
# 3.4.1: Examples

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Example 3.4.1.1: Navier-Stokes equations

The Navier-Stokes system for a viscous incompressible liquid is
\begin{eqnarray*}
v_t+(v\cdot\nabla_x)v&=&-\frac{1}{\rho}\nabla_x p+\gamma\triangle_x v\\
\text{div}_x\ v&=&0,
\end{eqnarray*}
where
$$\rho$$ is the (constant and positive) density of liquid,
$$\gamma$$ is the (constant and positive) viscosity of liquid,
$$v=v(x,t)$$ velocity vector of liquid particles, $$x\in\mathbb{R}^3$$ or in $$\mathbb{R}^2$$,
$$p=p(x,t)$$ pressure.

The problem is to find solutions $$v,\ p$$ of the above system.

Example 3.4.2.1: Linear elasticity

Consider the system

\label{elast}
\rho\frac{\partial^2u}{\partial t^2}=\mu\triangle_x u+(\lambda+\mu)\nabla_x(\text{div}_x\ u)+f.

Here is, in the case of an elastic body in $$\mathbb{R}^3$$,
$$u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$$ displacement vector,
$$f(x,t)$$ density of external force,
$$\rho$$ (constant) density,
$$\lambda,\ \mu$$ (positive) Lamé constants.

The characteristic equation is $$\det C=0$$ where the entries of the matrix $$C$$ are given by
$$c_{ij}=(\lambda+\mu)\chi_{x_i}\chi_{x_j}+\delta_{ij}\left(\mu|\nabla_x\chi|^2-\rho\chi_t^2\right).$$
The characteristic equation is
$$\left((\lambda+2\mu)|\nabla_x\chi|^2-\rho\chi_t^2\right)\left(\mu|\nabla_x\chi|^2-\rho\chi_t^2\right)^2=0.$$
It follows that two different speeds $$P$$ of characteristic surfaces $$\mathcal{S}(t)$$, defined by
$$\chi(x,t)=const.$$, are possible, namely
$$P_1=\sqrt{\frac{\lambda+2\mu}{\rho}},\ \ \mbox{and}\ \ P_2=\sqrt{\frac{\mu}{\rho}}.$$
We recall that $$P=-\chi_t/|\nabla_x\chi|$$.

### Contributors

• Integrated by Justin Marshall.