Flows of vector fields: existence and (non)uniqueness results Maria - PowerPoint PPT Presentation

Flows of vector fields: existence and (non)uniqueness results Maria Colombo EPFL SB, Institute of Mathematics 2020 Fields Medal Symposium October 19 - 23, 2020

Incipit - Particles of clouds Flows of vector fields Maria Colombo We want to describe the motion of some particles of clouds in a windy Flows and day. continuity equation Smooth vs nonsmooth theory Cauchy-Lipschitz thm Lack of uniqueness The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Incipit - Evolution of a single particle Flows of vector We model the clouds as a gas/fluid with given velocity v ( x ) for each fields Maria Colombo position x (direction and intensity). A single particle is transported along an integral curve of v Flows and continuity equation d Smooth vs dt γ ( t ) = v ( γ ( t )) for any t ∈ [0 , ∞ ) . nonsmooth theory Cauchy-Lipschitz thm If we consider many particles at the same time, each of them will Lack of uniqueness follow its own curve. The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Incipit - Evolution of a single particle Flows of vector We model the clouds as a gas/fluid with given velocity v ( x ) for each fields Maria Colombo position x (direction and intensity). A single particle is transported along an integral curve of v Flows and continuity equation d Smooth vs dt γ ( t ) = v ( γ ( t )) for any t ∈ [0 , ∞ ) . nonsmooth theory Cauchy-Lipschitz thm If we consider many particles at the same time, each of them will Lack of uniqueness follow its own curve. The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Incipit - Evolution of a distribution of particles Flows of vector fields If the particles are many, we model them as a distribution, namely Maria Colombo with a measure µ 0 . The distribution of pollutant µ t evolves according Flows and continuity to the PDE equation ∂ t µ t + v · ∇ µ t = 0 . Smooth vs nonsmooth theory Cauchy-Lipschitz thm Lack of uniqueness The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Incipit - Evolution of a distribution of particles Flows of vector fields If the particles are many, we model them as a distribution, namely Maria Colombo with a measure µ 0 . The distribution of pollutant µ t evolves according Flows and continuity to the PDE equation ∂ t µ t + v · ∇ µ t = 0 . Smooth vs nonsmooth theory Cauchy-Lipschitz thm Lack of uniqueness The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Incipit - Evolution of a distribution of particles Flows of vector fields If the particles are many, we model them as a distribution, namely Maria Colombo with a measure µ 0 . The distribution of pollutant µ t evolves according Flows and continuity to the PDE equation ∂ t µ t + v · ∇ µ t = 0 . Smooth vs nonsmooth theory Cauchy-Lipschitz thm Lack of uniqueness The nonsmooth theory: RLF A.e. uniqueness of integral curves Ideas Ambrosio’s superposition principle Interpolation Ill-posedness of CE by convex integration

Table of contents Flows of vector fields Maria Colombo Flow of vector fields and continuity equation 1 Flows and continuity equation Smooth vs nonsmooth theory 2 Smooth vs nonsmooth The Cauchy-Lipschitz theorem for smooth vector fields theory Lack of uniqueness of the flow for nonsmooth vector fields Cauchy-Lipschitz thm Regular Lagrangian Flows and the nonsmooth theory Lack of uniqueness The nonsmooth theory: RLF A.e. uniqueness A.e. uniqueness of integral curves 3 of integral curves Ideas Ambrosio’s Ideas of the proof 4 superposition principle Ambrosio’s superposition principle Interpolation Ill-posedness of CE Interpolation by convex integration Ill-posedness of CE by convex integration

The flow of a vector field Flows of vector fields Maria Colombo Flows and continuity equation Given a vector field b : [0 , ∞ ) × R d → R d , consider the flow X of b Smooth vs nonsmooth � theory d Cauchy-Lipschitz dt X ( t , x ) = b t ( X ( t , x )) ∀ t ∈ [0 , ∞ ) thm Lack of uniqueness X (0 , x ) = x . The nonsmooth theory: RLF A.e. uniqueness It can be seen of integral curves Ideas as a collection of trajectories X ( · , x ) labelled by x ∈ R d ; Ambrosio’s superposition as a collection of diffeomorphisms X ( t , · ) : R d → R d . principle Interpolation Ill-posedness of CE by convex integration

The flow of a vector field Flows of vector fields Maria Colombo Flows and continuity equation Given a vector field b : [0 , ∞ ) × R d → R d , consider the flow X of b Smooth vs nonsmooth � theory d Cauchy-Lipschitz dt X ( t , x ) = b t ( X ( t , x )) ∀ t ∈ [0 , ∞ ) thm Lack of uniqueness X (0 , x ) = x . The nonsmooth theory: RLF A.e. uniqueness It can be seen of integral curves Ideas as a collection of trajectories X ( · , x ) labelled by x ∈ R d ; Ambrosio’s superposition as a collection of diffeomorphisms X ( t , · ) : R d → R d . principle Interpolation Ill-posedness of CE by convex integration

Continuity/transport equation Flows of vector fields Consider the related PDE, named continuity equation Maria Colombo � Flows and continuity in (0 , ∞ ) × R d ∂ t µ t + div ( b t µ t ) = 0 equation µ 0 given . Smooth vs nonsmooth theory When b t is sufficiently smooth and µ t : R d × [0 , ∞ ) → R is a smooth Cauchy-Lipschitz thm Lack of uniqueness function, all derivatives can be computed. The nonsmooth theory: RLF Much less is needed to give a distributional sense to the PDE (e.g. b t A.e. uniqueness bounded and µ t finite measures). of integral curves When Ideas Ambrosio’s div b t ≡ 0 , superposition principle Interpolation the continuity equation is equivalent to the transport equation Ill-posedness of CE by convex integration ∂ t µ t + b · ∇ µ t = 0 .

Continuity/transport equation Flows of vector fields Consider the related PDE, named continuity equation Maria Colombo � Flows and continuity in (0 , ∞ ) × R d ∂ t µ t + div ( b t µ t ) = 0 equation µ 0 given . Smooth vs nonsmooth theory When b t is sufficiently smooth and µ t : R d × [0 , ∞ ) → R is a smooth Cauchy-Lipschitz thm Lack of uniqueness function, all derivatives can be computed. The nonsmooth theory: RLF Much less is needed to give a distributional sense to the PDE (e.g. b t A.e. uniqueness bounded and µ t finite measures). of integral curves When Ideas Ambrosio’s div b t ≡ 0 , superposition principle Interpolation the continuity equation is equivalent to the transport equation Ill-posedness of CE by convex integration ∂ t µ t + b · ∇ µ t = 0 .

Continuity/transport equation Flows of vector fields Consider the related PDE, named continuity equation Maria Colombo � Flows and continuity in (0 , ∞ ) × R d ∂ t µ t + div ( b t µ t ) = 0 equation µ 0 given . Smooth vs nonsmooth theory When b t is sufficiently smooth and µ t : R d × [0 , ∞ ) → R is a smooth Cauchy-Lipschitz thm Lack of uniqueness function, all derivatives can be computed. The nonsmooth theory: RLF Much less is needed to give a distributional sense to the PDE (e.g. b t A.e. uniqueness bounded and µ t finite measures). of integral curves When Ideas Ambrosio’s div b t ≡ 0 , superposition principle Interpolation the continuity equation is equivalent to the transport equation Ill-posedness of CE by convex integration ∂ t µ t + b · ∇ µ t = 0 .

Connection between continuity equation and flows Flows of vector fields Solutions of the CE flow along integral curves of b � R d � Maria Colombo Given b , its flow X an initial distribution of mass µ 0 ∈ P , a Flows and solution of the CE is continuity equation µ t := X ( t , · ) # µ 0 . Smooth vs nonsmooth theory Recall that the measure X ( t , · ) # µ 0 is defined by Cauchy-Lipschitz thm � � Lack of uniqueness ∀ ϕ : R d → R . The nonsmooth R d ϕ ( x ) d [ X ( t , · ) # µ 0 ]( x ) = R d ϕ ( X ( t , x )) d µ 0 ( x ) theory: RLF A.e. uniqueness of integral curves Indeed, for any test function ϕ ∈ C ∞ c ( R d ) we have Ideas Ambrosio’s superposition � � � principle d R d ϕ d µ t = d Interpolation R d ϕ ( X ( t , x )) d µ 0 ( x ) = R d ∇ ϕ ( X ) · ∂ t X d µ 0 Ill-posedness of CE dt dt by convex integration � � = R d ∇ ϕ ( X ) · b t ( X ) d µ 0 = R d ∇ ϕ · b t d µ t . This is the distributional formulation of the continuity equation.