# On a result of symmetry based on nonlinear flows

## Transcript Of On a result of symmetry based on nonlinear flows

On a result of symmetry based on nonlinear ﬂows

Jean Dolbeault http://www.ceremade.dauphine.fr/∼dolbeaul

Ceremade, Universit´e Paris-Dauphine February 24, 2017 ESI, Vienna

Geometric Transport Equations in General Relativity, February 20–24, 2017

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

The mexican hat potential in Schro¨dinger equations

Let us consider a nonlinear Schr¨odinger equation in presence of a radial external potential with a minimum which is not at the origin

−∆u + V (x) u − f (u) = 0

0.5

1.5

1.0

0.5

0.5

0.5

1.0

1.5

1.0

A one-dimensional potential V (x)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

A two-dimensional potential V (x) with mexican hat shape

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Radial solutions to −∆u + V (x) u − F (u) = 0

... give rise to a radial density of energy x → V |u|2 + F (u)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

symmetry breaking

... but in some cases minimal energy solutions

... give rise to a non-radial density of energy x → V |u|2 + F (u)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Symmetry breaking may occur... in presence of non-cooperative potentials in weighted equations or weighted variational problems in phase transition problems in evolution equations (instability of symmetric solutions) in various models of mathematical physics and quantum ﬁeld theory

etc.

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

BGK-type kinetic equation as a motivation for nonlinear diﬀusions – polytropes and fast diﬀusion / porous medium

ε2∂t f ε + εv · ∇x f ε − ε∇x V (x ) · ∇v f ε = Gf ε − f ε f ε(x, v , t = 0) = fI (x, v ) , x, v ∈ R3

|v |2

with the Gibbs equilibrium Gf := γ

+ V (x) − µρf (x, t)

2

The Fermi energy µρf (x, t) is implicitly deﬁned by

|v |2

γ

+ V (x) − µρf (x, t) dv =

f (x, v , t)dv =: ρf (x, t)

R3

2

R3

f ε(x, v , t) . . . phase space particle density V (x) . . . potential ε . . . mean free path

=⇒ µρf = µ¯(ρf )

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Diﬀusion limits

[J.D., P. Markowich, D. O¨ lz, C. Schmeiser] Theorem

For any ε > 0, the equation has a unique weak solution f ε ∈ C (0, ∞; L1 ∩ Lp(R6)) for all p < ∞. As ε → 0, f ε weakly converges to a local Gibbs state f 0 given by

f 0(x, v , t) = γ 1 |v |2 − µ¯(ρ(x, t)) 2

where ρ is a solution of the nonlinear diﬀusion equation

∂t ρ = ∇x · (∇x ν(ρ) + ρ ∇x V (x))

with initial data ρ(x, 0) = ρI (x) := R3 fI (x, v ) dv

ρ

ν(ρ) = s µ¯ (s) ds

0

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Outline

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Symmetry breaking and linearization The critical Caﬀarelli-Kohn-Nirenberg inequality A family of sub-critical Caﬀarelli-Kohn-Nirenberg inequalities Linearization and spectrum

Without weights: Gagliardo-Nirenberg inequalities and fast

diﬀusion ﬂows R´enyi entropy powers Self-similar variables and relative entropies The role of the spectral gap

With weights: Caﬀarelli-Kohn-Nirenberg inequalities and

weighted nonlinear ﬂows Towards a parabolic proof Large time asymptotics and spectral gaps A discussion of optimality cases

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Collaborations

Collaboration with...

M.J. Esteban and M. Loss (symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori (symmetry, subcritical case)

M. Bonforte, M. Muratori and B. Nazaret (linearization and large time asymptotics for the evolution problem)

M. del Pino, G. Toscani (nonlinear ﬂows and entropy methods) A. Blanchet, G. Grillo, J.L. V´azquez (large time asymptotics and

linearization for the evolution equations)

...and also

S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk ...

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Jean Dolbeault http://www.ceremade.dauphine.fr/∼dolbeaul

Ceremade, Universit´e Paris-Dauphine February 24, 2017 ESI, Vienna

Geometric Transport Equations in General Relativity, February 20–24, 2017

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

The mexican hat potential in Schro¨dinger equations

Let us consider a nonlinear Schr¨odinger equation in presence of a radial external potential with a minimum which is not at the origin

−∆u + V (x) u − f (u) = 0

0.5

1.5

1.0

0.5

0.5

0.5

1.0

1.5

1.0

A one-dimensional potential V (x)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

A two-dimensional potential V (x) with mexican hat shape

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Radial solutions to −∆u + V (x) u − F (u) = 0

... give rise to a radial density of energy x → V |u|2 + F (u)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

symmetry breaking

... but in some cases minimal energy solutions

... give rise to a non-radial density of energy x → V |u|2 + F (u)

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Symmetry breaking may occur... in presence of non-cooperative potentials in weighted equations or weighted variational problems in phase transition problems in evolution equations (instability of symmetric solutions) in various models of mathematical physics and quantum ﬁeld theory

etc.

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

BGK-type kinetic equation as a motivation for nonlinear diﬀusions – polytropes and fast diﬀusion / porous medium

ε2∂t f ε + εv · ∇x f ε − ε∇x V (x ) · ∇v f ε = Gf ε − f ε f ε(x, v , t = 0) = fI (x, v ) , x, v ∈ R3

|v |2

with the Gibbs equilibrium Gf := γ

+ V (x) − µρf (x, t)

2

The Fermi energy µρf (x, t) is implicitly deﬁned by

|v |2

γ

+ V (x) − µρf (x, t) dv =

f (x, v , t)dv =: ρf (x, t)

R3

2

R3

f ε(x, v , t) . . . phase space particle density V (x) . . . potential ε . . . mean free path

=⇒ µρf = µ¯(ρf )

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Diﬀusion limits

[J.D., P. Markowich, D. O¨ lz, C. Schmeiser] Theorem

For any ε > 0, the equation has a unique weak solution f ε ∈ C (0, ∞; L1 ∩ Lp(R6)) for all p < ∞. As ε → 0, f ε weakly converges to a local Gibbs state f 0 given by

f 0(x, v , t) = γ 1 |v |2 − µ¯(ρ(x, t)) 2

where ρ is a solution of the nonlinear diﬀusion equation

∂t ρ = ∇x · (∇x ν(ρ) + ρ ∇x V (x))

with initial data ρ(x, 0) = ρI (x) := R3 fI (x, v ) dv

ρ

ν(ρ) = s µ¯ (s) ds

0

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Outline

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Symmetry breaking and linearization The critical Caﬀarelli-Kohn-Nirenberg inequality A family of sub-critical Caﬀarelli-Kohn-Nirenberg inequalities Linearization and spectrum

Without weights: Gagliardo-Nirenberg inequalities and fast

diﬀusion ﬂows R´enyi entropy powers Self-similar variables and relative entropies The role of the spectral gap

With weights: Caﬀarelli-Kohn-Nirenberg inequalities and

weighted nonlinear ﬂows Towards a parabolic proof Large time asymptotics and spectral gaps A discussion of optimality cases

J. Dolbeault

A symmetry result based on nonlinear ﬂows

Symmetry breaking and linearization Entropy methods without weights

Weighted nonlinear ﬂows and CKN inequalities

Collaborations

Collaboration with...

M.J. Esteban and M. Loss (symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori (symmetry, subcritical case)

M. Bonforte, M. Muratori and B. Nazaret (linearization and large time asymptotics for the evolution problem)

M. del Pino, G. Toscani (nonlinear ﬂows and entropy methods) A. Blanchet, G. Grillo, J.L. V´azquez (large time asymptotics and

linearization for the evolution equations)

...and also

S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk ...

J. Dolbeault

A symmetry result based on nonlinear ﬂows