Hyperbolic Conservation Laws with Memory Cleopatra Christoforou - PowerPoint PPT Presentation

Hyperbolic Conservation Laws with Memory Cleopatra Christoforou Northwestern University USA July, 2006 Eleventh International Conference on Hyperbolic Problems Theory, Numerics, Applications Lyon, France

Hyperbolic Conservation Laws with Memory July, 06 Conservation Laws in one-space dimension: U t + F ( U ) x = 0 Elastic medium: the flux F is determined by the value U ( x, t ). Viscoelastic medium: the flux depends also on the past history of the medium U ( x, τ ) for τ < t . Materials with fading memory that correspond to constitutive relations with flux of the form: � t F ( U ( x, t )) + 0 k ( t − τ ) G ( U ( x, τ )) dτ (1) i.e. � t U t + F ( U ) x + 0 k ( t − τ ) G ( U ( x, τ )) x dτ = 0 (2) 1 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 � Classical solutions: Dafermos, Hrusa, MacCamy, Nohel, Re- nardy, Slemrod, Staffans... Main results: ◦ If the initial data are “small” and sufficiently smooth, then there exists a unique global smooth solution to (2) that decays to equilibrium as t → + ∞ . ��� in constrast to elastic media . ◦ If the initial data are “large”, then singularities develop in finite time. ��� as in elastic media . � Weak solutions: G.-Q. Chen, Dafermos, Nohel, Rogers, Tzavaras... Summary of results: Existence of global weak solutions in L ∞ ( bounded measurable functions ) is established, for special equations by the method of compansated compactness. 2 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 The aim is to treat entropy weak solutions of bounded variation ( BV ). Motivation: Hyperbolic Conservation Laws with Fading Memory in one-space dimension � t U t + F ( U ) x + 0 k ( t − τ ) G ( U ( x, τ )) x dτ = 0 (3) can be viewed as a linear Volterra equation under suitable choice of F and G . This was first observed by MacCamy [M] and later employed in Dafermos [D] and Nohel–Rogers–Tzavaras [NRT]. � t  U t + A ( U ) U x + g ( U ) = H ( t ) ¯ U − 0 K ( t − τ ) U ( τ ) dτ  (4) , U (0 , x ) = ¯ U ( x )  where, x ∈ R , U ( t, x ) ∈ R n , A ( U ) ∈ M n × n , g : R n → R n and H, K : [0 , + ∞ ) → M n × n . 3 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 The Vanishing Viscosity Method. � t  0 K ( t − τ ) U ε ( τ ) dτ + εU ε U ε t + A ( U ε ) U ε x + g ( U ε ) = H ( t ) ¯ U −  xx , U ε (0 , x ) = ¯ U ( x )  (5) ⇒ U ε → U L 1 in as ε → 0+ loc ♦ Scalar conservation law: g ≡ 0, H ≡ 0, K ≡ 0 ◦ Oleinik [O], 1957: One-space dimension ◦ Kruzkov [K], 1970: Several space dimensions ♦ Systems of conservation laws in one-space dimension: ◦ Bianchini and Bressan [BiB], 2005: g ≡ 0, H ≡ 0, K ≡ 0. ◦ Christoforou [C], 2006: g � = 0 and H ≡ 0, K ≡ 0. 4 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Assumptions ( ⋆ ) : Let U ∗ be a constant equilibrium solution to the hyperbolic problem (4), B ( U ∗ ) . = [ R ( U ∗ )] − 1 Dg ( U ∗ ) R ( U ∗ ) is strictly column diagonally 1. ˜ dominant , i.e. B ii ( U ∗ ) − B ji ( U ∗ ) | ≥ β > 0 ˜ | ˜ � i = 1 , ..., n. j � = i K ( s ) . = R ( U ∗ ) − 1 K ( s ) R ( U ∗ ) ∈ L 1 [0 , + ∞ ) is absolutely domi- ˜ 2. nated by ˜ B , i.e. there exists a positive constant κ ≥ 0, such that for each i = 1 , . . . , n � + ∞ n | ˜ � K ji ( s ) | ds < κ, and 0 ≤ κ < β. 0 j =1 3. H ( · ) ∈ L 1 [0 , + ∞ ). 5 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Theorem 1. (G-Q. Chen, Christoforou) Consider the Cauchy problem � t U ε t + A ( U ε ) U ε x + g ( U ε ) = H ( t ) ¯ 0 K ( t − τ ) U ε ( τ, x ) dτ + εU ε xx (6) U − U ε (0 , x ) = ¯ U ( x ) . (7) Assume that the system is strictly hyperbolic. Under Assump- U − U ∗ ∈ L 1 tions ( ⋆ ) , there exists a constant δ 0 > 0 such that if ¯ and TV { ¯ U } < δ 0 , then for each ε > 0 , (6) - (7) has a unique solution U ε , defined for all t ≥ 0 , that satisfies � t TV { U ε ( t, · ) } + 0 TV { U ε ( s, · ) } ds ≤ C TV { ¯ U } , (8) where C is a positive constant that is independent of t and ε . 6 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Moreover, if V ε is another solution of (6) with initial data ¯ V , then � t � U ε ( t ) − V ε ( t ) � L 1 + 0 � U ε ( τ ) − V ε ( τ ) � L 1 dτ ≤ L � ¯ U − ¯ V � L 1 . (9) Furthermore, the continuous dependence property with respect to time holds, i.e. √ � U ε ( t ) − U ε ( s ) � L 1 ≤ L ′ ( | t − s | + √ ε | t − √ s | ) , (10) for t , s > 0 . Finally, as ε ↓ 0+ , U ε → U in L 1 loc , where U is the admissible weak solution of the hyperbolic system with memory (4) . The proof follows closely the fundamental ideas in Bianchini– Bressan and the techniques in Christoforou in order to treat the source g ( u ). Additional estimates are employed to handle the integral term as lower order perturbation in terms of the damping effect of g due to Assumptions ( ⋆ ). 7 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Theorem 2. (G.-Q. Chen, Christoforou) Scalar equation: � t  u t + ( f ( u )) x + 0 k ( t − τ )( f ( u ( τ ))) x dτ = 0  (11) , u (0 , x ) = u 0 ( x )  Let r be the resolvent kernel associated with k : r + r ∗ k = − k. Assume r is nonnegative, nonincreasing in L 1 ( R + ) , then for each ε > 0 , consider � t � t � � u ε + u ε t + f ( u ε ) x + 0 k ( t − τ ) f ( u ε ( τ )) x dτ = ε 0 k ( t − τ ) u ε ( τ ) dτ . xx There exists a unique solution u ε defined globally with a uniform BV bound. 8 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Then as ε → 0+ , u ε converges in L 1 loc to an entropy solution u ∈ BV to � t  u t + ( f ( u )) x + 0 k ( t − τ )( f ( u ( τ ))) x dτ = 0  , u (0 , x ) = u 0 ( x )  which satisfies: � t TV { u ( t ) } + 0 r ( t − τ ) TV { u ( τ ) } dτ ≤ L M ( u 0 ) (12) � u ( t ) − u ( s ) � L 1 ≤ C M ( u 0 ) | t − s | , (13) � u � L ∞ ( R 2 + ) ≤ 2 � u 0 � L ∞ ( R ) , (14) where L = 1 + � r � L 1 ( R + ) , M ( u 0 ) = TV { u 0 } + 2 � u 0 � L ∞ ( R ) and C is a constant independent of ε and TV { u 0 } . 9 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Proof: By employing the resolvent kernel r , (11) can be written as follows � t 0 r ′ ( t − τ ) u ( τ ) dτ u t + f ( u ) x + r (0) u = r ( t ) u 0 − Note the similarity of the above equation with the one in the case of systems, (4). From here and on, the techniques are motivated by Vol’pert [V]- Kruzkov [K]. Let v = u x , � t v t + ( f ′ ( u ) v ) x + εr (0) v = εr ( εt ) u x − ε 2 0 r ′ ( ε ( t − τ )) v ( τ ) dτ + v xx d dt ( � v ( t ) � L 1 ) + εr (0) � v ( t ) � L 1 ≤ εr ( εt ) TV { u 0 } � t − ε 2 0 r ′ ( ε ( t − τ )) � v ( τ ) � L 1 dτ 10 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Integrating over t ∈ [0 , T ] and changing the order of integration � T � T � v ( T ) � L 1 + εr (0) 0 � v ( t ) � L 1 dt ≤ TV { u 0 } + ε 0 r ( εt ) dt · TV { u 0 } � T − ε 0 [ r ( ε ( T − τ )) − r (0)] � v ( τ ) � L 1 dτ. Thus, � t � � � v ( t ) � L 1 + ε 0 r ( ε ( t − τ )) � v ( τ ) � L 1 dτ ≤ TV { u 0 }· 1 + � r � L 1 [0 , + ∞ ) . Similarly, if w = u t , one can establish the L 1 time dependence estimate. � 11 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Theorem 3. (G.-Q. Chen, Christoforou). Uniqueness and Stability in L 1 . Let the resolvent kernel r ( t ) associated with k be a nonnegative and non-increasing function in L 1 ( R + ) . Let u, v ∈ BV ( R 2 + ) be entropy solutions to (11) with initial data u 0 , v 0 ∈ BV ( R ) , re- spectively. Then � t � u ( t ) − v ( t ) � L 1 ( R ) + 0 r ( t − τ ) � u ( τ ) − v ( τ ) � L 1 ( R ) dτ ≤ L � u 0 − v 0 � L 1 ( R ) . That is, any entropy solution in BV to (11) is unique and stable in L 1 . As a consequence, if u 0 is only in L ∞ , not necessarily in BV ( R ) , there exists a global entropy solution u ∈ L ∞ to (11) . 12 C. Christoforou

Hyperbolic Conservation Laws with Memory July, 06 Application: The kernel is a relaxation kernel (i.e. k ν depends on a small parameter ν > 0). Theorem 4. (G-Q. Chen, Christoforou) Let u ν be the unique entropy solution to: � t � u ν t + f ( u ν ) x + 0 k ν ( t − τ )( f ( u ν ( τ ))) x dτ = 0 (15) u ν (0 , x ) = u 0 ( x ) . u ν,ε → u ν as ε → 0+ Assume that r ν is uniformly bounded in L 1 independent of ν , TV { u ν } ≤ C TV { u 0 } . � r ν � L 1 ≤ M , = ⇒ u ν → u in L 1 If k ν ( t ) ⇀ ( α − 1) δ ( t ) as ν → 0+ , = (16) ⇒ loc � u t + α ( f ( u )) x = 0 local conservation law u (0 , x ) = u 0 ( x ) . 13 C. Christoforou