On nonlinear conservation laws with nonlocal diffusion term Franz - PowerPoint PPT Presentation

On nonlinear conservation laws with nonlocal diffusion term Franz Achleitner 1 Sabine Hittmeir 1 Christian Schmeiser 2 1 Vienna University of Technology 2 University of Vienna Padova, June 2012 financial support by the Austrian Science Fund (FWF)

outline 1 examples in physics 2 nonlocal operator 3 partial integro-differential equations 4 traveling wave solutions existence of traveling wave solutions asymptotic stability of traveling wave solutions 5 outlook 6 References

shallow water flow boundary conditions at free surface incompressible Navier-Stokes equations no-slip boundary conditions at rigid bottom Assumptions (Kluwick, Cox, Exner and Grinschgl (2010)) 1 Froude number 1 < U 0 √ gH << 2 2 length scales H << L 3 Reynolds number 1 << Re = L √ gH H 2 L 2 ν ⇒ viscous effects only important in boundary layer

incompressible Navier-Stokes equation u x + v y = 0 , � H 2 u t + uu x + vu y = − p x + 1 � L 2 u xx + u yy , Re v t + uv x + vv y = − L 2 � H 2 H 2 ( p y + 1) + 1 � L 2 v xx + v yy . Re x , y ...coordinates, u , v ...velocity components, t ...time, p ...pressure no-slip boundary condition y = s ( x , t ) : u = 0 , v = s t . boundary conditions at free surface � − 3 / 2 1 + H 2 � L 2 ( h x ) 2 y = h ( x , t ) : h t + uh x − v = 0 , p = − Th xx .

triple-deck structure: lower deck matching conditions Y →∞ U ( X , Y ) = Y + A , lim X →−∞ U ( X , Y ) = Y . lim no-slip b.c. at Y = 0: U = 0 , V = 0 . governing equations for ( U , V , P ) with X ∈ R , Y ∈ R + and t ∈ R + ∂ X U + ∂ Y V = 0 , U ∂ X U + V ∂ Y U = − ∂ X P + ∂ 2 Y U , ∂ t P + ∂ X P − P ∂ X P = K 1 ∂ X A + K 2 ∂ 3 X P .

linear flow response interaction equation for pressure P ∂ t P + ∂ X P = K 1 ∂ X D 1 / 3 P + K 2 ∂ 3 X P , X ∈ R , t ∈ R + , with constants K 1 and K 2 related to K 1 streamline curvature and boundary displacement effects K 2 detuning and surface tension and a nonlocal operator D α with 0 < α < 1 defined as � X P ′ ( ξ ) 1 ( D α P )( X ) := ( X − ξ ) α d ξ . Γ(1 − α ) −∞

Fowler equation: model for dune formation ∂ t u + 1 2 ∂ x u 2 = − ∂ x D 1 / 3 u + ∂ 2 x u , x ∈ R , t ∈ R + , where u ( x , t ) represents dune amplitude. no maximum principle (Alibaud, Azerad and Is` ebe (2010)) bore-like traveling wave solutions

nonlocal operator ∂ x D α For 0 < α < 1, the operator � x u ′ ( y ) 1 ( ∂ x D α u )( x ) = ∂ x ( x − y ) α dy Γ(1 − α ) −∞ is a Fourier multiplier operator 1 � F ( ∂ x D α u )( ξ ) = e − i ξ x ( ∂ x D α u )( x ) dx = Λ( ξ ) F u ( ξ ) √ 2 π R with u ∈ S ( R ) and symbol � απ απ �� � � � | ξ | α +1 , Λ( ξ ) = − sin − i sgn( ξ ) cos ξ ∈ R . 2 2

Fourier multiplier operator ( F Tf )( ξ ) = − ψ ( − ξ )( F f )( ξ ) Riesz-Feller operator T θ Id i sgn( ξ ) θπ ψ ( ξ ) = | ξ | a exp � � 2 2 real-valued parameters ∂ x − H 1 a index of stability ∂ x D α u 0 < a ≤ 2 1 − α ∂ 2 θ skewness parameter − Id − ∂ x H x 0 | θ | ≤ min( a , 2 − a ) Hilbert transform − H ∂ 2 ∂ 3 − ∂ x � ∞ H x x Hf := p . v . 1 f ( y ) a − 1 x − y dy π 0 1 1 + α 2 3 −∞

Fourier multiplier ψ a ,θ ( ξ ) := | ξ | a exp i sgn( ξ ) θ π � � 2 θ − 1 2 real-valued parameters i sgn( ξ ) i ξ 1 a index of stability − ( i ξ ) a 0 < a ≤ 2 1 − α | ξ | 2 θ skewness parameter | ξ | 1 0 | θ | ≤ min( a , 2 − a ) − ( i ξ ) 3 − i sgn( ξ ) − i ξ a − 1 0 1 1 + α 2 3

fractional diffusion equation ∂ t u = ∂ x D α u , x ∈ R , t ∈ R + , for some fixed α with 0 < α < 1. strongly continuous, convolution semigroup T t : L p ( R ) → L p ( R ) , u 0 �→ T t u 0 = u ( t , x ) = K ( t , · ) ∗ u 0 , with 1 ≤ p < ∞ and kernel K ( t , x ) = F − 1 (exp(Λ( . ) t ))( x ). Properties of K ( t , x ): for all x ∈ R , t > 0 and m ∈ N , non-negative K ( t , x ) ≥ 0 integrable � K ( t , . ) � L 1 ( R ) = 1 1 1 scaling K ( t , x ) = t − 1+ α K (1 , xt − 1+ α ) smooth K ( t , x ) is C ∞ smooth bounded there exists B m ∈ R + such that B m x K | ( t , x ) ≤ t − 1+ m | ∂ m 1+ α 2 1 + t − 1+ α | x | 2

L´ evy strictly stable distributions on R random variable X θ E [exp( i ξ X )] = exp( − ψ ( ξ )) ψ ( ξ ) = | ξ | a exp i sgn( ξ ) θ π � � 2 2 distributions L L´ evy-Smirnov 1 PDF x − 3 / 2 − 1 � � 2 √ π exp , 4 x 1 − α x > 0. C H N C Cauchy(-Lorentz) 0 PDF 1 1 L π 1+ x 2 H Holtsmark a − 1 N Normal (Gaussian) 0 1 1 + α 2 3 − x 2 1 � � PDF 2 πσ 2 exp √ 2 σ 2

approximate identity Theorem (Stein, Weiss) Suppose φ ∈ L 1 ( R n ) with � R n φ ( x ) dx = 1 and for ǫ > 0 let φ ǫ ( x ) = ǫ − n φ ( x /ǫ ) . If f ∈ L p ( R n ) , 1 ≤ p < ∞ , or f ∈ C 0 ⊂ L ∞ ( R n ) , then � f ∗ φ ǫ − f � p → 0 as ǫ → 0 . Theorem (Lieb, Loss) Let j be in L 1 ( R n ) with R n j = 1 . For ǫ > 0 , define j ǫ ( x ) := ǫ − n j ( x /ǫ ) , so � R n j ǫ = 1 and � j ǫ � 1 = � j � 1 . Let f ∈ L p ( R n ) for some 1 ≤ p < ∞ and � that define the convolution f ǫ := j ǫ ∗ f . Then f ǫ ∈ L p ( R n ) and � f ǫ � p ≤ � j � 1 � f � p . f ǫ → f strongly in L p ( R n ) as ǫ → 0 . If j ∈ C ∞ c ( R n ) , then f ǫ ∈ C ∞ ( R n ) and D α f ǫ = ( D α j ǫ ) ∗ f .

Theorem (L´ evy process) For every 0 < α < 1 , there exists a L´ evy process { X t | t ≥ 0 } such that the probability distribution µ t of X t has probability density function K ( t , x ) . Moreover, the associated transition semigroup { P t } is a strongly continuous semigroup on C 0 ( R ) with � P t � = 1 . The infinitesimal generator L α of { P t } is given for f ∈ C 2 0 ( R ) by � � ∞ f ( x + y ) − f ( x ) − yf ′ ( x ) (1 − α ) π � L α f ( x ) = cos d y y 2+ α 2 0 and has core C ∞ c ( R ) . transition function P t ( x , B ) = µ t ( B − x ) for t ≥ 0 , x ∈ R , B ∈ B ( R ) transition semigroup Define for f ∈ C 0 ( R ) � � ( P t f )( x ) = P t ( x , d y ) f ( y ) = f ( x + y ) K ( t , y ) d y = E [ f ( x + X t )] , R R then P t f ∈ C 0 ( R ) by the Lebesgue convergence theorem.

scalar conservation law with nonlocal diffusion ∂ t u + ∂ x f ( u ) = ∂ x D α u , x ∈ R , t ∈ R + , (1) where f ( u ) is a smooth flux function. Theorem (Droniou, Gallou¨ et and Vovelle (2003)) The Cauchy problem of (1) with initial datum u 0 ∈ L ∞ ( R ) has a unique global solution u ( t , x ) , in the sense that u ∈ L ∞ ((0 , ∞ ) × R ) satisfies � t � � u ( t , x ) = ( K ( t , . ) ∗ u 0 )( x ) − K ( t − τ, . ) ∗ ∂ x f ( u ( τ, . )) ( x ) d τ (2) 0 almost everywhere. Moreover, 1 u ∈ C ∞ ((0 , ∞ ) × R ) and u ∈ C ∞ b (( t 0 , ∞ ) × R ) for all t 0 > 0 . 2 u satisfies equation (1) in the classical sense. 3 for all t > 0 , � u ( t , . ) � ∞ ≤ � u 0 � ∞ and, in fact, u takes its values between the essential lower and upper bounds of u 0 . 4 u ( t ) t → 0 → u 0 in L ∞ ( R ) weak- ∗ and in L p − − loc ( R ) for all p ∈ [1 , ∞ ) .

sketch of proof Droniou, Gallou¨ et and Vovelle established the result in case of a nonlocal diffusion operator with symbol −| ξ | 1+ α for 0 < α < 1. existence Suppose u 0 ∈ C ∞ c ( R ). Construct approximate solution u δ , δ > 0, by a splitting method: Define u δ (0 , . ) = u 0 Define u δ ( t , x ) on ( t , x ) ∈ (2 n δ, (2 n + 1) δ ] × R , n ∈ N 0 , as the solution of ∂ t u = 2 ∂ x D α u with initial condition u δ (2 n δ, . ). Define u δ ( t , x ) on ( t , x ) ∈ ((2 n + 1) δ, (2 n + 2) δ ] × R , n ∈ N 0 , as the solution of ∂ t u + 2 ∂ x f ( u ) = 0 with initial condition u δ ((2 n + 1) δ, . ). For δ 0 > 0 sufficiently small, any compact set Q ⊂ R and T > 0, { u δ | δ ∈ (0 , δ 0 ] } is relatively compact in C ([0 , T ]; L 1 ( Q )). limit function u ∈ C ([0 , T ]; L 1 ( R )) satisfies mild formulation (2).

scalar conservation law with vanishing nonlocal diffusion ∂ t u ǫ + ∂ x f ( u ǫ ) = ǫ∂ x D α u ǫ , x ∈ R , t ∈ R + , (3) where f ( u ) is a smooth flux function. Theorem (Droniou (2003)) Suppose 0 < α ≤ 1 and u 0 ∈ L ∞ ( R ) . The solution u ǫ of ∂ t u ǫ + ∂ x f ( u ǫ ) = ǫ∂ x D α u ǫ , u ǫ (0 , x ) = u 0 ( x ) , converges as ǫ → 0 in C ([0 , T ]; L 1 loc ( R n )) for all T > 0 to the entropy solution of the Cauchy problem ∂ t u + ∂ x f ( u ) = 0 , u (0 , x ) = u 0 ( x ) . Moreover, if u 0 ∈ L ∞ ( R ) ∩ L 1 ( R ) ∩ BV ( R ) , then � u ǫ − u � C ([0 , T ]; L 1 ( R )) = O ǫ 1 / (1+ α ) � � .

traveling wave solutions of equation (1) Consider wave speed s ∈ R and traveling wave variable ξ := x − st . Definition A traveling wave solution of (1) is a solution of the form u ( t , x ) = ¯ u ( ξ ), for some function ¯ u that connects different endstates lim ξ →±∞ ¯ u ( ξ ) = u ± . traveling wave equation � x u ′ ( y ) 1 = D α u = � � h ( u ) := f ( u ) − su − f ( u − ) − su − ( x − y ) α dy Γ(1 − α ) −∞ (4) properties of specific traveling wave solutions Rankine-Hugoniot condition f ( u + ) − f ( u − ) = s ( u + − u − ) For convex flux function f ( u ) and a monotone solution, standard entropy condition u − > u + .