D. Serre Systemes de lois de conservation, I and II, Diderot, France, 1996. 26  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... p = 2 k[P; f ]uk L (R n ) kkfk Lip kuk H Finally we state here the compactness tool, due to Tartar and G erard ( 54] 13] The classical results concerning the applications of Compensated Compactness in the theory of hyperbolic systems are reported in the books of Dafermos [5] and Serre [50]. Let H, H denote separable Hilbert spaces, an open set, m 2 N. We have the following theorem taken from G erard [13] Theorem 2.4. Compensated Compactness Principle) Let P 2 OPS with principals symbol p(x; and fu k g be a bounded sequence of ; H) such that u k u. Assume ....

D. Serre, Systemes de lois de conservation. I,II Diderot Editeur, Paris, 1996.

....(w; z) w) 0 ; w; z) z) 0 ; for any (w) z) continuous. In particular, formulas (1. 7) imply strong convergence when both characteristic elds are genuinely nonlinear, or for the equations of elasticity with exactly one in ection point in the stress strain constitutive relation [Se 3 ] In Section 4, we derive a novel formula describing the coupling of oscillations among the two characteristic elds. We explain brie y the approach. First, from the compensated compactness bracket and the singular entropies, we derive the formula (1.8) where = w; z; is the ....

Serre D., Systemes de lois de conservation I, II. Fondations, Diderot, Paris, 1996.

....both mathematically and phenomenologically. These are partial differential equations of the type t u x J(u) 0 where u = u(t; x) takes its value in R n and J is a non linear function from R n into R n . The best known and most investigated examples are the following. See e.g. [7, 14, 15] for a comprehensive introduction and survey of the subject. 1 (1) Burgers equation (with no viscosity) n = 1 and t u x (u 2 =2) 0: 2) The isentropic gas dynamics equation in one space dimension: n = 2, the components are the density field ae(t; x) and momentum field m(t; x) ....

....get (7) under the same limiting procedure, provided that J(ae; u) aeu o(aeu) K(ae; u) ae o(ae) as ae; u 0: This indicates that (7) is valid for a wider class of microscopic systems. 4 Analysis of the PDE We are now going to see how the methods developed in the PDE literature (see [7, 14, 15]) can be applied to our system. In order to put things into perspective, we briefly recall general results and see how they can be applied in the context of our system (4) 4.1 Two component systems of hyperbolic conservation laws For a generic two component system we shall use the notation v = ....

D. Serre: Syst`emes de lois de conservation, vol. 1 and 2, Diderot Editeur, 1996.

....0 (x; k)dk (18) Where G(n; E) Z B v(k) F (n; E; k)dk: The regularity of F with respect to (n; E) as well as the assumptions on the electric field E insure the existence and uniqueness of a local in time regular solution of this equation. Indeed, by using the characteristic techniques (see [16] for instance) on can prove the following result Proposition 4.1 Let n 0 2 W 2;1 (IR d ) and E 2 W 2;1 loc (IR Theta IR d ) d . Then there exists T 0, such that the problem (18) admits a unique classical solution n 2 C 1 ( 0; T ] Theta (IR d ) Moreover, by chosing T ....

D. Serre, Syst`emes de lois de conservation. (vol. I and II) Fondations. Diderot Editeur, Paris, 1996.

....consider a weak limit u u. By a compensated compactness argument introduced by DiPerna [7] it then follows that u is actually a weak solution of the nonlinear system (1. 1) For a comprehensive discussion of the compensated compactness method and its applications to conservation laws, see [18]. 3) For n Theta n Temple class systems, a proof of the convergence of the viscous solutions u to a solution of (1.1) can be found in [17,18] 4) Assume that all characteristic fields of the system (1.1) are linearly degenerate. Then every solution with small total variation which is ....

....solution of the nonlinear system (1.1) For a comprehensive discussion of the compensated compactness method and its applications to conservation laws, see [18] 3) For n Theta n Temple class systems, a proof of the convergence of the viscous solutions u to a solution of (1. 1) can be found in [17,18]. 4) Assume that all characteristic fields of the system (1.1) are linearly degenerate. Then every solution with small total variation which is initially smooth remains smooth for all positive times [2] Clearly such solution can be obtained as limit of vanishing viscosity approximations. By a ....

D. Serre, Syst`emes de lois de conservation, Diderot Editor, Paris 1996.

....ff 0, typically f(u) u 2 =2 which corresponds to the usual inviscid Burgers equation [8] This equation generally produces shock waves in a finite time and solutions must be understood in a suitable weak sense. A good framework enforcing existence, uniqueness, L 1 stability, 7] 5] [6], is provided by the so called entropy solutions, obtained through the vanishing viscosity method by passing to the limit in the viscous approximation u t f(u) x Gamma ffiu xx = 0; 1.2) as ffi 0 tends to zero [7] Entropy solutions can be also characterized as the weak solutions of ....

Serre D., Syst`emes de Lois de Conservation. I, Diderot Editeur, Paris, 1996.

....the gradient with respect to U . A classical example of a strictly convex entropy pair for the homogeneous part of system (1) is given by: j(U) Gammanln p n fl ; q(U) Gammaj ln p n fl : 9) It can be shown that the physical entropy in (9) is strictly convex (see for instance [GR] or [S]) Furthermore, the entropy pair (9) is the unique non trivial entropy pair of the system [S] up to addition of conserved quantities and constants) We say that a weak solution (U; E) to the Cauchy problem (1) 2) 7) is an entropy solution if and only if for any entropy pair (j; q) with j ....

.... the homogeneous part of system (1) is given by: j(U) Gammanln p n fl ; q(U) Gammaj ln p n fl : 9) It can be shown that the physical entropy in (9) is strictly convex (see for instance [GR] or [S] Furthermore, the entropy pair (9) is the unique non trivial entropy pair of the system [S] (up to addition of conserved quantities and constants) We say that a weak solution (U; E) to the Cauchy problem (1) 2) 7) is an entropy solution if and only if for any entropy pair (j; q) with j convex, one has t j(U) x q(U) rj) T Delta G (U; E) in D 0 : Let us give a ....

D. Serre, Syst`emes de lois de conservation I, Diderot, Paris, 1996.

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D. Serre. Syst`emes de lois de conservation, II. Diderot arts & Sci. (1996.

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D. Serre. Syst`emes de lois de conservation, I. Diderot arts & Sci. (1996.

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D. Serre. Syst`emes de lois de conservation. Diderot, Paris, 1996.

....velocity s. Let u 0 2 U L 1 have a zero disturbance mass : Z R (u 0 Gamma U)dx = 0: Then lim t 1 ku(t) Gamma U( Delta Gamma st)k 1 = 0: The proof of this result splits in two parts, of equal importances. The first one considers the case where u 0 takes values in I(u l ; u r ) see [10]) It uses arguments of dynamical systems : compactness, limit set, Lyapunov functions and Lasalle s invariance principle. Let us note that thanks to the contraction property of the semi group, one always may restrict to a dense subset of data ; here it consists of those data which satisfy ....

D. Serre, Syst`emes de lois de conservation. Diderot, Paris, 1996.

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D. Serre Systemes de lois de conservation, I and II, Diderot, France, 1996. 26

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Serre, D. Systemes de Lois de Conservation (I)-(II), Diderot Editeur, Paris, 1996.

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D. Serre : Syst`emes de lois de conservation. I et II, Fondations, Diderot Editeur, Paris, 1996.

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