Past exam of the differential equations and applications course of the University of Oxford / 2011 / B / 4

For zero viscosity, the method of characteristics gives $u=u_0(a)e^{-t}$ and $x=a+u_0(a)(1-e^{-t})$. On the sloping part of the initial profile, $u_0(a)=-2a$, so $x=a(2e^{-t}-1)$. While this map is increasing, inversion givesThe sloping interval shrinks from width one to zero while its height decreases from two to one. Its spatial derivative is $-2e^{-t}/(2e^{-t}-1)$ and diverges as $2e^{-t}-1\downarrow 0$. Thus the first shock formation by characteristic intersection occurs atFor $0\le t<t_{\ast }$ the profile consists of a left plateau, a straight descending ramp ending at zero, and a right zero plateau. At $t_{\ast }$ the ramp becomes a vertical jump. This is a damped Burgers equation; viscosity smooths that jump into a thin transition.

In the moving stretched coordinates, ${\partial }_t={\partial }_{\tau }-s^{\prime }(\tau ){\epsilon }^{-1}{\partial }_{\xi }$ and ${\partial }_x={\epsilon }^{-1}{\partial }_{\xi }$. Multiplying the damped Burgers equation by $\epsilon$ givesDiscarding the order-$\epsilon$ terms yields the leading Viscous Burgers travelling shock equation $\boxed{(u-s^{\prime })u_{\xi }=u_{\xi \xi }}$. Integrate across a nonconstant transition whose derivative vanishes at either end:Multiplication by $u$ and a further integration by parts giveThe first identity is the Rankine-Hugoniot condition, giving $s^{\prime }=(u_{-}+u_{+})/2$. Substitution into the second makes its left side $(u_{+}-u_{-}{)}^3/12$, so the viscosity-selected entropy shock has $u_{-}>u_{+}$.

After the first characteristic collapse, the shock is fed by the left constant initial state and the right zero state. Their damped values are $u_{-}=2e^{-\tau }$ and $u_{+}=0$. Therefore $s^{\prime }=e^{-\tau }$. Matching to $s(\log 2)=0$ yieldsThese matching states are consistent with the characteristic feet: the left foot is $s-2(1-e^{-\tau })=-3/2+e^{-\tau }<-1$, while the right foot is $s>0$. The formula describes the leading thin-layer position in the small-viscosity limit.