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