Partial Regularity in Time for the Landau Equation (with Coulomb Interaction) François Golse CMLS, École polytechnique, Paris CIRM, October 21-25 2019 "The Analysis of Complex Quantum Systems: Large Coulomb Systems and Related Matters" Work in collaboration with M.P. Gualdani, C. Imbert and A. Vasseur arXiv:1906.02841 [math.AP] François Golse Partial Regularity for Landau
Landau Equation Landau equation with unknown f ≡ f ( t , v ) ≥ 0: � v ∈ R 3 ∂ t f ( t , v )=div v R 3 a ( v − w )( ∇ v −∇ w )( f ( t , v ) f ( t , w )) dw , with the notation: � � ⊗ 2 8 π ∇ 2 | z | = 1 1 z a ( z ) := 8 π | z | Π( z ) , Π( z ) := I − | z | Nonconservative form ∂ t f ( t , v ) = ( a ij ⋆ v f ( t , v )) ∂ v i ∂ v j f ( t , v ) + f ( t , v ) 2 Open question global existence of classical solutions or finite-time � blow-up for the Cauchy problem with f t = 0 = f in ? � François Golse Partial Regularity for Landau
Semilinear heat equation Finite time blow-up for u ≥ 0 soln of ∂ t u = ∆ x u + α u 2 Hint: Ricatti inequality ˙ L ( t ) ≥ − λ 0 L ( t ) + α L 2 ( t ) satisfied by � � − ∆ φ = λ 0 φ , B u ( t , x ) φ ( x ) dx φ > 0 on B L ( t ) := with � � B φ ( x ) dx φ ∂ B = 0 � “Isotropic Landau” global existence of radially symmetric nonin- creasing soln [Gressman-Krieger-Strain 2012, Gualdani-Guillen 2016] ∂ t u = (( − ∆) − 1 u )∆ u + α u 2 k solns with p > 3 t L p Conditional regularity L ∞ 2 and k > 5 are L ∞ t , v ([Silvestre 2017], radial solns [Gualdani-Guillen 2016]) François Golse Partial Regularity for Landau
Villani’s H-Solutions [1RMA1998] H-solution 0 ≤ f ∈ C ([ 0 , T ); D ′ ( R 3 )) ∩ L 1 (( 0 , T ); L 1 − 1 ( R 3 )) s.t. 1 1 � � f ( t , v ) dv = f in ( t , v ) dv v v R 3 | v | 2 R 3 | v | 2 � � R 3 f ( t , v ) ln f ( t , v ) dv ≤ R 3 f in ( v ) ln f in ( v ) dv for a.e. t ≥ 0, and � T � � R 3 f in ( v ) φ ( 0 , v ) dv + R 3 f ( t , v ) ∂ t φ ( t , v ) dv 0 � T � = R 6 (Φ( t , v ) − Φ( t , w )) · Π( v − w ) ( F ( ∇ v −∇ w ) F ) ( t , v , w ) dvdw 0 � f ( t , v ) f ( t , w ) with Φ( t , v ) := ∇ v φ ( t , v ) , F ( t , v , w ) := 8 π | v − w | � Notation � g � p ( 1 + | v | 2 ) k / 2 | g ( v ) | p dv with p ≥ 1 and k ∈ R k := L p François Golse Partial Regularity for Landau
Suitable Solutions Definition E ) -suitable solution on [ 0 , T ) × R 3 is an H-solution s.t. A ( N , q , C ′ � t 2 � 1 f ( t , v ) >κ ∇ v f ( t , v ) 1 / q � � 2 H + ( f ( t 2 , · ) | κ ) + C ′ � � L q ( R 3 ) dt E � t 1 � t 2 � ≤ H + ( f ( t 1 , · ) | κ ) + 2 κ R 3 ( f ( t , v ) − κ ) + dvdt t 1 for all t 1 < t 2 ∈ [ 0 , T ) \ N and κ ≥ 1, where � g ( v ) � � H + ( g | κ ) := R 3 κ h + dv , h + ( z ) := z (ln z ) + − ( z − 1 ) + κ François Golse Partial Regularity for Landau
Partial Regularity in Time Definition A regular time of f , suitable solution on I ⊂ ( 0 , + ∞ ) , is a time τ ∈ I s.t. f ∈ L ∞ (( τ − ǫ, τ ) × R 3 ) for some ǫ ∈ ( 0 , τ ) . The set of singular (i.e. nonregular) times of f on I is denoted S [ f , I ] . Main Thm Let f be a suitable solution to the Landau equation on [ 0 , T ) × R 3 for all T > 0 , with initial data f in satisfying � R 3 ( 1 + | v | k + | ln f in ( v ) | ) f in ( v ) dv < ∞ for all k > 3 Then Hausdorff dim S [ f , ( 0 , + ∞ )] ≤ 1 2 François Golse Partial Regularity for Landau
Existence Theory Prop 1 For all 0 ≤ f in ∈ L 1 ( R 3 ) s.t. � R 3 ( 1 + | v | k + | ln f in ( v ) | ) f in ( v ) dv < ∞ for some k > 3 there exists an ( N , q , C ′ E ) -suitable solution f on [ 0 , T ] with initial data f in and 2 k C ′ E ≡ C ′ E [ T , q , f in ] > 0 , q := k + 3 François Golse Partial Regularity for Landau
Desvillettes Theorem [JFA2015] �� � 1 / p R N ( 1 + | v | 2 ) k / 2 | g ( v ) | p Notation � g � L p k ( R N ) := Thm For each 0 ≤ f ∈ L 1 2 ( R 3 ) s.t. f ln f ∈ L 1 ( R 3 ) |∇ √ | Π( v − w )( ∇ v −∇ w ) √ � � f ( v ) | 2 dv f ( v ) f ( w ) | 2 ( 1 + | v | 2 ) 3 / 2 ≤ C D + C D dvdw | v − w | R 3 R 6 with �� � R 3 ( 1 , v , | v | 2 , | ln f ( v ) | ) f ( v ) dv C D ≡ C D > 0 Corollary Let 0 ≤ f in ∈ L 1 k ( R 3 )) with k > 2 s.t. f in | ln f in | ∈ L 1 ( R 3 ) . � ⇒ f ∈ L ∞ ( 0 , T ; L 1 k ( R 3 )) f H-solution s.t. f t = 0 = f in = � François Golse Partial Regularity for Landau
(Formal) H Theorem Assuming that f ( t , v ) > 0 a.e., one has � d R 3 f ( t , v ) ln f ( t , v ) dv dt � � � �� 2 f ( t , v ) f ( t , w ) ∇ v f ( t , v ) f ( t , v ) − ∇ w f ( t , w ) � � = − � Π( v − w ) dvdw 16 π | v − w | � f ( t , w ) R 6 François Golse Partial Regularity for Landau
(Formal) Truncated H Theorem One has d dt H + ( f ( t , · ) | κ ) � � � 1 f ( t , v ) >κ ∇ v f ( t , v ) �� 2 1 f ( t , w ) >κ ∇ w f ( t , w ) f ( t , v ) f ( t , w ) � � + � Π( v − w ) − dvdw 16 π | v − w | � f ( t , v ) f ( t , w ) � �� � D 1 � f ( t , v ) f ( t , w ) a ( v − w ): ∇ v (ln f ( t , v ) ) + ⊗∇ w (ln f ( t , w ) = − ) − dvdw κ κ � = − a ( v − w ): ∇ v f ( t , v ) 1 f ( t , v ) ≥ κ ⊗ ∇ w f ( t , w ) 1 f ( t , w ) <κ dvdw � − div v (div w a ( v − w )) ( f ( t , v ) − κ ) + ( κ − ( f ( t , w ) − κ ) − ) dvdw = � �� � ≥ 0 ( in fact = δ ( v − w ) ) � ≤ κ ( f ( t , v ) − κ ) + dv � �� � depleted NL François Golse Partial Regularity for Landau
Sketch of the Proof of Prop 1 • Replace a with its truncated variant 8 π ( 1 1 a n ( z ) = | z | ∧ n )Π( z ) , satisfying div(div a n ) ≥ 0 • Use the Desvillettes theorem to bound |∇ v √ � � f ( t , v ) | 2 1 1 f ( t , v ) >κ dv ≤ D 1 + R 3 ( f ( t , w ) − κ ) + dw C ′′ ( 1 + | v | ) 3 D R 3 • Using the Desvillettes corollary with p ′ = 2 / q (recall q ∈ ( 1 , 2 ) ) � � 1 f ( t , v ) >κ ∇ v f ( t , v ) 1 / q � q � � � L q ( R 3 ) |∇ v √ �� � 1 / p ′ f ( t , v ) | 2 1 f ( t , v ) ≥ κ ≤ ( 2 q ) q � f ( t , · ) � L p dv 3 p / 2 p ′ ( R 3 ) ( 1 + | v | 2 ) 3 / 2 R 3 François Golse Partial Regularity for Landau
The 1st De Giorgi Type Lemma Prop 2 Let f be a ( N , q , C ′ E ) –suitable solution to the Landau equa- E > 0 and q ∈ ( 6 tion for t ∈ [ 0 , 1 ] with C ′ 5 , 2 ) Then there exists η 0 ≡ η 0 [ q , C ′ E ] > 0 s.t. � 1 H + ( f ( t , · ) | 1 a.e. on [ 1 2 , 1 ] × R 3 2 ) dt < η 0 = ⇒ f ( t , v ) ≤ 2 1 / 8 François Golse Partial Regularity for Landau
Proof of Prop 2 Set � κ k := ( 1 + ( 2 1 / q − 1 )( 1 − 2 − k )) q t k := 1 4 · 2 − k , 2 − 1 k ( t , v ) := µ (( f ( t , v ) 1 / q − κ 1 / q f + with µ ( r ) := min( r , r 2 ) ) + ) k and observe that c h µ ( r ) ≤ h + ( r ) ≤ C ι ( r − 1 ) ι + Consider the quantity � c h R 3 f + k ( t , v ) q dv A k := ess sup 2 t k ≤ t ≤ 1 � 1 � 2 / q �� + 1 4 C ′ R 3 |∇ v f + k ( t , v ) | q dv dt E t k François Golse Partial Regularity for Landau
De Giorgi Nonlinearization [Mem. Accad. Sci. Torino 1957] • Observe first that k > µ (( 2 1 / q − 1 ) · 2 − k − 1 ) f + ⇒ f + k + 1 > 0 = � 1 � A k + 1 ≤ C q ,ι 4 ( k + 3 ) q ( 1 + ι ) k ( θ, v ) q ( 1 + ι ) dvd θ R 3 f + so that t k • Using the Hölder inequality + Sobolev embedding with ι = 2 3 5 q A k + 1 ≤ M Λ k A β β := 8 3 − 2 k , q > 1 and Λ := 2 · 4 3 with M ≡ M [ q , C ′ E ] > 0, so that 1 1 − ( β − 1 ) 2 = A 0 < M − β − 1 Λ ⇒ A k → 0 as k → + ∞ • Control A 0 by truncated entropy + conclude by Fatou’s lemma François Golse Partial Regularity for Landau
The Improved De Giorgi Type Lemma Prop 3 Let f be a ( N , q , C ′ E ) -suitable solution to the Landau equa- tion on [ 0 , 1 ] with q ∈ ( 4 3 , 2 ) . There exists η 1 ≡ η 1 [ q , C ′ E ] > 0 and δ 1 ∈ ( 0 , 1 ) such that � 1 � � 2 1 ǫ → 0 + ǫ γ − 3 � � lim � 1 f ( T , V ) >ǫ − γ ∇ V f ( T , V ) L q ( R 3 ) dT < η 1 q � 1 − ǫ γ ⇒ f ∈ L ∞ (( 1 − δ 1 , 1 ) × R 3 ) = with γ := 5 q − 6 2 q − 2 . François Golse Partial Regularity for Landau
Proof of Prop 3: (a) Scaling • 2-parameter group of invariance scaling transfo. for the Landau eq.: f λ,ǫ ( t , v ) := λ f ( λ t , ǫ v ) • let f be a ( N , q , C ′ E ) -suitable solution on [ 0 , 1 ] , with λ = ǫ γ H + ( f λ,ǫ ( t , · ) | ǫ γ κ ) = ǫ γ − 3 H + ( f ( ǫ γ t , · ) | ǫ γ κ ) � ǫ γ t 2 � t 2 � � ( f λ,ǫ ( t , v ) − ǫ γ κ ) + dvdt = 1 f ( T , V ) − κ ) + dVdT ǫ 3 ǫ γ t 1 t 1 while γ := 5 q − 6 2 q − 2 implies that � t 2 �� � 2 / q 1 λ,ǫ ( t , v ) | q dv q | 1 f λ,ǫ ≥ ǫ γ κ ∇ v f dt t 1 � ǫ γ t 2 �� � 2 / q 1 = ǫ γ − 3 q ( T , V ) | q dV | 1 f ≥ κ ∇ v f dT ǫ γ t 1 François Golse Partial Regularity for Landau
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