Burgers equation. 3

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9.3. Burgers equation. Shock formation

  1. Shock
  2. After shock wave formation
  3. Near shock wave formation

Shock

Recall that we consider Burgers equation \begin{align} &u_t+uu_x=\varepsilon u_{xx},\qquad t>0, -\infty < x<\infty \label{eq-9.3.1}\\ &u|_{t=0}=f(x). \label{eq-9.3.2} \end{align} assuming that

Condition 1.

$f(x)$ is a smooth function. and $f'(x)$ has at $x^*$ a non-degenerate negative minimum \begin{gather} f'(x^*)<0, \qquad f''(x ^*)=0, \qquad f'''(x^*) >0. \label{eq-9.3.3} \end{gather} For simplicity we assume that it is an only global minimum.

As $\varepsilon=0$ the proper solution has a shock $x=X(t)$ with $t\ge \bar{t}$, \begin{gather} \bar{t}= -\frac{1}{f'(x^*)}, \label{eq-9.3.4} \end{gather} described by (9.2.7)--(9.2.8) \begin{gather} \frac{dX}{dt}=\frac{1}{2}\bigl(f(z_-(X,t)) + f(z_+(X,t))\bigr), \label{eq-9.3.5}\\ X(0)=\bar{x}:=x^* + \bar{t} f(x^*) \label{eq-9.3.6} \end{gather} where $z_\pm (x,t)$ is a solution of (9.2.9) \begin{gather} x= z +tf(z), \label{eq-9.3.7}\\ z_\pm \gtrless x^* \label{eq-9.3.8} \end{gather} and one can prove that this equation has exactly one solution in $(-\infty,x^*)$ and one solution in $(x^*, \infty)$ as $t> \bar{t}$ while for $0< t \le \bar{t}$ there is just one solution at all.

Further, as $t> \bar{t}$ $u(x,t)=f (z_\pm (x,t))$ for $x\gtrless X(t)$.

Remark 1.

One can prove that \begin{gather} \frac{1}{2}\bigl(u_-(t)-u_+(t)\bigr) \sim -\sqrt{\frac{t-\bar{t}}{2f'''(x^*)}} f'(x^*) \qquad\text{as } t>\bar{t}. \end{gather}

After shock wave formation

The same construction as in Subsection 9.2.1 works for $\bar{t}+ T < t < T^*$ where $T^* >0$ is a sufficiently small constant and $T>0$ is an arbitrarily small constant. Formally the width of the inner zone is $(t-\bar{t})^{-\frac{1}{2}}\varepsilon $ due to Remark 1.

Without loss of the generality one can assume that $v:= \frac{1}{2}(u_+ + u_-)=0$ (one can reach it by change of variable $x\mapsto v -vt$).

Now scaling $t\mapsto (t-\bar{t})T^{-1}$, $x\mapsto (x-\bar{x}) T^{-\frac{3}{2}}$, $u\mapsto u T^{-\frac{1}{2}}$ we get $\varepsilon \mapsto \varepsilon ' := \varepsilon T^{-\frac{3}{2}}$ and therefore we arrive to

Remark 2.

  1. The same construction as in Subsection 9.2.1 works for \begin{gather} \bar{t} + T < t < \bar{t}+ T^* \qquad \text{with } \ T := \varepsilon ^{\frac{2}{3}-\delta}. \label{eq-9.3.10} \end{gather} In the original coordinates the width of the inner zone is $ T^{-\frac{1}{2}}\varepsilon ^{1-\delta}$.
  2. $u_\varepsilon \sim u_0$ for $t< \bar{t} -T$.

Near shock wave formation

Therefore we need to cover near shock wave formation \begin{gather} \bar{t} - T < t < \bar{t}+ T \qquad \text{with } \ \qquad T := \varepsilon ^{\frac{2}{3}-\delta}. \label{eq-9.3.11} \end{gather} The width of the inner zone is expected to be $\varepsilon ^{\frac{2}{3}(1-\delta)}$.

So, we get a zone $\{(x,t)\colon |x-\bar{x}| \le \varepsilon ^{\frac{2}{3}-\delta}, |t-\bar{t}| \le \varepsilon ^{\frac{2}{3}-\delta}\}$ which is a inner zone (rank 2) while $\{(x,t)\colon |x-\bar{x}| \le \varepsilon T^{-\frac{1}{2}-\delta}, |t-\bar{t}| \ge \varepsilon ^{\frac{2}{3}-\delta}\}$ is a prvious inner zone and the rest is an outer zone. This is an example of the hierarchy of inner zones.

For more details and for more general equation see

On the asymptotics of the solution of a problem with a small parameter, A. M. Il'in, Mathematics of the USSR-Izvestiya, 1990, Volume 34, Issue 2, 261–279


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