The Scalar Conservation Law
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1 The Scalar Conservation Law t + f() = 0 = conserved qantity, f() =fl d dt Z b a (t, ) d = Z b a t (t, ) d = Z b a f (t, ) d = f (t, a) f (t, b) = [inflow at a] [otflow at b] f((a)) f((b)) a b Alberto Bressan (Penn State) Scalar Conservation Laws 5 / 117
2 Weak soltions conservation eqation: t + f() = 0 qasilinear form: t + f 0 () = 0 Conservation eqation remains meaningfl for = (t, ) discontinos, in distribtional sense: ZZ t + f () ddt = 0 for all 2C 1 c Need only :, f () locally integrable Alberto Bressan (Penn State) Scalar Conservation Laws 6 / 117
3 Convergence of weak soltions t + f() = 0 Assme: n is a soltion, for every n 1, n!, f ( n )! f () in L 1 loc then ZZ ZZ t + f () ddt = lim n!1 for all 2C 1 c n t + f ( n ) ddt = 0 (no need to check convergence of derivatives) Alberto Bressan (Penn State) Scalar Conservation Laws 7 / 117
4 Scalar Eqation with Linear Fl t + f () = 0 f () = Eplicit soltion: (t, ) = ( t) t + = 0 (0, ) = () traveling wave with speed f 0 () = (0) λ t (t) Alberto Bressan (Penn State) Scalar Conservation Laws 8 / 117
5 The method of characteristics For each 0, consider the straight line t + f 0 () = 0 (0, ) = () t 7! (t, 0 ) = 0 + tf 0 ( ( 0 )) Set = ( 0 ) along this line, so that ẋ(t) =f 0 ((t, (t)). As long as characteristics do not cross, this yields a soltion: 0 = d dt (t, (t)) = t +ẋ = t + f 0 () t (t, 0) 0 Alberto Bressan (Penn State) Scalar Conservation Laws 9 / 117
6 Loss of Reglarity t + f 0 () = 0 Assme: characteristic speed f 0 () is not constant f () (0) (t) t f () Global soltions only in a space of discontinos fnctions (t, ) 2 BV Alberto Bressan (Penn State) Scalar Conservation Laws 10 / 117
7 Shocks _ + t + f() = 0 (t, ) = if < t + if > t is a weak soltion if and only if [ + ] = f ( + ) f ( ) Rankine - Hgoniot eqations [speed of the shock] [jmp in the state] = [jmp in the fl] Alberto Bressan (Penn State) Scalar Conservation Laws 11 / 117
8 Derivation of the Rankine - Hgoniot eqation ZZ t + f () ddt = 0 for all 2C 1 c 0 = v. = ZZ + [, f () div v ddt = Spp φ Ω + Ω n n + Z Z n + v ds + = =λt = n t v ds = Z + f ( + ) (t, t) dt + Z + f ( ) (t, t) dt = Z apple ( + ) (f ( + ) f ( )) (t, t) dt Alberto Bressan (Penn State) Scalar Conservation Laws 12 / 117
9 Geometric interpretation ( + ) = f ( + ) f ( ) = Z 1 0 f 0 + +(1 ) ( + ) d The Rankine-Hgoniot conditions hold if and only if the speed of the shock is = f (+ ) f ( ) + = Z 1 0 f 0 + +(1 ) d = [average characteristic speed] Alberto Bressan (Penn State) Scalar Conservation Laws 13 / 117
10 scalar conservation law: t + f () =0 λ f f () + + = f (+ ) f ( ) + = + 1 Z + f 0 (s) ds [speed of the shock] = [slope of secant line throgh, + on the graph of f ] = [average of the characteristic speeds between and + ] Alberto Bressan (Penn State) Scalar Conservation Laws 14 / 117
11 Points of approimate jmp The fnction = (t, ) hasanapproimate jmp at a point (, ) ifthere eists states 6= + and a speed sch that, calling. if < t, U(t, ) = + if > t, there holds lim! Z + Z + (t, ) U(t, ) ddt =0 t. = λ τ + ξ Theorem. If is a weak soltion to a conservation law then the Rankine-Hgoniot eqations hold at each point of approimate jmp. Alberto Bressan (Penn State) Scalar Conservation Laws 15 / 117
12 Weak soltions can be non-niqe Eample: a Cachy problem for Brgers eqation t +( 2 /2) = 0 (0, ) = Each 2 [0, 1] yields a weak soltion 1 if 0 0 if < 0 (t, ) = 8 < : 0 if < t/2 if t/2 apple < (1 + )t/2 1 if (1 + )t/2 1 α t = 0 = α t /2 = α = Alberto Bressan (Penn State) Scalar Conservation Laws 16 / 117
13 Stability conditions for shocks Pertrb the shock with left and right states, + by inserting an intermediate state 2 [, + ] Initial shock is stable () [speed of jmp behind] [speed of jmp ahead] f ( ) f ( ) f ( + ) f ( ) + + _ * _ * + Alberto Bressan (Penn State) Scalar Conservation Laws 17 / 117
14 speed of a shock = slope of a secant line to the graph of f f f * + + * Stability conditions: when when < + the graph of f shold remain above the secant line > +, the graph of f shold remain below the secant line Alberto Bressan (Penn State) Scalar Conservation Laws 18 / 117
15 The La admissibility condition t admissible t not admissible A shock connecting the states, +,travellingwithspeed = f (+ ) f ( ) + is admissible if f 0 ( ) f 0 ( + ) i.e. characteristics do not move ot from the shock from either side Alberto Bressan (Penn State) Scalar Conservation Laws 19 / 117
16 Eistence of soltions Cachy problem: t + f () = 0, (0, ) = ū() Polygonal approimations of the fl fnction (Dafermos, 1972) Choose a piecewise a ne fnction f n sch that f n () =f () = j 2 n, j 2 Z Approimate the initial data with a fnction ū n : R 7! 2 n Z f n n f f n Alberto Bressan (Penn State) Scalar Conservation Laws 20 / 117
17 Front tracking approimations piecewise constant approimate soltions: n (t, ) ( n ) t + f n ( n ) = 0 n (0, ) =ū n () t n Tot.Var.( n (t, )) apple Tot.Var.(ū n ) apple Tot.Var.(ū) =) as n!1, a sbseqence converges in L 1 loc ([0, T ] R) to a weak soltion = (t, ) Alberto Bressan (Penn State) Scalar Conservation Laws 21 / 117
18 Acontractivesemigropofentropyweaksoltions t + f () = 0 Two initial data in L 1 (R): 1 (0, ) =ū 1 (), 2 (0, ) =ū 2 () L 1 - distance between soltions does not increase in time: k 1 (t, ) 2 (t, )k L 1 (R) apple kū 1 ū 2 k L 1 (R) (not tre for the L p distance, p > 1) Alberto Bressan (Penn State) Scalar Conservation Laws 22 / 117
19 The L 1 distance between continos soltions remains constant f () 1(0) (0) 2 (t) 1 (t) 2 Alberto Bressan (Penn State) Scalar Conservation Laws 23 / 117
20 The L 1 distance decreases when a shock in one soltion crosses the graph of the other soltion f () (0) 1 (0) 2 (t) 1 (t) 2 Alberto Bressan (Penn State) Scalar Conservation Laws 24 / 117
21 ArelatedHamilton-Jacobieqation t + f () = 0 (0, ) = ū() U(t, ) = Z 1 (t, y) dy U t + f (U ) = 0 U(0, ) = U() = Z 1 ū(y) dy f conve =) U = U(t, ) is the vale fnction for an optimization problem Alberto Bressan (Penn State) Scalar Conservation Laws 25 / 117
22 Legendre transform 7! f () 2 R [{+1} conve f (p). = ma{p f ()} p f *(p) f() 0 0 η p Alberto Bressan (Penn State) Scalar Conservation Laws 26 / 117
23 Arepresentationformla U t + f (U ) = 0 U(0, ) = U() U(t, ) = inf z( ) Z t f (ż(s)) ds + U(z(0)) ; z(t) = 0 n = min tf y2r t y o + U(y) t (t,) z( ) y Alberto Bressan (Penn State) Scalar Conservation Laws 27 / 117
24 Ageometricconstrction U t + f (U ) = 0 define h(s) =. Tf s T _ U() U(0, ) = U() U(T,) f * 0 h U(T, ) = inf y n o U(y) h(y ) Alberto Bressan (Penn State) Scalar Conservation Laws 28 / 117
25 The La formla Cachy problem: t + f () = 0, (0, ) = ū() For each t > 0, and all bt at most contably many vales of 2 R, there eists a niqe y(t, ) s.t. n y(t, ) = argmin tf y Z y o + ū(s) ds y2r t 1 the soltion to the Cachy problem is (t, ) = (f 0 ) 1 y(t, ) t (1) Alberto Bressan (Penn State) Scalar Conservation Laws 29 / 117
26 t f() (t,) ω y(t,) n y(t, ) = argmin tf y2r t y Z y o + ū(s) ds 1 define the characteristic speed. = y(t, ) t if f 0 (!) = then (t, ) =! Alberto Bressan (Penn State) Scalar Conservation Laws 30 / 117
27 Initial-Bondary vale problem t + f () = 0 (0, ) = ū() > 0 (t, 0) = b(t) t > 0 t P. Le Floch, Eplicit formla for scalar non-linear conservation laws with bondary condition, Math. Models Appl. Sci. (1988) Alberto Bressan (Penn State) Scalar Conservation Laws 31 / 117
28 Systems of Conservation Laws 8 f 1( 1,..., n ) f n( 1,..., n ) = 0 t + f () =0 =( 1,..., n ) 2 R n conserved qantities f =(f 1,...,f n ):R n 7! R n fles Alberto Bressan (Penn State) Scalar Conservation Laws 32 / 117
29 Hyperbolic Systems t + f () = 0 = (t, ) 2 R n t + A() = 0 A() =Df () The system is strictly hyperbolic if each n n matri A() hasrealdistinct eigenvales 1() < 2 () < < n () right eigenvectors r 1 (),...,r n () (colmn vectors) left eigenvectors l 1 (),...,l n () (row vectors) Ar i = i r i l i A = i l i 1 if i = j Choose bases so that l i r j = 0 if i 6= j Alberto Bressan (Penn State) Scalar Conservation Laws 33 / 117
30 Alinearhyperbolicsystem t + A =0 (0, ) = () 1 < < n eigenvales r 1,...,r n eigenvectors Eplicit soltion: linear sperposition of travelling waves (t, ) = X i i( it)r i i (s) =l i (s) 1 2 Alberto Bressan (Penn State) Scalar Conservation Laws 34 / 117
31 Nonlinear e ects - 1 t + A() =0 eigenvales depend on =) waves change shape (0) (t) Alberto Bressan (Penn State) Scalar Conservation Laws 35 / 117
32 Nonlinear e ects - 2 eigenvectors depend on =) nontrivial wave interactions t t linear nonlinear Alberto Bressan (Penn State) Scalar Conservation Laws 36 / 117
33 Global soltions to the Cachy problem t + f () = 0 (0, ) = ū() Constrct a seqence of approimate soltions m Show that (a sbseqence) converges: m! in L 1 loc =) is a weak soltion 1 2 ν Need: a-priori bond on the total variation (J. Glimm, 1965) Alberto Bressan (Penn State) Scalar Conservation Laws 37 / 117
34 Bilding block: the Riemann Problem t + f () =0 (0, ) = if < 0 + if > 0 B. Riemann 1860: 2 2 system of isentropic gas dynamics P. La 1957: n n systems (+ special assmptions) T. P. Li 1975 n n systems (generic case) S. Bianchini 2003 (vanishing viscosity limit for general hyperbolic systems, possibly non-conservative) invariant w.r.t. symmetry: (t, ). = ( t, ) >0 Alberto Bressan (Penn State) Scalar Conservation Laws 38 / 117
35 Riemann Problem for Linear Systems t + A = 0 (0, ) = if < 0 + if > 0 t / t = λ 2 / t = λ 1 ω 1 ω 2 / t = λ 3 ω = 0 ω = nx + = c j r j (sm of eigenvectors of A) j=1. X intermediate states :! i = + c j r j i-th jmp:! i! i 1 = c i r i travels with speed i Alberto Bressan (Penn State) Scalar Conservation Laws 39 / 117 japplei
36 General soltion of the Riemann problem: concatenation of elementary waves t ω ω 2 1 ω 0 = ω = Alberto Bressan (Penn State) Scalar Conservation Laws 40 / 117
37 Constrction of a seqence of approimate soltions Glimm scheme: piecing together soltions of Riemann problems on a fied grid in the t- plane t 2 t * * θ = 1/3 2 t * * * θ = 1/ Alberto Bressan (Penn State) Scalar Conservation Laws 41 / 117
38 Front tracking scheme: piecing together piecewise constant soltions of Riemann problems at points where fronts interact t t 4 t 3 β α t 2 t 1 σ σ 0 Alberto Bressan (Penn State) Scalar Conservation Laws 42 / 117
39 Eistence of soltions t + f () =0, (0, ) =ū() Theorem (Glimm 1965). Assme: system is strictly hyperbolic (+ some technical assmptions) Then there eists >0 sch that, for every initial condition ū 2 L 1 (R; R n ) with Tot.Var.(ū) apple, the Cachy problem has an entropy admissible weak soltion = (t, ) defined for all t 0. Alberto Bressan (Penn State) Scalar Conservation Laws 43 / 117
40 Uniqeness and continos dependence on the initial data t + f () = 0 (0, ) =ū() Theorem (A.B.- R.Colombo, B.Piccoli, T.P.Li, T.Yang, ). For every initial data ū with small total variation, the front tracking approimations converge to a niqe limit soltion :[0, 1[ 7! L 1 (R). The flow map (ū, t) 7! (t, ). = S t ū is a niformly Lipschitz semigrop: S 0 ū =ū, S s (S t ū)=s s+t ū S t ū S s v L 1 apple L kū vk L 1 + t s for all ū, v, s, t 0 Theorem (A.B.- P. LeFloch, M.Lewicka, P.Goatin, ). Any entropy weak soltion to the Cachy problem coincides with the limit of front tracking approimations, hence it is niqe Alberto Bressan (Penn State) Scalar Conservation Laws 44 / 117
41 Vanishing viscosity approimations Claim: weak soltions of the hyperbolic system t + f () = 0 can be obtained as limits of soltions to the parabolic system " t + f ( " ) = " " letting the viscosity "! 0+ ε Alberto Bressan (Penn State) Scalar Conservation Laws 45 / 117
42 Theorem (S. Bianchini, A. Bressan, Annals of Math. 2005) Consider a strictly hyperbolic system with viscosity t + A() = " (0, ) =ū(). (CP) If Tot.Var.{ū} is s ciently small, then (CP) admits a niqe soltion " (t, ) =S t " ū, definedforallt 0. Moreover Tot.Var. S t " ū apple C Tot.Var.{ū}, (BV bonds) S " t ū S " t v L 1 apple L kū vk L 1 (L 1 stability) (Convergence) If A() =Df (), then as "! 0, the viscos soltions " converge to the niqe entropy weak soltion of the system of conservation laws t + f () = 0 Alberto Bressan (Penn State) Scalar Conservation Laws 46 / 117
43 Main open problems Global eistence of soltions to hyperbolic systems for initial data ū with large total variation Eistence of entropy weak soltions for systems in several space dimensions Alberto Bressan (Penn State) Scalar Conservation Laws 47 / 117
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