Conservation Laws in Continuum Mechanics

Abstract

A general fundamental mathematical framework at the base of the conservation laws of continuum mechanics is introduced. The notions of weak solutions, and the issues related to the entropy criteria are discussed in detail. The spontaneous creation of singularities, and the occurrence of diffusive limits are explained in view of their physical implications. A particular emphasis is given to the applications of hyperbolic conservation laws in the models of gas dynamics, nonlinear elasticity and traffic flows.

Appendix: BV Functions

In this section we collect some elementary facts about functions with bounded variations since their relevance in the study of conservation laws.

Definition 6.4.1

Let $I\subset \mathbb {R}$ be an interval and let $u:I\rightarrow \mathbb {R}$. The total variation of f over I is defined by

$$\displaystyle \begin{aligned} TV(u)=\sup\sum_{k=0}^q\vert u(t_{k+1})-u(t_{k})\vert \end{aligned} $$

(6.75)

where the supremum is taken over all finite sequences t₀ < …. < t_q so that t_i ∈ I, for every i. The function u is said to be of bounded variation on I, in symbol u ∈ BV (I), if TV (u) < ∞. It is easy to verify that the sum of two functions of bounded variations is also of bounded variation. Before proving the converse, let us introduce the notation V_u(a;x) to denote the total variation of the function u on the interval (a, x). Observe that if u is of bounded variation on [a, b] and x ∈ [a, b], then

$$\displaystyle \begin{aligned}\vert u(x)-u(a)\vert\leq V_u(a;x)\leq V_u(a;b)=TV(u).\end{aligned}$$

Theorem 6.4.2

If u is a function of bounded variation on [a, b], then u can be written as

$$\displaystyle \begin{aligned}u=u_1-u_2\end{aligned}$$

where u ₁ and u ₂ are nondecreasing functions.

Proof

Let x₁ < x₂ ≤ b and let a = t₀ < t₁ < … < t_k = x₁. Then

$$\displaystyle \begin{aligned}V_u(x_2)\geq \vert u(x_2)-u(x_1)\vert +\sum_{i=1}^k\vert u(t_i)-u(t_{i-1})\vert.\end{aligned}$$

Since by definition

$$\displaystyle \begin{aligned} V_u(x_1)=\sup\sum_{i=1}^k\vert u(t_i)-u(t_{i-1})\vert\end{aligned}$$

over all the sequences a = t₀ < t₁ < …t_k = x₁, we get

$$\displaystyle \begin{aligned}V_u(x_2)\geq \vert u(x_2)-u(x_1)\vert +V_u(x_1).\end{aligned}$$

Therefore

$$\displaystyle \begin{aligned}V_u(x_2)-u(x_2)\geq V_u(x_1)-u(x_1), \:\:\:V_u(x_2)+u(x_2)\geq V_u(x_1)+u(x_1).\end{aligned}$$

Hence V_u − u and V_u + u are nondecreasing functions. The claim follows by taking

$$\displaystyle \begin{aligned}u_1=\frac{1}{2}(V_u+u),\:\:\: u_2=\frac{1}{2}(V_u-u).\end{aligned}$$

□

Theorem 6.4.3

Let u be a function of bounded variation on [a, b]. Then u is Borel measurable and has at most a countable number of discontinuities. Moreover, the following statements hold true

• (i) u′ exists a.e. on [a, b];

• (ii) u′ is Lebesgue measurable;

• (iii) for a.e. x ∈ [a, b]

  $$\displaystyle \begin{aligned} \vert u'(x)\vert=V_{u}^{\prime}(x); \end{aligned}$$
• (iv)
  $$\displaystyle \begin{aligned} \int_a^b\vert u'(x)\vert\,dx\leq V_u(b); \end{aligned}$$

• (v) if u is nondecreasing on [a, b], then

  $$\displaystyle \begin{aligned} \int_a^b u'(x)\,dx\leq u(b)-u(a). \end{aligned}$$

The following theorem due to Helly is a fundamental result in the theory of bounded variation functions.

Theorem 6.4.4

Let $u_n:[a,b]\rightarrow \mathbb {R}$ be a sequence of functions satisfying the condition

$$\displaystyle \begin{aligned} \sup_n TV(u_n) <\infty. \end{aligned} $$

(6.76)

Then there exists a subsequence, still denoted by u_nand a function u of bounded variation such that u_n(x) → u(x) as n →∞ for every x ∈ [a, b] and

$$\displaystyle \begin{aligned} TV(u)\leq\liminf_n\,TV(u_n). \end{aligned} $$

(6.77)