Scalar conservation law ($u_t+{\partial }_{x}f(u)=0$)

A scalar conservation law in one space dimension has the form $u_t+{\partial }_{x}f(u)=0$. While a classical solution exists, it obeys the quasilinear transport equation $u_t+f^{\prime }(u)u_x=0$.

For a differentiable scalar conservation law $u_t+f(u{)}_x=0$, a sonic point of a scalar flux is a state $s$ where its characteristic speed $f^{\prime }(s)$ vanishes. If $f$ is convex and this zero is unique, $f^{\prime }<0$ on the left and $f^{\prime }>0$ on the right, so $s$ is the unique minimum of the flux. The definition refers to a vanishing transport speed in a scalar conservation law. In compressible fluid mechanics, a sonic point instead refers to flow at the local sound speed.

In the common PDE convention, a uniformly convex scalar flux is a $C^2$ flux satisfying $F^{\prime \prime }(z)\ge \kappa >0$ throughout the specified state interval, with one constant $\kappa$. Equivalently, $F(z)-\kappa z^2/2$ is a convex function. This guarantees strict inequality below nontrivial chords and strict ordering of the two endpoint derivatives around the chord slope. On the whole real line it is incompatible with a globally bounded first derivative.

Positive $\epsilon$ adds diffusion to a one-dimensional scalar conservation law. For a sufficiently decaying smooth solution, the divergence structure gives $\frac{1}{2}\frac{d}{dt}\Vert u{\Vert }_2^2+\epsilon \Vert u_x{\Vert }_2^2=0$, so the $L^2$ energy estimate is uniform in $\epsilon$. If $|F^{\prime }|\le M$, testing against $-u_{xx}$ also yields $\Vert u_x(t){\Vert }_2^2\le e^{M^{2}t/\epsilon }\Vert u_x(0){\Vert }_2^2$. Thin travelling waves explain why a uniform gradient bound generally fails.

This approximation studies the limit of a viscous scalar conservation law as its positive diffusion coefficient tends to zero. Under hypotheses giving compactness and appropriate convergence, the viscous entropy dissipation selects an entropy solution of the inviscid scalar conservation law. Fixed-phase decreasing travelling waves for a strictly convex flux converge to an entropy shock. Translating each profile differently can change or destroy the limit, so phase or initial data must specify the family.

Let $f$ be smooth and strictly concave, with $f(0)=0$ and $f^{\prime }$ bijective, and let $u_0$ be smooth with compact support. For $0<t<t_{\ast }$ in the smooth concave-flux characteristic lifespan, the primitive $U_x=u$ solves the Hamilton-Jacobi equation $U_t+f(U_x)=0$ and obeyswhere $g$ is the concave Legendre dual. The tangent inequality for a concave function bounds every candidate by $U(t,x)$, and the backward characteristic curve attains equality. Its foot $x_0$ is the unique maximizer, and $u(t,x)=h((x-x_0)/t)$ for $h=(f^{\prime }{)}^{-1}$. This statement concerns the smooth solution, without asserting a post-crossing continuation.

Under the inverse-flux quadratic bounds, smooth compactly supported data in a scalar conservation law satisfyAt the maximizing foot $x_0$, the maximum representation for a concave conservation law bounds $U$ below by $-\Vert u_0{\Vert }_1$ and above by $\Vert u_0{\Vert }_1+k_{+}(x-x_0-ct{)}^2/t$. This bounds the deviation of the characteristic speed from $c$; the linear inverse-flux bound then controls $u$. The time interval ends at the first characteristic crossing.

Before characteristic crossing, the scalar conservation law $u_t+f(u{)}_x=0$ has solution $u(t,X_t(\xi ))=u_0(\xi )$ with $X_t(\xi )=\xi +t{f}^{\prime }(u_0(\xi ))$. When this map is a diffeomorphism, the solution is $u_0\circ X_t^{-1}$. Its spatial derivative is $u_0^{\prime }(\xi )/(1+t{f}^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi ))$.

For smooth compactly supported data in the scalar conservation law $u_t+f(u{)}_x=0$, put $m={\min }_{\xi }{f}^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi )$. Its smooth characteristic solution of a scalar conservation law exists until $t_{\ast }=-1/m$ if $m<0$, and for all times if $m\ge 0$. The characteristic flow map $X_t(\xi )=\xi +t{f}^{\prime }(u_0(\xi ))$ has derivative $1+t{f}^{\prime \prime }(u_0)u_0^{\prime }$; its first zero produces gradient blowup. The formula applies to smooth fluxes of either concavity, although it is particularly useful for a concave function flux.

This $C^1$ flux has $|f^{\prime }|\le 1$ globally and agrees with the Inviscid Burgers equation flux on $[-1,1]$. It allows Burgers examples to satisfy a global bounded-speed hypothesis. For Riemann data $-1$ to the left and $1$ to the right, both the stationary expansion discontinuity and the fan $u=x/t$ on $|x|<t$, with exterior values $\pm 1$, are bounded weak solutions. Only the fan is an entropy solution.

A shock wave is a moving discontinuity of a weak solution to a conservation law. Its speed is fixed by the Rankine-Hugoniot condition, while an entropy condition selects the physically admissible direction of compression.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

For a scalar equation with strictly convex flux and states $u_l>u_r$, the Rankine-Hugoniot condition determines a discontinuity speed strictly between the right and left characteristic speeds. Characteristic curves enter this compressive discontinuity from both sides. Its entropy flux for a scalar conservation law inequalities make it an entropy solution. It is the limit of a phase-normalized decreasing travelling wave in the vanishing viscosity approximation.

A rarefaction wave is a continuous expanding fan of characteristics that connects two constant states of a hyperbolic conservation law.

For $\gamma >1$ and positive mass density and pressure, let $\xi =x/t$ in one-dimensional adiabatic compressible flow. The smooth equations are $(u-\xi ){\rho }^{\prime }+\rho u^{\prime }=0$, $(u-\xi )u^{\prime }+p^{\prime }/\rho =0$ and $(u-\xi )p^{\prime }+\gamma p{u}^{\prime }=0$. Away from a constant branch, their coefficient determinant forces $\xi =u+\sigma c$ with $\sigma =\pm 1$ and adiabatic sound speed $c=\sqrt{\gamma p/\rho }$. The entropy equation gives $p=K{\rho }^{\gamma }$, and combining continuity with $c^{\prime }/c=(\gamma -1){\rho }^{\prime }/(2\rho )$ givesThus $u=[2\xi +(\gamma -1)J]/(\gamma +1)$, $c=\sigma (\gamma -1)(\xi -J)/(\gamma +1)$, $\rho =(c^2/(\gamma K){)}^{1/(\gamma -1)}$ and $p=K{\rho }^{\gamma }$, on intervals where $c>0$. The specific entropy is constant. Together with uniform branches, these exhaust smooth positive-gas self-similar solutions; a contact discontinuity is not a smooth fan.

The Inviscid Burgers equation is the scalar conservation lawFor a classical solution, the chain rule makes this equivalent to $u_t+u{u}_x=0$, a nonlinear transport equation with characteristic speed $u$. The conservative form remains meaningful for a discontinuous weak solution, whereas the product $u{u}_x$ cannot simply be evaluated pointwise at a jump. Its discontinuities obey the Rankine-Hugoniot condition $s=(u_L+u_R)/2$; this condition alone permits inadmissible expansion jumps. For bounded initial conditions on the infinite line, imposing the Kruzhkov entropy inequalities and the initial trace selects a unique entropy solution. For step data, incoming characteristic curves select $u_L>s>u_R$, while $u_L<u_R$ produces a rarefaction wave.^[ref]

Before characteristic crossing, the method of characteristics for the Inviscid Burgers equation transports each initial value without changing it:Here $\xi$ labels the starting position. Suppose $u_0$ is a bounded continuously differentiable function with bounded derivative. As long as $1+t{u}_0^{\prime }(\xi )>0$ for every label, the characteristic flow map is invertible and differentiating it gives the displayed formula for $u_x$.^[ref]

If the negative minimum $m={\min }_{\xi }{u}_0^{\prime }(\xi )$ is attained, the first breaking time is $t_{\ast }=-1/m$. Two labels ${\xi }_1<{\xi }_2$ with $u_0({\xi }_1)>u_0({\xi }_2)$ have formal intersection timeTheir neighboring-label limit explains why the most negative initial slope breaks first. As $t\uparrow t_{\ast }$, the amplitude remains bounded but $u_x$ diverges: this is gradient blow-up. Continuing intersecting lines would assign incompatible values at one position; the post-breaking entropy solution instead uses shock waves. Nondecreasing data have no forward characteristic crossing by this mechanism.

For a concrete calculation in dimensionless quantities on the infinite line, choose $u_0(\xi )=1-\tanh \xi$, with limiting states $2$ and $0$. Then $u_0^{\prime }(\xi )=-(1-{\tanh }^2\xi )$ has minimum $-1$ at $\xi =0$, so $t_{\ast }=1$ and the breaking point is $(x,t)=(1,1)$. Along the central characteristic curve $x=t$, $u=1$ and $u_x=-1/(1-t)$ before breaking. The labels $\xi =\pm 1$ give intersecting formal lines at $t=1/\tanh 1\approx 1.313$, after the classical solution has already broken.

Consider the Riemann problem on $\mathbb{R}$ with initial condition $u_0(x)=u_L$ for $x<0$ and $u_0(x)=u_R$ for $x>0$. No endpoint boundary conditions are imposed. When $u_L>u_R$, its entropy solution is a shock wave with constant speed $s=(u_L+u_R)/2$. In the original example $u_L=2$, $u_R=0$, the Rankine-Hugoniot condition givesThe left characteristic speed is $2$ and the right characteristic speed is $0$, so $2>1>0$: both sides enter the entropy shock. The Viscous Burgers travelling shock gives an exact smooth family tending to this solution.

Reversing the states to $u_L=0$, $u_R=2$ still gives a Rankine-Hugoniot condition speed of $1$, but now characteristic curves move away from the discontinuity. That expansion jump is a weak solution and fails entropy admissibility. For general $u_L<u_R$, the selected entropy solution is the continuous self-similar solutionThis rarefaction wave fills the expanding gap between the two exterior characteristic curves. Inside its fan, $u_t=-x/t^2$ and $u{u}_x=x/t^2$, verifying the Inviscid Burgers equation; its values agree continuously with both exterior states. For $0$ to $2$, the fan occupies $0<x<2t$. If $u_L=u_R$, the solution is constant.^[ref]

For the negative-flux equation $f_Z-f{f}_{\theta }=0$, a period-two increasing initial ramp develops rarefaction fans at odd points and steepens at even points. Before $Z=1$, its central ramp is $f=(\theta -2m)/(1-Z)$ on $|\theta -2m|<1-Z$, and its odd-point fans are $f=-(\theta -(2m+1))/Z$. After $Z=1$, stationary compressive shocks occupy the even points and $f=(2m+1-\theta )/Z$ between consecutive shocks. The amplitude decays as $1/Z$. The Rankine-Hugoniot condition and incoming characteristic speeds certify the post-breaking entropy solution.

For left state zero and right state $U$, negative-flux Inviscid Burgers equation gives a shock at $\theta =-UZ/2$ when $U>0$, and a rarefaction wave $f=-\theta /Z$ on $0<\theta <-UZ$ when $U<0$. This is the standard positive-flux equation after setting $u=-f$; entropy solution selection reverses which step direction is compressive.

An entropy solution is a weak solution of a nonlinear scalar conservation law that also satisfies entropy inequalities selecting physically admissible shocks. For convex fluxes, this excludes expansion shocks and restores uniqueness for bounded initial data.

For a scalar conservation law law $u_t+f(u{)}_x=0$, the Kruzhkov entropy inequalities requirein distributions for every real constant $c$, together with the initial trace. This means the integral against every nonnegative compactly supported test function is nonpositive after applying the distributional derivatives. For a smooth solution the entropy balance is an equality away from $u=c$. For a viscous approximation $u_t+f(u{)}_x=\epsilon u_{xx}$, smooth convex approximations $\eta$ of $|u-c|$ give ${\partial }_t\eta (u)+{\partial }_{x}q(u)=\epsilon {\partial }_{xx}\eta (u)-\epsilon {\eta }^{\prime \prime }(u)u_x^2\le \epsilon {\partial }_{xx}\eta (u)$, with $q^{\prime }={\eta }^{\prime }{f}^{\prime }$. Letting the entropy smoothing vanish and then taking a bounded, strongly locally convergent zero-viscosity limit proves the inequality. The discrete entropy inequality of a monotone conservative scheme is its numerical analogue.

For a scalar conservative monotone update with flux $F$, define $Q(a,b;c)=F(a\vee c,b\vee c)-F(a\wedge c,b\wedge c)$. Monotonicity and preservation of constant states give $|U_j^{n+1}-c|\le |U_j^n-c|-(k/d)(Q_{j+1/2}-Q_{j-1/2})$. At equal states $Q(u,u;c)=\mathrm{sgn}(u-c)(f(u)-f(c))$, the Kruzhkov entropy flux. Such discrete inequalities connect stability of a monotone conservative scheme with the admissibility of its limiting entropy solution.

For a bounded entropy solution with its initial entropy inequalities, the displayed limit holds on every compact spatial set. Test a constant-level entropy with $(1-t/\delta {)}_{+}\rho (x)$. The flux remainder is $O(\delta )$, giving a bound by $\int \rho |u_0-k|$. A finite smooth partition and constant levels approximate the initial function in local $L^1$; the triangle inequality makes the limsup arbitrarily small. This averaged trace suffices for one-sided time mollifiers supported in $[\delta ,2\delta ]$.

A convex entropy $\eta$ is paired with any flux antiderivative satisfying $q^{\prime }(z)={\eta }^{\prime }(z)f^{\prime }(z)$ almost everywhere. Its admissibility inequality is ${\partial }_t\eta (u)+{\partial }_{x}q(u)\le 0$ distributionally, with the matching initial entropy trace. Affine entropies $\eta (z)=\pm z$ recover the weak formulation, while bounded classical solutions satisfy equality for smooth entropies and then by approximation for piecewise smooth convex entropies.

For ${\eta }_k(z)=|z-k|$, the displayed flux is continuous at $z=k$ and has derivative $\mathrm{sgn}(z-k)f^{\prime }(z)$ almost everywhere. On a bounded state range with $|f^{\prime }|\le M$, its symmetric two-state version $Q(a,b)$ obeys $|Q(a,b)|\le M|a-b|$ and is Lipschitz in both arguments. The vanishing factor at $a=b$ removes any sign-function jump contribution.

Apply the entropy solution inequality for $u(t,x)$ with level $v(s,y)$ and for $v(s,y)$ with level $u(t,x)$, integrate both in the other pair of variables, and add them. Symmetry of the Kruzhkov entropy flux produces derivatives ${\partial }_t+{\partial }_s$ and ${\partial }_x+{\partial }_y$. A nonnegative mollifier concentrated near the diagonal then yields the Kato inequality for scalar conservation laws. Boundary-time terms require an initial trace argument, not merely interior translation continuity.

For bounded entropy solutions of the same scalar law, $Q(u,v)=\mathrm{sgn}(u-v)(f(u)-f(v))$ satisfies the displayed distributional inequality, with initial value $|u_0-v_0|$ in its test-function form. The doubling of variables for scalar conservation laws, local translation continuity and the averaged initial trace of an entropy solution prove it. Its significance is comparison between two solutions rather than an inequality against a constant state.

If $|f^{\prime }|\le M$ on the solutions' state range, the Kato inequality for scalar conservation laws implies the displayed estimate for almost every $s\in [0,t]$. Approximate the shrinking interval by smooth cutoffs $W$ satisfying $W_s+M|W_x|\le 0$ and multiply by a temporal cutoff ending at $s$. Since $|Q|\le M|u-v|$, the lateral flux cannot increase the integral. The estimate gives uniqueness and a finite propagation speed without global $L^1$ assumptions on the data.

Subtract the weak equation for $u-v$ from its Kato inequality for scalar conservation laws and divide by two. The result is an inequality for $(v-u{)}_{+}$ with a flux bounded in magnitude by $M(v-u{)}_{+}$. The same shrinking-interval argument proves $u_0\ge v_0\Rightarrow u\ge v$ almost everywhere. Choosing the constant solution $v=0$ proves preservation of nonnegativity even when $f(0)\ne 0$. Two-sided absolute-value contraction alone should not be mistaken for this one-sided comparison proof.

Characteristic crossing occurs when the map from an initial point to its position at time $t$ loses injectivity. For $u_t+f(u{)}_x=0$, its Jacobian is $1+t{f}^{\prime \prime }(u_0)u_0^{\prime }$ in the autonomous case, so a negative value of $f^{\prime \prime }(u_0)u_0^{\prime }$ causes finite-time crossing.

Gradient blow-up means that a solution stays bounded while the norm of a spatial derivative tends to infinity. In a scalar conservation law this happens as characteristics meet, because the derivative contains the reciprocal of the characteristic-map Jacobian.