3.4.1: Ejemplos

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Ejemplo 3.4.1.1: Ecuaciones de Navier-Stokes

El sistema Navier-Stokes para un líquido viscoso incompresible es
\ 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*}
donde
$\rho$ es la densidad (constante y positiva) del líquido,
$\gamma$ es la viscosidad (constante y positiva) del líquido, vector de
$v=v(x,t)$ velocidad de las partículas líquidas,$x\in\mathbb{R}^3$ o en$\mathbb{R}^2$,
$p=p(x,t)$ presión.

El problema es encontrar soluciones$v,\ p$ al sistema anterior.

Ejemplo 3.4. 2.1: Elasticidad lineal

Considera el sistema
\ begin {ecuación}
\ label {elast}
\ rho\ frac {\ parcial^2u} {\ parcial t^2} =\ mu\ triangle_x u+ (\ lambda+\ mu)\ nabla_x (\ text {div} _x\ u) +f.
\ end {ecuación}
Aquí está, en el caso de un cuerpo elástico en$\mathbb{R}^3$,
$u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$vector de desplazamiento,
$f(x,t)$ densidad de fuerza externa, densidad
$\rho$ (constante), constantes de Lamé
$\lambda,\ \mu$ (positivas).

La ecuación característica es$\det C=0$ donde las entradas de la matriz$C$ están dadas por
$$
c_ {ij} =(\ lambda+\ mu)\ chi_ {x_i}\ chi_ {x_j} +\ delta_ {ij}\ left (\ mu|\ nabla_x\ chi|^2-\ rho\ chi_t^2\ right).
$$
La ecuación característica es
$$
\ left ((\ lambda+2\ mu) |\ nabla_x\ chi|^2-\ rho\ chi_t^2\ right)\ left (\ mu|\ nabla_x\ chi|^2-\ rho\ chi_t^2\ right) ^2=0.
$$ De
ello se deduce que son posibles dos velocidades diferentes$P$ de superficies características$\mathcal{S}(t)$
$\chi(x,t)=const.$, definidas por, a saber,
$$
P_1=\ sqrt {\ frac {\ lambda+2\ mu} {\ rho}},\\\ mbox {y}\\ P_2=\ sqrt {\ frac {\ mu} {\ rho}}.
$$ Eso
lo recordamos$P=-\chi_t/|\nabla_x\chi|$.

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