Nonlinear hyperbolic systems in one dimension

Most equations of the form

\[ \newcommand{\U}{\mathbf{U}} \newcommand{\f}{\mathbf{f}} \newcommand{\r}{\mathbf{r}} \begin{equation} \begin{aligned} \partial_t \U + \partial_x \f(\U) &= 0 \\ \U(x, 0) &= \U_0(x) \end{aligned} \label{eq:nonlinear_hyperbolic_system} \end{equation} \]

from physics are nonlinear hyperbolic systems. We will now look at

• structural properties and
• well-posedness,

in particular, for the Riemann problem

\[ \begin{equation} \begin{aligned} \partial_t \U + \partial_x \f(\U) &= 0 \\ \U(x, 0) &= \begin{cases} \U_L, & x < 0 \\ \U_R, & x > 0, \end{cases} \end{aligned} \label{eq:riemann_problem} \tag{RP} \end{equation} \]

an determine the general form of the entropy solution. We will assume \(\U \in L_\text{loc}^1(\R \times \R_+, \mathcal U)\) for some \(\mathcal U \subset \R^m\), for which \(\eqref{eq:nonlinear_hyperbolic_system}\) makes sense.

As with the scalar case, \(\U\) can develop discontinuities in finite time, so we need to interprate \(\eqref{eq:nonlinear_hyperbolic_system}\) in the weak sense:

\[ \begin{equation} \int_{\R_+} \int_\R \U \partial_t \phi + \f(\U) \partial_x \phi \, \dd x \, \dd t + \int_\R \U_0(x) \phi(x, 0) \, \dd x = 0 \label{eq:weak_nonlinear_hyperbolic_system} \end{equation} \]

for all test functions \(\phi \in C_c^\infty(\R \times \R_+)\). Similarly, one can derive the rankine-hugoniot condition: If \(\U\) is piecewise \(C^1\) with only jump discontinuities, then \(\U\) is a weak solution of \(\eqref{eq:nonlinear_hyperbolic_system}\) if and only if it is a classical solution wherever it is \(C^1\) and satisfies

\[ \begin{equation} s = \frac{[\![\f(\U)]\!]}{[\![\U]\!]} \label{eq:rankine_hugoniot} \tag{RH} \end{equation} \]

where \(s = \gamma'(t)\) is the speed of the discontinuity. We now have \(2m + 1\) unknowns, \(\U_L\), \(\U_R\), and \(s\), but only \(m\) equations. The entropy condition will deal with the last \(m+1\) unknowns.

Structural properties

To develop existence and uniqueness results, we will need to impose some assumptions on \(\f\).

Definition (Hyperbolic system)

The system of equations \(\eqref{eq:nonlinear_hyperbolic_system}\) is called hyperbolic if the Jacobian matrix \(\f'(\U)\) is real diagonizable for all \(\U \in \mathcal U\):

\[\f'(\U) = R(\U) \Lambda(\U) R(\U)^{-1}\]

It is called strictly hyperbolic if

\[\lambda_1(\U) < \lambda_2(\U) < \ldots < \lambda_m(\U).\]

Definition

The \(j\)-th wave-family is genuinely nonlinear if for all \(\U \in \mathcal U\),

\[\nabla \lambda_j(\U) \cdot \r_j(\U) \neq 0.\]

---

It is called linearly degenerate if for all \(\U \in \mathcal U\),

\[\nabla \lambda_j(\U) \cdot \r_j(\U) = 0.\]

Example (Scalar conservation law)

For \(m=1\), the system reduces to a scalar conservation law. Then, we only have a single eigenvalue \(\lambda(U) = f'(U) \in \R\) and can choose \(r(U) = 1\). Thus, \(\eqref{eq:nonlinear_hyperbolic_system}\) is always hyperbolic.

Additionally,

\[\nabla \lambda(U) \cdot r(U) = f''(U).\]

We therefore have that it is

• genuinely nonlinear if \(f\) is strictly convex or concave,
• linearly degenerate if \(f\) is affine.

Example (Shallow water equations)

A model for water waves in a shallow body of water is given by

\[\begin{aligned} \partial_t h + \partial_x (hv) &= 0, \\ \partial_t (hv) + \partial_x \qty(\frac{1}{2} g h^2 + hv^2) &= 0 \end{aligned}\]

where the height \(h\) and the momentum \(m:=hv\) are the conserved quantities. We can write this as \(\eqref{eq:nonlinear_hyperbolic_system}\) with

\[\U = \begin{pmatrix} h \\ m \end{pmatrix}, \quad \f(\U) = \begin{pmatrix} m \\ \frac{1}{2} g h^2 + \frac{m^2}{h} \end{pmatrix}.\]

This gives the Jacobian

\[\f'(\U) = \begin{pmatrix} 0 & 1 \\ gh - \frac{m^2}{h^2} & \frac{2m}{h} \end{pmatrix},\]

and the eigenvalues and eiegenvectors

\[\lambda_\pm = v \pm c,\quad \r_\pm = \begin{pmatrix} 1 \\ v \pm c \end{pmatrix}.\]

The values if \(\U\) for which the equation makes sense are

\[\mathcal U = \qty{ (h, m) \in \R^2 \mid h > 0 }.\]

The families of waves are genuinely nonlinear, as

\[\nabla \lambda_\pm \cdot \r_\pm = \mp \frac{3}{2} \sqrt{\frac{g}{h}} \neq 0.\]

Simple solutions

From now on, we will assume that all the families of waves are either genuinely nonlinear or linearly degenerate. Additionally, if the \(j\)-th family is genuinely nonlinear, we will normalize \(\r_j\) such that

\[\nabla \lambda_j(\U) \cdot \r_j(\U) = 1 \quad \forall \U \in \mathcal U\]

We will now fix \(\U_L \in \mathcal U\) and find all \(U_R\) such that we get a rarefaction wave or a shock wave in the Riemann problem \(\eqref{eq:riemann_problem}\).

Rarefaction waves

A rarefaction wave is a smooth solution of \(\eqref{eq:nonlinear_hyperbolic_system}\) connecting \(\U_L\) and \(\U_R\). Recall that the solution is invariant under the transformation \((x, t) \mapsto (\alpha x, \alpha t)\) so it takes the form \(\newcommand{\u}{\mathbf{u}}\U(x, t) = \u(x/t)\), where \(\u \in C^1(\R, \R^m)\). Inserting into \(\eqref{eq:nonlinear_hyperbolic_system}\) gives

\[\u'(\xi) f'(u(\xi)) = \xi \u'(\xi).\]

Then, we either have \(\u'(\xi) = 0\) or \(\xi\) is an eigenvalue of the Jacobian \(f'(\u(\xi))\). In the latter case, we have that

\[\u'(\xi) = \r_j(\u(\xi)), \quad \xi = \lambda_j(\u(\xi))\]

for some \(j\). Now, if there are some \(\xi_L, \xi_R \in \R\) such that \(\u(\xi_L) = \U_L\) and \(\u(\xi_R) = \U_R\), they must be \(\xi_L = \lambda_j(\U_L)\) and \(\xi_R = \lambda_j(\U_R)\). The rarefaction solution of \(\eqref{eq:riemann_problem}\) is then given by

\[\U(x, t) = \left\{\begin{aligned} \U_L, &&& \frac{x}{t} < \lambda_j(\U_L) \\ \u\qty(\frac{x}{t}), && \lambda_j(\U_L) < &\frac{x}{t} < \lambda_j(\U_R) \\ \U_R, && \lambda_j(\U_R) < & \frac{x}{t} \end{aligned}\right.\]