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2018 | OriginalPaper | Buchkapitel

Lagrangian Representation for Systems of Conservation Laws: An Overview

verfasst von : Stefano Modena

Erschienen in: Theory, Numerics and Applications of Hyperbolic Problems II

Verlag: Springer International Publishing

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Abstract

We present an overview on some recent works in collaboration with S. Bianchini (see Bianchini and Modena in Lagrangian representation for solution to general systems of conservation laws [9] and the Ph.D. thesis Modena in Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for hyperbolic systems of conservation laws [15]), in which we propose a way to describe BV solutions to hyperbolic systems of conservation laws in one space dimension from a Lagrangian point of view.

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Vorheriges Kapitel A Numerical Approach of Friedrichs’ Systems Under Constraints in Bounded Domains

Nächstes Kapitel Kinematical Conservation Laws in Inhomogeneous Media

If A, B are sets, $\mathscr {A}, \mathscr {B}$ are $\sigma $-algebras on A, B, respectively, and $f: A \rightarrow B$ is a measurable function, then for any measure $\mu $ on $(A, \mathscr {A})$, the push-forward $f_\sharp \mu $ is the measure on $(B, \mathscr {B})$, defined by $f_\sharp \mu (E) = \mu (f^{-1}(E))$ for any $E \in \mathscr {B}$.

By entropic solution, we mean a solution obtained as limit of vanishing viscosity approximations; see [3].

See [11] for the definition of genuinely nonlinear or linearly degenerate characteristic fields. Roughly speaking, it amounts to say that the flux F has some strong convexity property.

The Vol’pert’s rule (see, for instance, [1, Theorem 3.96]) is the chain rule for the derivative of the composition F(u(x)) of a Lipschitz function F with a BV function u.

This can be done, for instance, in the following way. Assume for simplicity $u^q(t, \cdot )$ is right continuous. Set $\bar{U}^q(x) := \text { Tot.Var.}(\bar{u}^q; (-\infty , x])$. Set $W^q:= (0, \text { Tot.Var.}(\bar{u}^q)]$,

$$\begin{aligned} \mathtt X^q(0, w) := (\bar{U}^q)^{-1}(w), \quad \rho ^q(w) := {\left\{ \begin{array}{ll} 1 &{} \text {if } {u^q} \text { has a positive jump at } \mathtt X^q(0, w), \\ -1 &{} \text {if } {u^q} \text { has a negative jump at }\mathtt X^q(0, w). \end{array}\right. } \end{aligned}$$

Set also for simplicity $\mathtt u^q(w) := \int _0^w \rho ^q(w') dw'$. Denote by $\{(t_j, x_j)\}_j$ the points in the (t, x)-plane where two wavefronts in $u^q$ collide (the discontinuity points at $t=0$ are treated as collision points). By recursion, assume $\mathtt X^q(t, \cdot )$ is defined on $[0, t_j]$ and let us define it on $(t_j, t_{j+1}]$. Assume that at $(t_j, x_j)$ the outgoing Riemann problem is $(u^L, u^R)$ with $u^L < u^R$ (the case $u^R < u^L$ is completely similar). Set $\mathtt A(w) := \min \{\max \{ \mathtt u^q(w') \ | \ w' \le w\}, u^R\}$ for any $ w \in \mathtt X^q(t_j)^{-1}(x_j) $ and then

$$\begin{aligned} \mathtt X^q(t, w) := x_j + \bigg [\frac{d \mathrm {conv}_{[u^L, u^R]} F^q}{du}( \mathtt A(w))\bigg ] (t - t_j) \quad \text { for any } w \in \mathtt X^q(t_j)^{-1}(x_j) \text { and any } t \in (t_j, t_{j+1}]. \end{aligned}$$

If $s_k <0$ the convex envelope ${{\mathrm{conv}}}_{[0, s_k]} f_k(\tau )$ must be substituted by the concave envelope ${{\mathrm{conv}}}_{[s_k, 0]} f_k(\tau )$.

L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 2000)

S. Bianchini, Interaction estimates and Glimm functional for general hyperbolic systems. DCDS-A 9(1), 133–166 (2003)MathSciNetCrossRef

S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005)MathSciNetCrossRef

S. Bianchini, E. Marconi, On the concentration of entropy for scalar conservation laws. DCDS-S 9(1), 73–88 (2016)MathSciNetMATH

S. Bianchini, E. Marconi, On the structure of $L^\infty $ solutions to scalar conservation laws in one-space dimension. Preprint SISSA (2016)

S. Bianchini, S. Modena, On a quadratic functional for scalar conservation laws. J. Hyperb. Differ. Equ. 11(2), 355–435 (2014)MathSciNetCrossRef

S. Bianchini, S. Modena, Quadratic interaction functional for systems of conservation laws: a case study. Bull. Inst. Math. Acad Sinica (New Series) 9(3), 487–546 (2014)

S. Bianchini, S. Modena, Quadratic interaction functional for general systems of conservation laws. Comm. Math. Phys. 338, 1075–1152 (2015)MathSciNetCrossRef

S. Bianchini, S. Modena, Lagrangian representation for solution to general systems of conservation laws. In preparation (2016)

10.

S. Bianchini, L. Yu, Structure of entropy solutions to general scalar conservation laws in one space dimension. J. Math. Anal. Appl. 428(1), 356–386 (2015)MathSciNetCrossRef

11.

A. Bressan, Hyperbolic systems of conservation laws The one dimensional Cauchy problem (Oxford University Press, 2000)

12.

C. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107, 127–155 (1989)MathSciNetCrossRef

13.

R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 130, 312–366 (1989)MathSciNetMATH

14.

T.-P. Liu, The deterministic version of the Glimm scheme. Comm. Math. Phys. 57, 135–148 (1977)MathSciNetCrossRef

15.

S. Modena, Interaction functionals, Glimm approximations and Lagrangian structure of $BV$ solutions for hyperbolic systems of conservation laws. Ph.D. Thesis, Sissa digital library, 2015

Titel: Lagrangian Representation for Systems of Conservation Laws: An Overview
verfasst von: Stefano Modena
Verlag: Springer International Publishing
Buch: Theory, Numerics and Applications of Hyperbolic Problems II
Print ISBN: 978-3-319-91547-0

Electronic ISBN: 978-3-319-91548-7

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-91548-7_26

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