Balance Laws as Quasidifferential Equations in Metric Spaces
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1 Balance Laws as Quasidifferential Equations in Metric Spaces Rinaldo M. Colombo 1 Graziano Guerra 2 1 Department of Mathematics Brescia University 2 Department of Mathematics and Applications Milano - Bicocca University June 12th, 2008
2 Outline 1 Balance Laws 2 Quasidifferential Equations 3 Back to Balance Laws 4 Other Applications
3 Conservation Laws t u + x f (u) = 0 u(0, x) = u o (x) t time t [0, + [ x space x R u unknown u Ω, Ω R n f hyperbolic flow f : Ω R n smooth u o initial datum u o : R R n in L 1 BV
4 Balance Laws t u + x f (u) = G[u] u(0, x) = u o (x) t time t [0, + [ x space x R u unknown u Ω, Ω R n f hyperbolic flow f : Ω R n smooth u o initial datum u o : R R n in L 1 BV G[u] source possibly non local
5 Balance Laws With Boundary t u + x f (u) = G[u] b ( u(t, γ(t) ) = g(t) u(0, x) = u o (x) t time t [0, + [ x space x R or x > γ(t) u unknown u Ω, Ω R n f hyperbolic flow f : Ω R n smooth u o initial datum u o : R R n in L 1 BV G[u] source possibly non local γ boundary γ : [0, + [ R + g boundary data g : [0, + [ R n l γ is l noncharacteristic: λ l (u) + c γ(t) λ l+1 (u) c
6 Balance Laws Radiating Gas (Vincenti Kruger, 1965) t ρ + x (ρv) = 0 t (ρv) + x (ρv 2 + p) = 0 ( ) ( ) t ρe ρv 2 + x [v ρe ρv 2 + p 2 xxq + q + x θ 4 = 0 ] + q = 0 ρ density v speed p pressure e internal energy θ temperature q heat flux
7 Balance Laws Radiating Gas (Vincenti Kruger, 1965) t ρ + x (ρv) = 0 t (ρv) + x (ρv 2 + p) = 0 ( ) ( t ρe ρv 2 + x [v ρe ρv 2 + p) ] = θ 4 + Q θ 4 ρ density v speed p pressure e internal energy θ temperature q heat flux Q kernel
8 Balance Laws Gas Networks with Compressors flow direction x2 α2(0+) Tube 2 α1(0+) x1 Tube 1 x Tube 1 Tube 2 x
9 Balance Laws Gas Networks with Compressors flow direction x2 α1(0+) x1 Tube 1 Tube 2 α2(0+) t ρ l + x q l = 0 l = 1, 2 ) (ql 2 t q l + x + p(ρ l ) = ν q l q l ρ l ρ l x Tube 1 Tube 2 gρ l sin α l x
10 Balance Laws Gas Networks with Compressors flow direction x2 α1(0+) x1 Tube 1 Tube 2 α2(0+) t ρ l + x q l = 0 l = 1, 2 ) (ql 2 t q l + x + p(ρ l ) = ν q l q l ρ l ρ l x Tube 1 Tube 2 gρ l sin α l x ρ density q linear momentum p pressure ν friction g gravity α inclination
11 Balance Laws Gas Networks with Compressors flow direction x2 α1(0+) x1 Tube 1 Tube 2 α2(0+) t ρ l + x q l = 0 l = 1, 2 ) (ql 2 t q l + x + p(ρ l ) = ν q l q l ρ l ρ l x Tube 1 Tube 2 gρ l sin α l x q 1 (t, 0+) + q 2 (t, 0+) = 0 (Mass conservation) ( ( ) (γ 1)/γ p(ρ 2 (t,0+)) q 2 (t, 0+) 1) = Π(t) (Steinbach: 2007) p(ρ 1 (t,0+)) γ from the γ-law Π compressor power
12 Balance Laws Gas Networks with Compressors Tube 2 t [ t [ ρ 1 ρ 2 q 1 q 2 ] + x [ ] + x q 1 q 2 q 2 1 b ] = 0 ρ 1 + p(ρ 1 ) q 2 2 ρ 2 + p(ρ 2 ) ρ 1 (t, 0+) ρ 2 (t, 0+) q 1 (t, 0+) q 2 (t, 0+) x = Tube 1 ν q 1 q 1 ρ 1 gρ 1 sin α 1 ν q 2 q 2 ρ l gρ 2 sin α 2 = g(t) R2
13 Quasidifferential Equations (X, d) metric space. For all u X, assign a curve exiting u at t = t o
14 Quasidifferential Equations (X, d) metric space. Assign F : [0, δ] [0, T ] X X with F(0, t o )u = u
15 Quasidifferential Equations (X, d) metric space. Assign F : [0, δ] [0, T ] X X with F(0, t o )u = u F (t, to)u u
16 Quasidifferential Equations (X, d) metric space. Assign F : [0, δ] [0, T ] X X with F(0, t o )u = u Find sufficient conditions on F (local flux) that imply the existence of a P : [0, T ] [0, T ] X X such that F (t, to)u u
17 Quasidifferential Equations (X, d) metric space. Assign F : [0, δ] [0, T ] X X with F(0, t o )u = u Find sufficient conditions on F (local flux) that imply the existence of a P : [0, T ] [0, T ] X X such that P process: P(0, t o ) = Id, P(t + s, t o ) = P(s, t o + t) P(t, t o ) F (t, to)u u
18 Quasidifferential Equations (X, d) metric space. Assign F : [0, δ] [0, T ] X X with F(0, t o )u = u Find sufficient conditions on F (local flux) that imply the existence of a P : [0, T ] [0, T ] X X such that P process: P(0, t o ) = Id, P(t + s, t o ) = P(s, t o + t) P(t, t o ) P tangent to F: lim t 0+ 1 t d ( P(t, t o )u, F(t, t o )u ) = 0 F (t, to)u P (t, to)u. u
19 Quasidifferential Equations A.I. Panasyuk. Quasididifferential equations in a metric space. Differ. Uravn., A.I. Panasyuk and D. Bentsman. Application of quasidifferential equations to the description of discontinuous processes. Differ. Uravn., J.-P. Aubin. Mutational and morphological analysis C. Calcaterra and D. Bleecker. Generating Flows on metric spaces. J. Math. Anal. Appl., 2000.
20 Quasidifferential Equations A.I. Panasyuk. Quasididifferential equations in a metric space. Differ. Uravn., A.I. Panasyuk and D. Bentsman. Application of quasidifferential equations to the description of discontinuous processes. Differ. Uravn., J.-P. Aubin. Mutational and morphological analysis C. Calcaterra and D. Bleecker. Generating Flows on metric spaces. J. Math. Anal. Appl., A common requirement is that for small t one has d ( F(t, t o )u, F(t, t o )v ) d (u, v) + Ctd (u, v) e Ct d (u, v)
21 Quasidifferential Equations Euler ε-polygonals in a metric space: t [0, ε] F ε (t, t o )u = F(t, t o )u t [ε, 2ε] F ε (t, t o )u = F(t ε, t o ) F ε (ε, t o )u t [2ε, 3ε] F ε (t, t o )u = F(t 2ε, t o ) F ε (2ε, t o )u...
22 Quasidifferential Equations Euler ε-polygonals in a metric space: t [0, ε] F ε (t, t o )u = F(t, t o )u t [ε, 2ε] F ε (t, t o )u = F(t ε, t o ) F ε (ε, t o )u t [2ε, 3ε] F ε (t, t o )u = F(t 2ε, t o ) F ε (2ε, t o )u... F ε (t, t o ) u = F (t kε, t o + kε) k 1 F(ε, t o + hε) u h=0 where n i=1 f i = f 1 f 2... f n and k =integer part of t/ε.
23 Quasidifferential Equations Euler ε-polygonals in a metric space: t [0, ε] F ε (t, t o )u = F(t, t o )u t [ε, 2ε] F ε (t, t o )u = F(t ε, t o ) F ε (ε, t o )u t [2ε, 3ε] F ε (t, t o )u = F(t 2ε, t o ) F ε (2ε, t o )u... F ε (t, t o ) u = F (t kε, t o + kε) k 1 F(ε, t o + hε) u h=0 where n i=1 f i = f 1 f 2... f n and k =integer part of t/ε. Remark: F process = F ε = F and Euler polygonal = orbit
24 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied:
25 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied: 1 F is Lipschitz;
26 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied: 1 F is Lipschitz; If F were already a process, then: ( d F ( ) ) (k + 1)t, t o u, F(kt, to + t) F(t, t o )u = 0
27 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied: 1 F is Lipschitz; The simple Lipschitz continuity would imply: ( d F ( ) ) (k + 1)t, t o u, F(kt, to + t) F(t, t o )u C(k + 1)t
28 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied: 1 F is Lipschitz; 2 the defect in the semigroup defect of F is bounded by ( d F ( ) ) (k + 1)t, t o u, F(kt, to + t) F(t, t o )u (k +1)t ω(t) with t 0 ω(s) s ds < +, for instance ω(t) = t or ω(t) = t;
29 Quasidifferential Equations Theorem (R.M. Colombo, G. Guerra: DCDS, to appear) Given a local flux F, there exists a unique Lipschitz process P tangent to F if the following three conditions are satisfied: 1 F is Lipschitz; 2 the defect in the semigroup defect of F is bounded by ( d F ( ) ) (k + 1)t, t o u, F(kt, to + t) F(t, t o )u (k +1)t ω(t) with t 0 ω(s) s ds < +, for instance ω(t) = t or ω(t) = t; 3 F defines uniformly Lipschitz continuous Euler polygonals, i.e. d ( F ε (t, t o )u, F ε (t, t o )w ) L d(u, w)
30 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property)
31 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property) 2 Lipschitz
32 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property) 2 Lipschitz 1 3 tangent to F, i.e. lim t 0+ t d ( P(t, t o )u, F(t, t o )u ) = 0
33 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property) 2 Lipschitz 1 3 tangent to F, i.e. lim t 0+ t d ( P(t, t o )u, F(t, t o )u ) = 0 Moreover, if a curve γ : [0, T ] X is: 1 Lipschitz,
34 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property) 2 Lipschitz 1 3 tangent to F, i.e. lim t 0+ t d ( P(t, t o )u, F(t, t o )u ) = 0 Moreover, if a curve γ : [0, T ] X is: 1 Lipschitz, 2 tangent at γ(t o ) to t F(t, t o )u, i.e. for all t o [0, T [ 1 lim t 0 t d ( γ(t o + t), F (t, t o )u ) = 0
35 Quasidifferential Equations P is uniquely characterized in its domain by being: 1 a process (semigroup property) 2 Lipschitz 1 3 tangent to F, i.e. lim t 0+ t d ( P(t, t o )u, F(t, t o )u ) = 0 Moreover, if a curve γ : [0, T ] X is: 1 Lipschitz, 2 tangent at γ(t o ) to t F(t, t o )u, i.e. for all t o [0, T [ 1 lim t 0 t d ( γ(t o + t), F (t, t o )u ) = 0 then, γ(t) = P(t, 0)γ(0)
36 Back to Balance Laws t u + x f (u) = G[u] u(t o, x) = u o (x)
37 Back to Balance Laws t u + x f (u) = G[u] t u + x f (u) = 0 u(t o, x) = u o (x) u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x)
38 Back to Balance Laws t u + x f (u) = 0 u(t o, x) = u o (x) t u + x f (u) = G[u] u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = S(t, t o )u o u(t) = u o + tg[u o ]
39 Back to Balance Laws t u + x f (u) = 0 u(t o, x) = u o (x) t u + x f (u) = G[u] u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = S(t, t o )u o u(t) = u o + tg[u o ]
40 Back to Balance Laws t u + x f (u) = 0 u(t o, x) = u o (x) t u + x f (u) = G[u] u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = S(t, t o )u o u(t) = u o + tg[u o ] F(t, t o )u = S(t, t o )u + tg[u]
41 Back to Balance Laws t u + x f (u) = G[u] b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = u o + tg[u o ]
42 Back to Balance Laws t u + x f (u) = 0 b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) t u + x f (u) = G[u] b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = u o + tg[u o ]
43 Back to Balance Laws t u + x f (u) = 0 b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) t u + x f (u) = G[u] b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = S(t, t o )u o u(t) = u o + tg[u o ]
44 Back to Balance Laws t u + x f (u) = 0 b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) t u + x f (u) = G[u] b ( u(t, γ(t) ) = g(t) u(t o, x) = u o (x) { t u = G[u] u(t o, x) = u o (x) u(t) = S(t, t o )u o u(t) = u o + tg[u o ] F(t, t o )u = S(t, t o )u + tg[u]
45 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for
46 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources)
47 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) C. Dafermos, L. Hsiao: Indiana Univ. Math. J., 1982 G. Crasta, B. Piccoli: Discrete and Contin. Dynam. Systems, 1997 D. Amadori, G. Guerra: Nonlinear Anal., 2002 Christoforou, C: J. Differential Equations 2006
48 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative)
49 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) R.M. Colombo, G. Guerra: Comm. P.D.E.s, 2007
50 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source
51 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source R.M. Colombo, G.Guerra: C.P.A.A., to appear
52 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source 4 balance laws with boundary and a non local source
53 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source 4 balance laws with boundary and a non local source R.M. Colombo, G. Guerra, M. Herty, V. Sachers: preprint, 2008 R.M. Colombo, G. Guerra: in preparation
54 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source 4 balance laws with boundary and a non local source In all these cases:
55 Back to Balance Laws F(t, t o )u = S(t, t o )u + tg[u] satisfies the assumption of the theorem for 1 standard balance laws (dissipative/non dissipative sources) 2 balance laws with a non local source (non dissipative) 3 balance laws with a dissipative non local source 4 balance laws with boundary and a non local source In all these cases: existence (local/global in time), uniqueness and L 1 -Lipschitz continuous dependence. with a unique approach.
56 Other Applications The Theorem in Metric spaces applies also to
57 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u)
58 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) F(t, t, o )u = u + t v(t o, u)
59 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D
60 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D ( F(t)u ) (x) = u(x) + u(x 2 t) 2u(x) + u(x + 2 t) 4
61 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D 3 Hille Yosida Theorem
62 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D 3 Hille Yosida Theorem ( ) 1 1 F(t)u = t R t, A u t > 0, u t = 0.
63 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D 3 Hille Yosida Theorem 4 nonlinear Trotter formula in a metric space
64 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D 3 Hille Yosida Theorem 4 nonlinear Trotter formula in a metric space F(t)u = S 1 t S 2 t u
65 Other Applications The Theorem in Metric spaces applies also to 1 standard o.d.e.s u = v(t, u) 2 the heat equation t u = 2 xxu in 1D 3 Hille Yosida Theorem 4 nonlinear Trotter formula in a metric space Thanks for your attention!!
66 Thanks for your attention!!
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