Dissipative dynamical systems and their attractors

Grzegorz Šukaszewicz, University of Warsaw

MIM Qolloquium, 05.11.2020 Plan of the talk

Context: conservative and dissipative systems

3 Basic notions

10 Important problems of the theory.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 2 / 14 Newtonian mechanics ∼ conservative system of ODEs in Rn system reversible in time ∼ groups ∼ deterministic chaos From P. S. de Laplace to H. Poincaré, and ...

Evolution of a conservative system

Example 1. Motivation: Is our solar system stable? (physical system)

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 3 / 14 Evolution of a conservative system

Example 1. Motivation: Is our solar system stable? (physical system)

Newtonian mechanics ∼ conservative system of ODEs in Rn system reversible in time ∼ groups ∼ deterministic chaos From P. S. de Laplace to H. Poincaré, and ...

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 3 / 14 Mech. of continuous media ∼ dissipative system of PDEs in a Hilbert phase space system irreversible in time ∼ semigroups ∼ innite dimensional dynamical systems ∼ deterministic chaos Since O. Ladyzhenskaya's papers on the NSEs (∼ 1970)

Evolution of a dissipative system

Example 2. Motivation: How does turbulence in uids develop?

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 4 / 14 Evolution of a dissipative system

Example 2. Motivation: How does turbulence in uids develop?

Mech. of continuous media ∼ dissipative system of PDEs in a Hilbert phase space system irreversible in time ∼ semigroups ∼ innite dimensional dynamical systems ∼ deterministic chaos Since O. Ladyzhenskaya's papers on the NSEs (∼ 1970)

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 4 / 14 Let us compare the above problems

Example 1. Solar system (physical system)

Newtonian mechanics ∼ conservative system of ODEs in Rn system reversible in time ∼ groups ∼ deterministic chaos From P. S. de Laplace to H. Poincaré, and ...

Example 2. Turbulence in uids (physical system)

Mech. of continuous media ∼ dissipative system of PDEs in a Hilbert phase space system irreversible in time ∼ semigroups ∼ innite dimensional dynamical systems ∼ deterministic chaos Since O. Ladyzhenskaya's papers on the NSEs (∼ 1970)

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 5 / 14 Clarifying example: (k > 0)

du 0 −kt 0 dt = −ku, u( ) = u0 ∈ H → u(t) = e u0 = S(t)u0, t >

→ global attractor A = {0}. Application in hydrodynamics:

Navier-Stokes eq. → Properties of {S(t)}→A∼ Time asymptotics of the uid ow

(Physics → Modelling → Qualitative theory of dierential eqs. → →Topological dynamics → Phenomenology → Physics)

Structure of the theory (simplications for this talk)

Autonomous dierential eq. → Semigroup → Global attractor

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 6 / 14 Application in hydrodynamics:

Navier-Stokes eq. → Properties of {S(t)}→A∼ Time asymptotics of the uid ow

(Physics → Modelling → Qualitative theory of dierential eqs. → →Topological dynamics → Phenomenology → Physics)

Structure of the theory (simplications for this talk)

Autonomous dierential eq. → Semigroup → Global attractor

Clarifying example: (k > 0)

du 0 −kt 0 dt = −ku, u( ) = u0 ∈ H → u(t) = e u0 = S(t)u0, t >

→ global attractor A = {0}.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 6 / 14 (Physics → Modelling → Qualitative theory of dierential eqs. → →Topological dynamics → Phenomenology → Physics)

Structure of the theory (simplications for this talk)

Autonomous dierential eq. → Semigroup → Global attractor

Clarifying example: (k > 0)

du 0 −kt 0 dt = −ku, u( ) = u0 ∈ H → u(t) = e u0 = S(t)u0, t >

→ global attractor A = {0}. Application in hydrodynamics:

Navier-Stokes eq. → Properties of {S(t)}→A∼ Time asymptotics of the uid ow

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 6 / 14 Structure of the theory (simplications for this talk)

Autonomous dierential eq. → Semigroup → Global attractor

Clarifying example: (k > 0)

du 0 −kt 0 dt = −ku, u( ) = u0 ∈ H → u(t) = e u0 = S(t)u0, t >

→ global attractor A = {0}. Application in hydrodynamics:

Navier-Stokes eq. → Properties of {S(t)}→A∼ Time asymptotics of the uid ow

(Physics → Modelling → Qualitative theory of dierential eqs. → →Topological dynamics → Phenomenology → Physics)

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 6 / 14 Dissipative system, time asymptotics, global attractor

DISSIPATIVENESS ∼ existence of a universal absorbing set B0.

B0 is open and bounded.

GLOBAL ATTRACTOR A (if it exists!) is a subset of the absorbing set B0.

A is invariant, compact, and attracts all bounded sets of the phase space H.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 7 / 14 The simplest example (trivial dynamics)

Example 3. For k > 0,

du −kt 0 dt = −ku, u(t) = e u0 = S(t)u0, A = { }, B0 = O(A).

For k ≤ 0 the system is not dissipative (for k = 0 is conservative). Here {S(t): t ∈ R} is a group, S(0) = Id, S(t + s) = S(t)S(s).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 8 / 14 More interesting structure of the global attractor

Example 4.

du 1 0 1 dt = |u|( − u), u(t) = S(t)u0, A = [ , ], B0 = O(A).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 9 / 14 Basic properties of the global attractor

A is maximal bounded invariant subset of H. A is minimal closed attracting subset of H. A is connected. If dim H = ∞ then A has empty interior.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 10 / 14 Existence of a unique global in time solution for every u0 ∈ H, (to dene the semigroup u(t) = S(t)u0, t ≥ 0).

Existence of the global attractor A. (It depends solely on the properties of {S(t)}).

Properties of the semigroup equivalent to the existence of the global attractor. (It depends on the phase space H).

Structure of A.

Some problems of the theory of dissipative dierential eqs.

Example 5. Context: Study of turbulent uid ows - Navier-Stokes equation as a dynamical system. We are interested in the evolution of the uid velocity eld.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 11 / 14 Existence of the global attractor A. (It depends solely on the properties of {S(t)}).

Properties of the semigroup equivalent to the existence of the global attractor. (It depends on the phase space H).

Structure of A.

Some problems of the theory of dissipative dierential eqs.

Example 5. Context: Study of turbulent uid ows - Navier-Stokes equation as a dynamical system. We are interested in the evolution of the uid velocity eld.

Existence of a unique global in time solution for every u0 ∈ H, (to dene the semigroup u(t) = S(t)u0, t ≥ 0).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 11 / 14 Properties of the semigroup equivalent to the existence of the global attractor. (It depends on the phase space H).

Structure of A.

Some problems of the theory of dissipative dierential eqs.

Example 5. Context: Study of turbulent uid ows - Navier-Stokes equation as a dynamical system. We are interested in the evolution of the uid velocity eld.

Existence of a unique global in time solution for every u0 ∈ H, (to dene the semigroup u(t) = S(t)u0, t ≥ 0).

Existence of the global attractor A. (It depends solely on the properties of {S(t)}).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 11 / 14 Structure of A.

Some problems of the theory of dissipative dierential eqs.

Example 5. Context: Study of turbulent uid ows - Navier-Stokes equation as a dynamical system. We are interested in the evolution of the uid velocity eld.

Existence of a unique global in time solution for every u0 ∈ H, (to dene the semigroup u(t) = S(t)u0, t ≥ 0).

Existence of the global attractor A. (It depends solely on the properties of {S(t)}).

Properties of the semigroup equivalent to the existence of the global attractor. (It depends on the phase space H).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 11 / 14 Some problems of the theory of dissipative dierential eqs.

Example 5. Context: Study of turbulent uid ows - Navier-Stokes equation as a dynamical system. We are interested in the evolution of the uid velocity eld.

Existence of a unique global in time solution for every u0 ∈ H, (to dene the semigroup u(t) = S(t)u0, t ≥ 0).

Existence of the global attractor A. (It depends solely on the properties of {S(t)}).

Properties of the semigroup equivalent to the existence of the global attractor. (It depends on the phase space H).

Structure of A.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 11 / 14 Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

Existence of invariant measures on A (related to properties of statistical solutions).

Does the semigroup become a group on A ? (reversibility on the global attractor?)

How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

How the dimension of A is related to boundary conditions? (related to testing the model equations).

Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 Existence of invariant measures on A (related to properties of statistical solutions).

Does the semigroup become a group on A ? (reversibility on the global attractor?)

How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

How the dimension of A is related to boundary conditions? (related to testing the model equations).

Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 Does the semigroup become a group on A ? (reversibility on the global attractor?)

How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

How the dimension of A is related to boundary conditions? (related to testing the model equations).

Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

Existence of invariant measures on A (related to properties of statistical solutions).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

How the dimension of A is related to boundary conditions? (related to testing the model equations).

Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

Existence of invariant measures on A (related to properties of statistical solutions).

Does the semigroup become a group on A ? (reversibility on the global attractor?)

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 How the dimension of A is related to boundary conditions? (related to testing the model equations).

Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

Existence of invariant measures on A (related to properties of statistical solutions).

Does the semigroup become a group on A ? (reversibility on the global attractor?)

How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 Some problems of the theory of dissipative dierential eqs.

Best estimates the fractal dimension of A (related to the complexity of the uid ow).

Existence of regular projections of A to RN (related to a hypothesis of Kolmogorov that turbulence can be described in terms of a nite number of parameters).

Existence of invariant measures on A (related to properties of statistical solutions).

Does the semigroup become a group on A ? (reversibility on the global attractor?)

How fast A attracts bounded sets? (related to numerical analysis of the uid ow).

How the dimension of A is related to boundary conditions? (related to testing the model equations).

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 12 / 14 Some extensions of the theory

There are several theories. For example:

We do not need the uniqueness of solutions. We can consider multivalued ows.

We do not need the system to be autonomous. The system can be nonautonomous.

Reference: e.g. my book with Piotr Kalita: G.Š, P. Kalita: Navier-Stokes Equations. An Introduction with Applications, Springer 2016.

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 13 / 14 THANK YOU FOR YOUR ATTENTION !

谢谢

G.Šukaszewicz Dissipative dynamical systems and their attractors Qolloquium 14 / 14