Riemann invariant

For one-dimensional homentropic ideal-gas flow,is constant along the characteristic $dx/dt=u\pm c$.

For $U_x+A(U)U_y=(1,\sqrt{v}{)}^T$ with $A=\left(\begin{array}{cc}u& 2\\ 2v& u\end{array}\right)$ and $v>0$, the variables $R=u-2\sqrt{v}$ and $S=u+2\sqrt{v}$ decouple into an unforced and a uniformly forced Inviscid Burgers equation. Smooth data for $u,v$ need not be smooth data for these variables when $v$ vanishes. For example $u(0,y)=2y$, $v(0,y)=y^2$ gives $R_0=4\min (y,0)$ and $S_0=4\max (y,0)$. Their derivative corners travel along different characteristic families. In the positive quadrant, $R=0$ and $S=2x$ below $y=x^2$, whereas $S=2(x+2x^2+2y)/(1+4x)$ above. Reconstructing $u=(R+S)/2$, $v=(S-R{)}^2/16$ gives a continuous piecewise classical continuation whose $y$ derivative jumps at the parabola. Treating $\sqrt{y^2}$ as $y$ for all real $y$ would miss this obstruction. The degeneracy at zero density and the regularity class are essential parts of the Cauchy problem.

For a smooth hyperbolic system $U_t+A(U)U_x=S(U)$, suppose $\nabla R$ is a left eigenvector of $A$ with eigenvalue $\lambda$. The chain rule gives $({\partial }_t+\lambda {\partial }_x)R=\nabla R\cdot S$. Hence the homogeneous invariant remains an exact Riemann invariant when its gradient annihilates the source. For $U=(u,v)$, $A=\left(\begin{array}{cc}v& u\\ u^{-3}& v\end{array}\right)$, $S=(-u^3,u)$ and $u>0$, the function $R=v-1/u$ has $\nabla R=(u^{-2},1)$, eigenvalue $v+1/u$, and $\nabla R\cdot S=0$. On its zero level, the other characteristic speed vanishes and $v_t=1/v$ along constant spatial positions. Source terms need not destroy every invariant.

For a barotropic pressure law with $c^2=dp/d\rho$, the invariantsobey $[{\partial }_t+(u\pm c){\partial }_x]R_{\pm }=0$.

An initially resting perfect gas with adiabatic index $\gamma >1$ next to a piston $X(t)=-f{t}^2/2$ develops a smooth rarefaction. The incoming invariant gives $c_p=c_0-(\gamma -1)ft/2$. Pressure falls as $p_p=p_0(1-t/t_v{)}^{2\gamma /(\gamma -1)}$ until $t_v$, when the piston speed is $2c_0/(\gamma -1)$. Thereafter the gas edge moves at that limiting constant speed while the piston accelerates away, opening a vacuum gap.

A simple wave is a region of a hyperbolic flow in which one Riemann invariant is constant throughout, leaving the other invariant to determine all fluid variables.

For positive smooth tube area $A$, the equation $q_Z-q{q}_{\theta }+q(\log A{)}^{\prime }/2=0$ becomes $Q_{\zeta }-Q{Q}_{\theta }=0$ under the displayed transformation. Indeed, $g^{\prime }/g=-A^{\prime }/(2A)$ cancels geometrical spreading and ${\zeta }^{\prime }=g$ makes the remaining nonlinear and transport coefficients equal. Starting with $q(\theta ,Z_0)=f(\theta )$ gives $Q=f(\xi )$ and $\theta =\xi -f(\xi )\zeta$; characteristic crossing first occurs when $\zeta \sup f^{\prime }=1$, if that supremum is positive.

For area $A(R)=R^2$ and smooth data $q(\theta ,R_0)=f(\theta )$ at $R_0>0$, the transformed distance is $\zeta =R_0\log (R/R_0)$. If $M=\sup f^{\prime }>0$, the first shock wave occurs at the displayed radius; if $M\le 0$, there is no forward characteristic crossing. Nontrivial bounded data cannot instead be imposed at $R=0$ in this outgoing simple-wave model: along every characteristic, $Rq$ is constant, so a bounded limit at the origin forces that constant to vanish. With distance measured from a finite emitting surface, use $A(Z)=(R_0+Z{)}^2$ and subtract $R_0$ from the shock radius.

A characteristic field is linearly degenerate if the directional derivative of its speed along its right eigenvector vanishes. Its simple waves have constant propagation speed along the state curve, so they do not steepen by amplitude-dependent transport.

An ideal-MHD simple wave traces a one-parameter state-space curve whose tangent is a right eigenvector of the characteristic matrix. Its parameter obeys $q_t+v(q)q_x=0$ and its characteristic speed obeys the Inviscid Burgers equation. Nonlinear Alfvén waves are a linearly degenerate example.

For a one-dimensional velocity perpendicular to a transverse magnetic field, $c_f^2=\gamma p/\rho +B^2/({\mu }_0\rho )$. The primitive-variable characteristic matrix has two advective eigenvalues and the two speeds $u\pm c_f$. Integrating the fast eigenvectors keeps $p/{\rho }^{\gamma }$ and $B/\rho$ constant. The state parameter obeys ${\rho }_t+(u\pm c_f){\rho }_x=0$, so the characteristic speed satisfies the Inviscid Burgers equation.

With $B_x=0$, a fast simple wave in magnetohydrodynamics has constant $u_y$ and $B_y/\rho =K$. Its longitudinal velocity satisfies $d{u}_x/d\rho =\pm c_f/\rho$, and ${\rho }_t+(u_x\pm c_f){\rho }_x=0$. Along this state curve the speed derivative is $\pm [c_f/\rho +K^2/(2{\mu }_0{c}_f)]$, which is nonzero. Compressive profiles therefore exhibit shock formation by characteristic intersection, unlike a linearly degenerate characteristic field.

For $u_t+(c_0+bu)u_x=-\alpha u$, $b,\alpha >0$, the characteristic curves are $u=e^{-\alpha t}{u}_0(\xi )$ and $x=c_{0}t+\xi +(b/\alpha )(1-e^{-\alpha t})u_0(\xi )$. If $m=\max [-u_0^{\prime }]>0$, finite-time gradient blow-up occurs precisely for $\alpha <bm$, at $t_s=-{\alpha }^{-1}\log (1-\alpha /(bm))$. Equality $\alpha =bm$ still gives a positive Jacobian at each finite time; only the limiting map degenerates.

For a piston receding from initially stationary homentropic gas, the right-moving simple wave has $R_{-}=0$, so $c=c_0+(\gamma -1)u/2$. A $C_{+}$ characteristic emitted by the piston at time $s$ carries $u={\dot{X}}_p(s)$ and is the straight line

The gas state at a receding piston reaches vacuum when ${\dot{X}}_p=-2c_0/(\gamma -1)$, because then the sound speed and mass density both vanish. If the piston subsequently recedes faster, a vacuum gap opens between it and the gas boundary, which continues with velocity $-2c_0/(\gamma -1)$.

For $X_p(t)=-t^{1+\alpha }/(1+\alpha )$ with $0<\alpha \ll 1$, a point on the ray $x=(c_0-a)t$, $0<a<(\gamma +1)/2$, receives the characteristic emitted at the exponentially small timeAlthough $s\to 0$, the piston speed $-s^{\alpha }$ has the finite limit $-2a/(\gamma +1)$, producing the centered-rarefaction profile of the singular limit $\alpha =0$.

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, soThe isentropic relationconverts the velocity disturbance into its nonlinear pressure disturbance.

A smooth compressive simple wave forms a shock when characteristics first intersect. If characteristics are parametrized by their emission time $\tau$, the first shock occurs at the minimum positive time for which $\partial x(\tau ,t)/\partial \tau =0$.

A smooth solution undergoes a gradient catastrophe when a spatial derivative becomes unbounded while the solution itself remains bounded. For $u_t+q^{\prime }(u)u_x=0$, characteristics have $x=s+t{q}^{\prime }(u_0(s))$ and $u=u_0(s)$; thus $u_x=u_0^{\prime }(s)/[1+t{q}^{\prime \prime }(u_0(s))u_0^{\prime }(s)]$. Vanishing of the characteristic projection derivative gives the first such time. A weak solution with an entropy condition can continue after this breakdown of the smooth description.