) | u ′ | ( t ) := lim for a.e. t ∈ (0 , T ) , | h | h → 0 ( � u ′ ( t ) � � | u ′ | ( t )), and satisfies � t | u ′ | ( r ) d r d ( u ( s ) , u ( t )) ≤ ∀ 0 ≤ s ≤ t ≤ T . s Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Metric derivatives • Setting: A complete metric space ( X , d ) Metric derivative & geodesics Given an absolutely continuous curve u : (0 , T ) → X ( u ∈ AC (0 , T ; X )), its metric derivative is defined by d ( u ( t ) , u ( t + h )) | u ′ | ( t ) := lim for a.e. t ∈ (0 , T ) , | h | h → 0 ( � u ′ ( t ) � � | u ′ | ( t )), and satisfies � t | u ′ | ( r ) d r d ( u ( s ) , u ( t )) ≤ ∀ 0 ≤ s ≤ t ≤ T . s A curve u is a (constant speed) geodesic if d ( u ( s ) , u ( t )) = | t − s || u ′ | ∀ s , t ∈ [0 , 1] . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Slopes • Setting: A complete metric space ( X , d ) Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Slopes • Setting: A complete metric space ( X , d ) Local slope Given a proper functional φ : X → ( −∞ , + ∞ ] and u ∈ D ( φ ), the local slope of φ at u is ( φ ( u ) − φ ( v )) + | ∂φ | ( u ) := lim sup u ∈ D ( φ ) d ( u , v ) v → u Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Slopes • Setting: A complete metric space ( X , d ) Local slope Given a proper functional φ : X → ( −∞ , + ∞ ] and u ∈ D ( φ ), the local slope of φ at u is ( φ ( u ) − φ ( v )) + | ∂φ | ( u ) := lim sup u ∈ D ( φ ) d ( u , v ) v → u ( � − ∇ φ ( u ) � � | ∂φ | ( u )). Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Slopes • Setting: A complete metric space ( X , d ) Local slope Given a proper functional φ : X → ( −∞ , + ∞ ] and u ∈ D ( φ ), the local slope of φ at u is ( φ ( u ) − φ ( v )) + | ∂φ | ( u ) := lim sup u ∈ D ( φ ) d ( u , v ) v → u ( � − ∇ φ ( u ) � � | ∂φ | ( u )). To fix ideas Suppose that X is a Banach space B , and φ : B → ( −∞ , + ∞ ] is l.s.c. and convex (or a C 1 -perturbation of a convex functional), with subdifferential (in the sense of Convex Analysis) ∂φ . Then | ∂φ | ( u ) = min {� ξ � B ′ : ξ ∈ ∂φ ( u ) } ∀ u ∈ D ( φ ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Slopes • Setting: A complete metric space ( X , d ) Local slope Given a proper functional φ : X → ( −∞ , + ∞ ] and u ∈ D ( φ ), the local slope of φ at u is ( φ ( u ) − φ ( v )) + | ∂φ | ( u ) := lim sup u ∈ D ( φ ) d ( u , v ) v → u ( � − ∇ φ ( u ) � � | ∂φ | ( u )). Definition: chain rule The local slope satisfies the chain rule if for any absolutely continuous curve v : (0 , T ) → D ( φ ) the map t �→ ( φ ◦ ) v ( t ) is absolutely continuous and satisfies d d t φ ( v ( t )) ≥ −| v ′ | ( t ) | ∂φ | ( v ( t )) for a.e. t ∈ (0 , T ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of Curve of Maximal Slope (w.r.t. the local slope) ( 2 -)Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a ( 2 -)curve of maximal slope for φ (w.r.t. the local slope) if Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of Curve of Maximal Slope (w.r.t. the local slope) ( 2 -)Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a ( 2 -)curve of maximal slope for φ (w.r.t. the local slope) if dt φ ( u ( t )) = − 1 d 2 | u ′ | 2 ( t ) − 1 2 | ∂φ | 2 ( u ( t )) a.e. in (0 , T ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of Curve of Maximal Slope (w.r.t. the local slope) ( 2 -)Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a ( 2 -)curve of maximal slope for φ (w.r.t. the local slope) if dt φ ( u ( t )) = − 1 d 2 | u ′ | 2 ( t ) − 1 2 | ∂φ | 2 ( u ( t )) a.e. in (0 , T ) . • If | ∂φ | satisfies the chain rule, it is sufficient to have dt φ ( u ( t )) ≤ − 1 d 2 | u ′ | 2 ( t ) − 1 2 | ∂φ | 2 ( u ( t )) a.e. in (0 , T ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of p -Curve of Maximal Slope Consider p , q ∈ (1 , + ∞ ) with 1 p + 1 q = 1. Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of p -Curve of Maximal Slope Consider p , q ∈ (1 , + ∞ ) with 1 p + 1 q = 1. p -Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a p -curve of maximal slope for φ if Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of p -Curve of Maximal Slope Consider p , q ∈ (1 , + ∞ ) with 1 p + 1 q = 1. p -Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a p -curve of maximal slope for φ if dt φ ( u ( t )) = − 1 d p | u ′ | p ( t ) − 1 q | ∂φ | q ( u ( t )) a.e. in (0 , T ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Definition of p -Curve of Maximal Slope Consider p , q ∈ (1 , + ∞ ) with 1 p + 1 q = 1. p -Curve of Maximal Slope We say that an absolutely continuous curve u : (0 , T ) → X is a p -curve of maximal slope for φ if dt φ ( u ( t )) = − 1 d p | u ′ | p ( t ) − 1 q | ∂φ | q ( u ( t )) a.e. in (0 , T ) . • If | ∂φ | satisfies the chain rule, it is sufficient to have dt φ ( u ( t )) ≤ − 1 d p | u ′ | p ( t ) − 1 q | ∂φ | q ( u ( t )) a.e. in (0 , T ) . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. To fix ideas... ◮ 2-curves of maximal slope in P 2 ( R n ) lead (for a suitable φ ) to the linear transport equation ∂ t ρ − div ( ρ ∇ V ) = 0 Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. To fix ideas... ◮ 2-curves of maximal slope in P 2 ( R n ) lead (for a suitable φ ) to the linear transport equation ∂ t ρ − div ( ρ ∇ V ) = 0 ◮ p -curves of maximal slope in P p ( R n ) lead (for a suitable φ ) to a nonlinear version of the transport equation ∂ t ρ − ∇ · ( ρ j q ( ∇ V )) = 0 � | r | q − 2 r r � = 0 , j q ( r ) := 1 p + 1 0 r = 0 , q = 1 . Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Approximation of p − curves of maximal slope Given an initial datum u 0 ∈ X , does there exist a p − curve of maximal slope u on (0 , T ) fulfilling u (0) = u 0 ? Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Approximation of p − curves of maximal slope Existence is proved by passing to the limit in an approximation scheme by time discretization Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Approximation of p − curves of maximal slope Existence is proved by passing to the limit in an approximation scheme by time discretization ◮ Fix time step τ > 0 partition P τ of (0 , T ) � Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Approximation of p − curves of maximal slope Existence is proved by passing to the limit in an approximation scheme by time discretization ◮ Fix time step τ > 0 partition P τ of (0 , T ) � ◮ Discrete solutions u 0 τ , u 1 τ , . . . , u N τ : solve recursively τ ∈ Argmin u ∈ X { 1 p τ d p ( u , u n − 1 u 0 u n ) + φ ( u ) } , τ := u 0 τ For simplicity, we take p = 2. Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
Evolution equations & Wasserstein distance Curves of Maximal Slope Existence of a Global Attractor Applications More.. Approximation of p − curves of maximal slope Existence is proved by passing to the limit in an approximation scheme by time discretization ◮ Fix time step τ > 0 partition P τ of (0 , T ) � ◮ Discrete solutions u 0 τ , u 1 τ , . . . , u N τ : solve recursively τ ∈ Argmin u ∈ X { 1 p τ d p ( u , u n − 1 u 0 u n ) + φ ( u ) } , τ := u 0 τ For simplicity, we take p = 2. This variational formulation of the implicit Euler scheme still makes sense in a purely metric framework Riccarda Rossi Long-time behaviour of gradient flows in metric spaces
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