High frequency waves and the maximal smoothing effect for nonlinear - PowerPoint PPT Presentation

High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws Stéphane Junca Labo JAD, Université de Nice Sophia-Antipolis & Coffee Team INRIA June 25 2012 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 1 / 33

Sommaire Conjecture : Lions, Perthame, Tadmor 1994 1 High frequency waves 2 dimension d = 1 d>1 Sobolev estimates Characterization of Nonlinear Flux 3 Bound of the uniform maximal smoothing effect 4 Recent Works 5 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 2 / 33

Sommaire Conjecture : Lions, Perthame, Tadmor 1994 1 High frequency waves 2 dimension d = 1 d>1 Sobolev estimates Characterization of Nonlinear Flux 3 Bound of the uniform maximal smoothing effect 4 Recent Works 5 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 3 / 33

I) Lions, Perthame, Tadmor 1994 ∂ u ∂ t + ∇ · F ( u ) = 0 u ( 0 , X ) = u 0 ( X ) R d → I R d , R d X ∈ I u : [ 0 , + ∞ ) × I R , F : I R → I Smoothing effect for NONLINEAR flux M > 0 , ∃ s = s ( F , M ) > 0 u ∈ W s , 1 R d ) ∩ W s , 1 R d ) , | u 0 ( X ) | ≤ M ⇒ loc (] 0 , + ∞ [ × I loc ( I ∀ t > 0 sup X Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 4 / 33

II) Lions, Perthame, Tadmor 1994 NONLINEAR flux velocity : a ( v ) = F ′ ( v ) : ∃ α > 0 , ∃ C > 0 , C δ α measure {| v | ≤ M , | τ + ξ · a ( v ) | ≤ δ } ≤ sup τ 2 + | ξ | 2 = 1 α Sobolev exponent : ∀ s < 2 + α α Improvement : Tadmor, Tao, 2007 : ∀ s < 1 + 2 α Conjecture : Lions, Perthame, Tadmor 1994 s sup = α sup Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 5 / 33

III) Lions, Perthame, Tadmor 1994 s uniform UNIFORM Sobolev bound and boundary layers Let M > 0 , ε > 0 , A > 0 , s < s ( F , M ) , ∃ C = C ( ε, s , � u 0 � ∞ , A ) � u 0 � L ∞ ≤ M � u � W s , 1 ([ ε, A ] × [ − A , A ] d ) + sup t >ε � u ( t , . ) � W s , 1 ([ − A , A ] d ) ≤ C Proof : Kinetic formulation & averaging lemmas ∂ t f + a ( v ) . ∇ x f = ∂ v m f ( t , x , v ) , v ∈ I R , m ( t , x , v ) ≥ 0 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 6 / 33

Sommaire Conjecture : Lions, Perthame, Tadmor 1994 1 High frequency waves 2 dimension d = 1 d>1 Sobolev estimates Characterization of Nonlinear Flux 3 Bound of the uniform maximal smoothing effect 4 Recent Works 5 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 7 / 33

One dimensional case : d = 1 Linear case : a ′ ≡ 0 ∂ u ∂ t + c ∂ u ∂ x = 0 No smoothing effect : u ( t , x ) = u 0 ( x − ct ) Propagations of high oscillations with large amplitude : � x − ct � � x � u 0 ε ( x ) = u 0 ⇒ u ε ( t , x ) = u 0 , ε ε where u 0 is periodic. Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 8 / 33

The strongest Nonlinear case : strictly convex flux � 1 with u 0 ( x + 1 ) ≡ u 0 ( x ) ∈ L ∞ ( I R ) and mean u 0 = u 0 ( θ ) d θ 0 f ( u ) = u 2 ∂ u ∂ t + ∂ f ( u ) R , a ′ ( v ) � = 0 , , ∀ v ∈ I ∂ x 2 Smoothing effect, O. Oleinik ; P . D. Lax 50’ C On ( 0 , 1 ) Total variation u ( t , . ) ≤ t . ⇒ Decay of the amplitude, | u ( t , x ) − u 0 | ≤ C t ⇒ No propagations of high oscillations with large amplitude : � x | u ε ( t , x ) − u 0 | ≤ ε C � u 0 ε ( x ) = u 0 ⇒ ε t Proof : y = x ε , s = t ε Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 9 / 33

Propagations of high oscillations with small amplitude � x � u 0 ε ( x ) = ε u 0 ε q � x  � ε u 0 if q < 1 linear propagation ε q    t , x � � u ε ( t , x ) ≃ ε U if q = 1 nonlinear propagation ε   ε u 0 if q > 1 nonlinear smoothing effect  Critical exponent, q = 1 = α , R. Diperna, A. Majda, C.M.P ., 1985, U 2 ∂ U ∂ t + ∂ 2 = 0 , U ( 0 , θ ) = u 0 ( θ ) ∂θ t , x � � u ε ( t , x ) = ε U . ε Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 10 / 33

Degenerate Nonlinear case p ≥ 2, a ( v ) = v p , α sup = 1 p < 1 � u 1 + p ∂ u ∂ t + ∂ � = 0 ∂ x 1 + p � x  � ε u 0 if q < p � x ε q  �  u 0 t , x � � ε ( x ) = ε u 0 ⇒ u ε ( t , x ) ≃ ε U if q = p ε q ε q  ε u 0 if q > p  Profile equation for critical exponent, q = p , U 1 + p ∂ U ∂ t + ∂ 1 + p = 0 , U ( 0 , θ ) = u 0 ( θ ) ∂θ t , x � � u ε ( t , x ) = ε U ε p Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 11 / 33

Burgers 2D is not non-linear ∂ t u + ∂ x u 2 + ∂ y u 2 = 0 � x − y � u ( 0 , x , y ) = u 0 ε 2012 Panov : ξ = ( 1 , − 1 ) , F = ( u 2 , u 2 ) , ξ · F ( u ) ≡ 0 Lions-Perthame-Tadmor : ξ · F ′ ( v ) = ξ · a ( v ) ≡ 0, α sup = 0 Enguist-E : ξ · F ′′ ( v ) ≡ 0, stationary solutions without smoothing effect u ( t , x , y ) = u 0 ( x − y ) Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 12 / 33

2D genuine nonlinear example u 2 u 3 ∂ t u + ∂ x 2 + ∂ y = 0 3 � φ 0 ( x , y ) � u ( 0 , x , y ) = 0 + ε u 0 ε 2 Genuine nonlinear flux : a ( u ) = ( u , u 2 ) � a ′ ( u ) � 1 � � 2 u = a ′′ ( u ) 0 2 φ 0 ( x , y ) = x cancellations φ 0 ( x , y ) = y propagations : ( 0 , 1 ) = ∇ φ 0 ⊥ a ( u = 0 ) = ( 1 , 0 ) Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 13 / 33

High frequency waves ∂ u ∂ t + ∇ x . F ( u ) = 0 � φ 0 ( x ) � u ( 0 , x ) = u + ε U 0 ε q φ 0 ( x ) = v · x Propagation : u ε ( t , x ) = u + ε U ( t , ε − q φ ( t , x )) + . . . 1 Cancellation u ε ( t , x ) = u + ε U + . . . 2 Chen,J, Rascle (JDE 06), multiphase : u ( 0 , x ) = u + ε U 0 ( ε − q 1 φ 1 ( x ) , · · · , ε − q d φ d ( x )) Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 14 / 33

Propagation of smooth high oscillations Theorem q > 1 an integer, F ∈ C ∞ ( I R d ) , U 0 ∈ C 1 ( I R , I R / Z Z , I R ) , v � = ( 0 , · · · , 0 ) , d k a du k ( u ) � v = 0 , k = 1 , · · · , q − 1 (1) then ∃ T 0 > 0 such that, for all ε ∈ ] 0 , 1 ] , u ε smooth on [ 0 , T 0 ] × I R : � � t , φ ( t , x ) u ε ( t , x ) = u + ε U + O ( ε 2 ) in C 1 ([ 0 , T 0 ] × I R d ) , ε q ∂ t + b ∂ U q + 1 ∂ U = 0 , U ( 0 , θ ) = U 0 ( θ ) . ∂θ 1 a ( q ) ( u ) � v � � b = , φ ( t , x ) = v · ( x − t a ( u )) . ( q + 1 )! Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 15 / 33

Proof : WKB expansions � � t , φ ( t , x ) u ε ( t , x ) = u + ε U ε , U ε ( 0 , θ ) = U 0 ( θ ) ε q Taylor expansions : v = ∇ φ � � t , φ ( t , x ) ∂ t U ε − ε − q ( a ( u ) · v ) ∂ θ U ε ∂ t U ε = ε q q ∂ θ U k + 1 1 � ε div x F ( u ε ) ( k + 1 )! a ( k ) ( u ) · v + ε q + 2 div x G ε = q ( U ε ) ε q − ( k + 1 ) k = 0 ε 1 − q ( a ( u ) · v ) ∂ θ U ε + ε b ∂ θ U q + 1 + ε 2 ∂ θ g ε = q ( U ε ) , ε Simplification and characteristics 0 = ∂ t u ε + div x F ( u ε ) � � ∂ t U ε + b ∂ θ U q + 1 + ε∂ θ g ε = ε q ( U ε ) . ε Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 16 / 33

Uniform Sobolev bounds sequence bounded in L ∞ for 0 < ε ≤ 1 � t , φ ( t , x ) � u ε ( t , x ) = u + ε U + . . . ε q and if d d θ U 0 � = 0 a.e. there exists C > 0, 0 ≤ t ≤ T 0 1 C ≤ � u ε ( t , . ) � W s , 1 s = 1 R d ) ≤ C , q loc ( I q the Sobolev norm W s , 1 For s > 1 R d ) blows up. loc ( I Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 17 / 33

Sobolev estimates 0 < s < 1 , � x d “ s ′′ ε � �� ε sq = ε 1 − sq ε V ∼ dx “ s ′′ ε q Upper bound : interpolation between L 1 and W 1 , 1 order ε 1 − q in W 1 , 1 order ε in L 1 loc loc order ε ( 1 − s ) 1 + s ( 1 − q ) in W s , 1 loc Lower bound : intrinsic semi-norm W s , p ( I R d ) � � | V ( x ) − V ( y ) | p dxdy | x − y | d + sp Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 18 / 33

If the orthogonality condition : d k a du k ( u ) � v = 0 , k = 1 , · · · , q − 1 , is violated then Cancellation of high oscillations, Chen, J, Rascle, 06’ Let F belongs to C q + 1 and U 0 ∈ L ∞ ( I R / Z Z , I R ) , If for some 0 < j < q d j a du j ( u ) � v � = 0 then u ε ( t , x ) in L 1 R d ) . = u + ε U 0 + o ( ε ) loc (] 0 , + ∞ [ × I proof : compactness with kinetic formulation Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 19 / 33

Sommaire Conjecture : Lions, Perthame, Tadmor 1994 1 High frequency waves 2 dimension d = 1 d>1 Sobolev estimates Characterization of Nonlinear Flux 3 Bound of the uniform maximal smoothing effect 4 Recent Works 5 Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 20 / 33

Nonlinear flux Lax : d=1, f ′′ = a ′ � = 0 ⇐ ⇒ genuine nonlinear Tartar : d=1, not a linear function on any interval Engquist-E : independence of functions F ′′ 1 ( v ) , · · · , F ′′ d ( v ) Lions-Perthame-Tadmor : related to a ( v ) = F ′ ( v ) : α sup Panov : ∀ ξ � = 0 , ξ · F is non constant on any interval genuine nonlinearity condition for smooth multi-D flux F : det ( a ′ ( v ) , a ′′ ( v ) , · · · , a ( d ) ( v )) � = 0 everywhere Chen, J., Rascle 2006, Crippa ; Otto ; Westdickenberg 2008. Stéphane Junca (Nice) HYPERBOLIC PDE’s 2012 June 25 2012 21 / 33