3.3 Non-linear hyperbolic systems of conservation laws

Let us consider the system of $p$ equations of conservation laws

$$\frac{\partial {\bf u}}{\partial t} + \sum_{j=1}^{d} \frac{\... ...\bf u})}{\partial x_j} = 0 \left(= {\bf s}({\bf u})\right)$$ (83)

with ${\bf x} = (x_1,x_2,\ldots,x_d)\in R^d$ and where ${\bf u} = (u_1,u_2,\ldots,u_p)^T$ is the vector of unknowns, ${\bf u} = {\bf u}({\bf x},t)$ and ${\bf f}_j({\bf u}) = (f_{1j},f_{2j},\ldots,f_{pj})^T$ is the vector of fluxes.

Formally, system (84) expresses the conservation of the vector ${\bf u}$. Let $D$ be an arbitrary domain of $R^d$ and let ${\bf n} = (n_1,\ldots,n_d)$ the outward unit normal to the boundary $\partial D$ of $D$. Then, from (84), it follows that

$$\frac{d}{dt} \int_D {\bf u} d{\bf x} + \sum_{j=1}^d \int_{\partial D} {\bf f}_j({\bf u}) n_j dS = 0.$$ (84)

This balance equation establishes that the time variation of $$\int_D {\bf u} d{\bf x}$$ is equal to the losses through the boundary $\partial D$.

Now, for all $j=1,\ldots,d$ let

$${\bf A}_j({\bf u}) = \frac{\partial {\bf f}_j({\bf u})}{\partial {\bf u}}$$ (85)

be the Jacobian matrix of ${\bf f}_j({\bf u})$.

Definition: The system (84) is called hyperbolic if, for any ${\bf u}$ and any ${\bf\omega} = (\omega_1, \ldots,\omega_d) \in R^d$, the matrix

$${\bf A}({\bf u},{\bf\omega}) = \sum_{j=1}^d \omega_j {\bf A}_j({\bf u})$$ (86)

has $p$ real eigenvalues $\lambda_1({\bf u},{\bf\omega}) \leq \lambda_2({\bf u},{\bf\omega}) \leq \cdots \leq \lambda_p({\bf u},{\bf\omega})$ and $p$ linearly independent (right) eigenvectors ${\bf r}_1({\bf u},{\bf\omega})$, ${\bf r}_2({\bf u},{\bf\omega})$, $\ldots$, ${\bf r}_p({\bf u},{\bf\omega})$. If, in addition, the eigenvalues $\lambda_k({\bf u},{\bf\omega})$ are all different, the system (84) is called strictly hyperbolic.

In most situations we will consider the so-called initial value problem (IVP), i.e., the solution of system (84) with the initial condition

$${\bf u}({\bf x},0) = {\bf u}_0({\bf x}).$$ (87)

A key property of hyperbolic systems is that features in the solution propagate at the characteristic speeds given by the eigenvalues of the Jacobian matrices. The characteristic curves associated to system (84) are the integral curves of the differential equations

$$\frac{dx}{dt} = \lambda_k({\bf u}(x,t)), \mbox{$k=1,\ldots,p$},$$ (88)

($d=1$). The so-called characteristic variables (a combination of the components of ${\bf u}$) are constant along these curves. Essentially, characteristic curves give information about the propagation of the initial data, which formally permits to reconstruct the future solution for the initial value problem (84) with (88) at $t>0$.

Continuous and differentiable solutions that satisfy (84) and (88) pointwise are called classical solutions. However, for non-linear systems, classical solutions do not exist in general even when the initial condition ${\bf u}_0$ is a smooth function, and discontinuities develop after a finite time.

Then we seek generalized solutions that satisfy the integral form of the conservation system (85) which are classical solutions where they are continuous and have a finite number of discontinuities (weak solutions)