Scalar Conservation Laws: Shocks, Rankine-Hugoniot, and Entropy Solutions

Intuition Beginner

Picture traffic on a single-lane road, or water piling up in a shallow channel, or a band of gas being squeezed. In each case there is a quantity that is neither created nor destroyed, only moved around: the number of cars, the amount of water, the mass of gas. A conservation law is the bookkeeping statement that whatever flows out of a stretch of road must equal whatever the count inside that stretch drops by. The catch is that the speed at which the stuff moves depends on how much of it there is. Dense traffic crawls; light traffic flies. That single fact, speed depending on density, is what makes these equations wild.

Here is the trouble it causes. Suppose the back of a wave is moving faster than the front. The fast part catches up to the slow part. A smooth hump of density steepens, like a wave approaching a beach, until the profile becomes vertical: in a finite amount of time the density would have to take two values at once, which is impossible. The smooth solution simply stops existing. Nature resolves this by forming a sharp jump, a moving wall where the density leaps from one value to another. In traffic this is the back end of a jam you slam into; in gas it is a shock wave; in water it is the wall of a breaking bore.

Once jumps are allowed, a new question appears. The plain conservation rule, by itself, permits jumps that never actually happen. It would let a traffic jam un-form spontaneously, cars leaping from stopped to fast across a backward-moving wall, which no real road does. The bare equation cannot tell a physical jump from this time-reversed fiction, because across a jump there is no slope to plug in, just as a corner had no slope in the front-tracking problem you met earlier. An extra rule is needed to keep only the jumps that occur.

That extra rule is the entropy condition, and it has the same origin as before. Add a whisker of diffusion, the same smoothing that smears a sharp temperature spike, and every jump rounds into a steep but smooth ramp with an honest slope. Shrink the diffusion to zero. The smoothed solutions settle onto jumps of one kind only, the kind that compress the flow, never the kind that would rarefy across a wall. The selected jumps are the shocks; where the flow wants to spread out instead, the smooth limit fans it open gradually into what is called a rarefaction. The admissible-jump condition is exactly the fingerprint that vanishing diffusion leaves behind.

Why insist on all this? Because the conserved count is the real, measurable thing, and getting the jump wrong throws the count off. The entropy solution is the one bookkeeping that stays honest at every shock: it tracks the true mass, the true traffic, the true gas, and it is unique, so two people computing it must agree.

Visual Beginner

The picture to hold is a smooth density profile steepening until it tips into a vertical jump that then travels as a shock.

Read the panels left to right as a film. In the first, the density is a smooth hump and the arrows record the central fact: where the stuff is dense it moves slowly, where it is sparse it moves fast, so the trailing edge gains on the leading edge. In the second panel the trailing edge has caught up completely and the front face stands vertical; a function cannot be vertical and single-valued, so the classical solution ends exactly here, at a finite tip-over time.

The third panel shows the resolution. The vertical face becomes a genuine jump, a step that travels rightward at one definite speed, the shock speed. The size of the step and the speed are not independent; the conservation bookkeeping ties them together, which is the jump relation you will compute next. The blurred inset is the tell: a little diffusion turns the step into a steep smooth ramp, and as the diffusion vanishes the ramp converges back to this step but never to the upside-down version, where a fan would open instead of a wall closing. That asymmetry between a closing wall and an opening fan is the whole content of admissibility.

Worked example Beginner

We compute the speed of a shock for the simplest model, where the flow rate is one half the square of the density. This is Burgers' flow, the textbook stand-in for traffic and gas. The conservation law says the density obeys a rule in which each level of density travels at a speed equal to that level itself: a density value of $2$ moves at speed $2$, a value of $1$ moves at speed $1$.

Step 1. Set up a jump. Start with density equal to $2$ everywhere to the left and equal to $0$ everywhere to the right, a single downward step at the origin at time zero. The left level $2$ wants to move right at speed $2$; the right level $0$ sits still. So the left material charges into the stationary right material, and they cannot pass through each other, since density is single-valued. A shock must form immediately.

Step 2. Use conservation to get the speed. In a small time the shock moves some distance to the right. The high block of density $2$ has advanced, the low block of density $0$ has been overrun. Counting the conserved stuff: the flow rate just left of the jump is one half of $2$ squared, which is $2$; the flow rate just right is one half of $0$ squared, which is $0$. The jump must travel at a speed that balances the count, and that balancing speed is the difference in flow rates divided by the difference in densities: $(2-0)$ divided by $(2-0)$, which equals $1$.

Step 3. Read off the answer. The shock speed is $1$. Notice it is exactly the average of the two density levels, $(2+0)$ divided by $2$, which also equals $1$. That averaging is special to this square-law flow and is a handy check: a Burgers shock travels at the mean of the levels on its two sides.

Step 4. Confirm it is the right kind of jump. The level behind the shock, $2$, is faster than the level ahead, $0$. So material is piling into the shock from behind and being swallowed from in front; the wall is a compression, the kind a little diffusion keeps. The reversed setup, density $0$ on the left and $2$ on the right, would have the slow material in front and the fast behind pulling apart, and there the smooth limit opens a gradual fan instead of a jump.

What this tells us: the conservation count alone fixes the speed of a jump once a jump is allowed, giving here a shock at speed $1$, and the compression check tells us this particular jump is the physical one rather than the forbidden time-reversed fiction.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $u:{\mathbb{R}}^n\times (0,\infty )\to \mathbb{R}$ is the scalar unknown and $F:\mathbb{R}\to {\mathbb{R}}^n$ is the flux, assumed $C^1$ (and, where stated, with $F^{\prime }$ Lipschitz or convex). The scalar conservation law is $$u_t+\mathrm{div}F(u)=0\text{in }{\mathbb{R}}^n\times (0,\infty ),u(\cdot ,0)=u_0,$$ where $\mathrm{div}F(u)={\sum }_{i=1}^n{\partial }_{x_i}(F^i(u))=F^{\prime }(u)\cdot Du$ when $u$ is $C^1$. In one space dimension this is $u_t+F(u{)}_x=0$ with characteristic speed $F^{\prime }(u)$. The model flux is Burgers, $F(u)=\frac{1}{2}{u}^2$, with $F^{\prime }(u)=u$.

Definition (weak solution). A function $u\in L^{\infty }({\mathbb{R}}^n\times (0,\infty ))$ is a weak solution of the conservation law with initial data $u_0\in L^{\infty }$ if for every test function $\phi \in C_c^1({\mathbb{R}}^n\times [0,\infty ))$, $${\int }_0^{\infty }{\int }_{{\mathbb{R}}^n}(u{\phi }_t+F(u)\cdot D\phi )dxdt+{\int }_{{\mathbb{R}}^n}{u}_0(x)\phi (x,0)dx=0.$$ This is the conservation law integrated against $\phi$ with the derivatives moved onto $\phi$; it makes sense for merely bounded $u$ and encodes the initial data through the boundary term at $t=0$.

Definition (Rankine-Hugoniot jump). Suppose a weak solution is $C^1$ on each side of a $C^1$ hypersurface $\Sigma =\{x_1=\sigma (x^{\prime },t)\}$ across which $u$ jumps from $u_{\ell }$ to $u_r$. In one space dimension, with the shock curve $x=s(t)$ and speed $\dot{s}=s^{\prime }$, the weak formulation forces the Rankine-Hugoniot condition $$s^{\prime }(u_{\ell }-u_r)=F(u_{\ell })-F(u_r),\text{i.e.}{s}^{\prime }=\frac{[[F(u)]]}{[[u]]},$$ where $[[q]]=q_{\ell }-q_r$ denotes the jump. The speed of a discontinuity is the jump in flux divided by the jump in the conserved quantity.

Definition (Lax entropy condition). A jump from $u_{\ell }$ to $u_r$ travelling at the Rankine-Hugoniot speed $s^{\prime }$ satisfies the Lax entropy condition if $$F^{\prime }(u_{\ell })\ge s^{\prime }\ge F^{\prime }(u_r).$$ For convex $F$ this is equivalent to $u_{\ell }>u_r$ (a compressive jump): characteristics run into the shock from both sides. The reversed inequality describes a rarefaction shock, which is inadmissible.

Definition (entropy/entropy-flux pair and entropy solution). A pair of functions $(\eta ,q)$ with $\eta :\mathbb{R}\to \mathbb{R}$ convex and $q:\mathbb{R}\to {\mathbb{R}}^n$ is an entropy/entropy-flux pair if $q^{\prime }={\eta }^{\prime }{F}^{\prime }$ (so that $C^1$ solutions satisfy $\eta (u{)}_t+\mathrm{div}q(u)=0$). A bounded $u$ is an entropy solution if for every such pair with $\eta$ convex, $$\eta (u{)}_t+\mathrm{div}q(u)\le 0\text{in the sense of distributions}.$$ Equivalently (Kruzhkov), using the family ${\eta }_k(u)=|u-k|$, $q_k(u)=\mathrm{sgn}(u-k)(F(u)-F(k))$ indexed by all constants $k\in \mathbb{R}$, $${\partial }_t|u-k|+\mathrm{div}(\mathrm{sgn}(u-k)(F(u)-F(k)))\le 0\text{for all }k.$$ The sign convention is fixed here: the inequality is $\le 0$, matching the dissipation of entropy under the vanishing-viscosity limit $u_t^{\epsilon }+\mathrm{div}F(u^{\epsilon })=\epsilon \Delta u^{\epsilon }$, $\epsilon \downarrow 0$, where convex $\eta$ produces a nonpositive distributional remainder.

Counterexamples to common slips Intermediate+

• Weak solutions are not unique. For Burgers with $u_0=0$ on the left and $1$ on the right, both the spreading rarefaction fan and a single jump at speed $\frac{1}{2}$ are weak solutions satisfying Rankine-Hugoniot. Only the rarefaction is the entropy solution; the jump violates Lax. The bare weak formulation does not select.

• Rankine-Hugoniot is necessary, never sufficient. The jump relation merely balances the conserved count across a discontinuity; it is satisfied by both the admissible shock and its forbidden time-reverse. Admissibility is an extra, strict inequality, not an equality.

• The sign of the entropy inequality is load-bearing. Replacing $\le 0$ by $\ge 0$ selects the opposite (anti-dissipative) solutions, the rarefaction shocks. The convention $\eta (u{)}_t+\mathrm{div}q(u)\le 0$ for convex $\eta$ is the one consistent with $\epsilon \Delta u$, $\epsilon \downarrow 0$, exactly as the touching-side convention was load-bearing for viscosity solutions 02.18.05.

• Entropy is not the PDE holding a.e. A function can solve $u_t+F(u{)}_x=0$ at every point off a jump set, hence a.e., and still be inadmissible; the distributional entropy inequality at the measure-zero shock set is the extra information, exactly as the touching condition at the corner was for Hamilton-Jacobi 02.18.05.

Key theorem with proof Intermediate+

Theorem (Rankine-Hugoniot from the weak formulation). Let $u$ be a weak solution of $u_t+F(u{)}_x=0$ that is $C^1$ on each side of a $C^1$ curve $\Gamma =\{(x,t):x=s(t)\}$, with one-sided limits $u_{\ell }(t),u_r(t)$ and one-sided classical satisfaction of the PDE off $\Gamma$. Then the speed satisfies $s^{\prime }(t)(u_{\ell }-u_r)=F(u_{\ell })-F(u_r)$ at every $t$ ^{[Lax 1957] [Evans 2010 §3.4]}.

Proof. Fix a test function $\phi \in C_c^1$ supported in a small open set $V$ that $\Gamma$ divides into a left region $V_{\ell }$ (where $x<s(t)$) and a right region $V_r$, with $V$ chosen so small that $u$ is $C^1$ up to $\Gamma$ on each side and $\phi$ vanishes on $\partial V$. The weak identity (with no $t=0$ contribution, since $V$ is interior) reads $${\iint }_V(u{\phi }_t+F(u){\phi }_x)dxdt=0.$$ Split the integral over $V_{\ell }$ and $V_r$. On $V_{\ell }$, where $u$ is $C^1$, integrate by parts: ${\iint }_{V_{\ell }}(u{\phi }_t+F(u){\phi }_x)=-{\iint }_{V_{\ell }}(u_t+F(u{)}_x)\phi +{\int }_{\partial V_{\ell }}\phi (u{\nu }_t+F(u){\nu }_x)d{\mathcal{H}}^1$, where $\nu =({\nu }_x,{\nu }_t)$ is the outward unit normal to $\partial V_{\ell }$. The interior term vanishes because $u$ solves the PDE classically on $V_{\ell }$, and $\phi =0$ on $\partial V\cap \partial V_{\ell }$, so only the part of the boundary integral along $\Gamma$ survives.

Along $\Gamma$ parametrized by $t$, the curve $x=s(t)$ has tangent $(s^{\prime },1)$ and outward normal (from $V_{\ell }$, pointing into increasing $x$) proportional to $(1,-s^{\prime })$; normalizing, the line element contributes $\phi (u_{\ell }\cdot (-s^{\prime })+F(u_{\ell })\cdot 1)dt$ after clearing the common normalization. Hence $${\iint }_{V_{\ell }}(u{\phi }_t+F(u){\phi }_x)={\int }_{\Gamma }\phi (F(u_{\ell })-s^{\prime }{u}_{\ell })dt.$$ The right region $V_r$ has the opposite outward normal along $\Gamma$, giving $-{\int }_{\Gamma }\phi (F(u_r)-s^{\prime }{u}_r)dt$. Adding, the weak identity becomes $${\int }_{\Gamma }\phi \left[\right(F(u_{\ell })-s^{\prime }{u}_{\ell })-(F(u_r)-s^{\prime }{u}_r\left)\right]dt=0.$$ Since this holds for every $\phi \in C_c^1$ supported near $\Gamma$, the bracket vanishes identically: $F(u_{\ell })-F(u_r)=s^{\prime }(u_{\ell }-u_r)$, which is Rankine-Hugoniot. $\square$

Bridge. Rankine-Hugoniot is the foundational reason a jump can be admitted into the solution class at all: it is exactly the conservation bookkeeping made precise, the statement that the flux imbalance across a discontinuity must be carried off at the jump's own speed, and this is exactly the one-dimensional shadow of the divergence-form structure that the wave equation's energy identity 02.13.04 exploits on a smooth front. The condition is necessary but blind to direction, which is why it builds toward the Lax and entropy criteria that supply the missing sign; this is dual to the viscosity-solution story 02.18.05, where the Rankine-Hugoniot speed of $u_x$ is the velocity at which a corner of the Hamilton-Jacobi solution travels. The central insight is that differentiating a one-dimensional Hamilton-Jacobi equation in $x$ turns its solution gradient into a conservation law, so a corner there becomes a shock here and the touching test becomes the entropy inequality. Putting these together, the same vanishing-diffusion limit that rounded corners rounds shocks, and the asymmetry it leaves becomes the admissibility condition; the jump relation appears again in the Riemann problem, where it fixes shock speeds, and in Kruzhkov's theorem, where the entropy pairs upgrade it to a full uniqueness statement.

Exercises Intermediate+

Advanced results Master

The entropy formulation organizes a complete well-posedness theory: Kruzhkov gives uniqueness and $L^1$ contraction, Lax-Oleinik gives existence and an explicit formula for convex flux, vanishing viscosity gives the selection principle its name, and the Riemann problem gives the local building block from which general solutions are assembled.

Theorem 1 (Kruzhkov uniqueness and $L^1$ contraction). Let $F\in C^1$ with $F^{\prime }$ Lipschitz on the range of the data, and let $u,v$ be entropy solutions with initial data $u_0,v_0\in L^{\infty }\cap L^1$. Then for every $t>0$, $${\int }_{{\mathbb{R}}^n}|u(x,t)-v(x,t)|dx\le {\int }_{{\mathbb{R}}^n}|u_0(x)-v_0(x)|dx,$$ so the entropy solution is unique and the solution map is an $L^1$ contraction ^{[Kruzhkov 1970] [Evans 2010 §11.4]}. The proof doubles the variables: one applies the Kruzhkov inequality for $u(x,t)$ with constant $v(y,\tau )$ and for $v(y,\tau )$ with constant $u(x,t)$, adds them against a test function concentrating on the diagonal $\{x=y,t=\tau \}$, and passes to the limit; the flux modulus is controlled by the mollification scale exactly as the Hamiltonian modulus was in the viscosity comparison proof (Exercise 8). The finite-speed-of-propagation cone built into $\chi$ localizes the estimate, and the diagonal limit reduces the eight-fold integral to the contraction. Uniqueness is immediate: equal data force $\Vert u-v{\Vert }_{L^1}=0$.

Theorem 2 (Lax-Oleinik formula). Let $F$ be $C^2$, uniformly convex ($F^{\prime \prime }\ge \theta >0$), with $G=(F^{\prime }{)}^{-1}$, and $u_0\in L^{\infty }$ with primitive $w_0(y)={\int }_0^y{u}_0$. Then the entropy solution is $$u(x,t)=G\left(\frac{x-y(x,t)}{t}\right),y(x,t)=\underset{y\in \mathbb{R}}{\mathrm{argmin}}\left\{tL\right(\frac{x-y}{t})+w_0(y)\},$$ with $L=F^{\ast }$ the Legendre transform; $y(\cdot ,t)$ is nondecreasing, $u$ is the a.e. derivative of the Hopf-Lax solution $w$ of $w_t+F(w_x)=0$, and the one-sided Oleinik bound $u(x+z,t)-u(x,t)\le z/(\theta t)$ holds ^{[Oleinik 1957] [Lax 1954]}. The formula is the conservation-law differentiate of the Hamilton-Jacobi value function 02.18.05: where the minimizer $y$ jumps the gradient $u$ jumps downward, producing exactly the admissible shocks, and convexity of $F$ (equivalently of $L$) is what makes the minimizer monotone and hence the jumps one-signed.

Theorem 3 (existence by vanishing viscosity). For $F\in C^2$ and $u_0\in L^{\infty }\cap BV$, the parabolic regularizations $$u_t^{\epsilon }+\mathrm{div}F(u^{\epsilon })=\epsilon \Delta u^{\epsilon },u^{\epsilon }(\cdot ,0)=u_0,$$ have smooth solutions with $\Vert u^{\epsilon }{\Vert }_{L^{\infty }}$ and $\mathrm{TV}(u^{\epsilon }(\cdot ,t))$ bounded uniformly in $\epsilon$ 02.13.04. By Helly's selection theorem a subsequence converges in $L_{\mathrm{loc}}^1$, $u^{\epsilon }\to u$, and multiplying the equation by ${\eta }^{\prime }(u^{\epsilon })$ for convex $\eta$ yields $\eta (u^{\epsilon }{)}_t+\mathrm{div}q(u^{\epsilon })=\epsilon \Delta \eta (u^{\epsilon })-\epsilon {\eta }^{\prime \prime }(u^{\epsilon })|D{u}^{\epsilon }{|}^2\le \epsilon \Delta \eta (u^{\epsilon })$, so the limit satisfies the entropy inequality ^{[Hopf 1950]}. This is the construction that names the theory: the term $-\epsilon {\eta }^{\prime \prime }|D{u}^{\epsilon }{|}^2\le 0$ is the entropy dissipation, the sign that survives the limit, and Kruzhkov (Theorem 1) makes the limit independent of the subsequence. The artificial viscosity is the exact mechanism the Beginner picture invoked.

Theorem 4 (Riemann problem, convex flux). For uniformly convex $F$ and piecewise-constant data $u_0=u_{\ell }$ for $x<0$, $u_0=u_r$ for $x>0$, the entropy solution is self-similar, $u(x,t)=R(x/t)$, and is given by exactly two cases ^{[Lax 1957] [Dafermos 2016]}. If $u_{\ell }>u_r$, a single shock at speed $s^{\prime }=[[F]]/[[u]]$ satisfying Lax. If $u_{\ell }<u_r$, a rarefaction wave $u=G(x/t)$ on the fan $F^{\prime }(u_{\ell })<x/t<F^{\prime }(u_r)$, constant outside it. These two elementary waves are the building blocks of the general theory: the front-tracking and Glimm schemes assemble approximate solutions by solving a Riemann problem at every jump and propagating the resulting shocks and rarefactions, and the entropy solution of arbitrary $BV$ data is the limit of such assemblies.

Theorem 5 (entropy solutions are $BV$ and have at most countably many shock curves). For uniformly convex $F$ and $u_0\in L^{\infty }$, the Lax-Oleinik solution is $B{V}_{\mathrm{loc}}$ for each $t>0$, its jump set is a countable union of Lipschitz shock curves along which Rankine-Hugoniot and Lax hold, and away from shocks $u$ is the classical solution carried by straight characteristics $x=x_0+t{F}^{\prime }(u_0(x_0))$ ^{[Dafermos 2016] [Serre 1999]}. The structure theorem identifies the singular support: characteristics emanating from the data are straight lines of slope $F^{\prime }(u_0)$, they collide precisely where the Oleinik minimizer $y(x,t)$ becomes multivalued, and the locus of collisions is the shock set. This is the rigorous form of the Beginner tip-over picture: the catching-up of characteristics is the focusing that the convex Legendre minimum resolves into a single admissible front.

Synthesis. The entropy theory is the foundational reason scalar conservation laws have a well-posed weak solution theory at all, and the entire edifice is generated by one asymmetric device: across an admissible jump characteristics run inward and the convex entropies dissipate, $\eta (u{)}_t+\mathrm{div}q(u)\le 0$, and the sign distinguishing this from the forbidden rarefaction shock is the trace of a vanishing diffusion. Putting these together, Kruzhkov (Theorem 1) converts the one-signed entropy inequalities into a global $L^1$ ordering by doubling the variables, and this is exactly the mechanism that delivers uniqueness and contraction; existence is then supplied three ways — explicitly by Lax-Oleinik for convex $F$ (Theorem 2), constructively by vanishing viscosity for general $F$ (Theorem 3), and combinatorially by Riemann-problem assembly (Theorem 4) — each of which generalises a classical idea past the tip-over time where smooth solutions die.

The central insight is that the convex case is dual to Hamilton-Jacobi through the Legendre transform: $u$ is the spatial gradient of a Hopf-Lax value function 02.18.05, the Oleinik minimizer is the optimal characteristic, and the shock is the locus where two characteristics tie for the minimum. This is dual to the viscosity-solution picture exactly as differentiation is dual to integration: the bridge is that a corner of $w$ is a jump of $u=w_x$, the touching test becomes the entropy inequality, and putting these together the same doubling-of-variables argument proves uniqueness on both sides of the derivative. The selection principle appears again in the systems theory of gas dynamics and in the numerical conservation laws that must respect it to converge.

Full proof set Master

Proposition 1 (Lax condition implies a single entropy inequality across the jump). Let $F$ be convex and consider a jump from $u_{\ell }$ to $u_r$ at Rankine-Hugoniot speed $s^{\prime }$. If $u_{\ell }>u_r$ (so the Lax condition $F^{\prime }(u_{\ell })\ge s^{\prime }\ge F^{\prime }(u_r)$ holds), then for every convex entropy $\eta$ with flux $q^{\prime }={\eta }^{\prime }{F}^{\prime }$, the jump dissipates: $s^{\prime }[[\eta (u)]]-[[q(u)]]\ge 0$.

Proof. The distributional entropy inequality $\eta (u{)}_t+q(u{)}_x\le 0$, evaluated across a single $C^1$ shock curve exactly as in the Rankine-Hugoniot theorem (test against $\phi \in C_c^1$ near $\Gamma$ and integrate by parts on each side), reduces to the algebraic jump inequality $-s^{\prime }[[\eta (u)]]+[[q(u)]]\le 0$, i.e. $s^{\prime }(\eta (u_{\ell })-\eta (u_r))\ge q(u_{\ell })-q(u_r)$. Define $E(u_{\ell },u_r):=s^{\prime }(\eta (u_{\ell })-\eta (u_r))-(q(u_{\ell })-q(u_r))$ with $s^{\prime }=(F(u_{\ell })-F(u_r))/(u_{\ell }-u_r)$ fixed. Differentiate $E$ in $u_{\ell }$ holding $u_r$ fixed: using $s_{u_{\ell }}^{\prime }(u_{\ell }-u_r)=F^{\prime }(u_{\ell })-s^{\prime }$ and $q^{\prime }={\eta }^{\prime }{F}^{\prime }$, $${\partial }_{u_{\ell }}E=s_{u_{\ell }}^{\prime }(\eta (u_{\ell })-\eta (u_r))+s^{\prime }{\eta }^{\prime }(u_{\ell })-{\eta }^{\prime }(u_{\ell })F^{\prime }(u_{\ell }).$$ A direct computation (or Oleinik's chord argument: $E$ measures the area between the graph of $F$ and its chord, weighted by ${\eta }^{\prime \prime }$) shows $E\ge 0$ precisely when the chord from $u_r$ to $u_{\ell }$ lies below the graph of $F$, which for convex $F$ is exactly the case $u_{\ell }>u_r$. Hence the convex entropy dissipates across every Lax-admissible shock. $\square$

Proposition 2 (uniqueness from $L^1$ contraction). If $u_1,u_2$ are entropy solutions of $u_t+F(u{)}_x=0$ with the same initial data $u_0\in L^{\infty }\cap L^1$ and $F^{\prime }$ Lipschitz on the data range, then $u_1=u_2$ a.e.

Proof. The Kruzhkov contraction (Theorem 1) gives $\Vert u_1(\cdot ,t)-u_2(\cdot ,t){\Vert }_{L^1}\le \Vert u_0-u_0{\Vert }_{L^1}=0$ for every $t>0$, so $u_1(\cdot ,t)=u_2(\cdot ,t)$ in $L^1$, hence a.e., for each $t$; since both are $L^{\infty }$ functions of $(x,t)$ this gives $u_1=u_2$ a.e. on ${\mathbb{R}}^n\times (0,\infty )$. $\square$

Proposition 3 (Lax-Oleinik is a weak solution). Let $F$ be uniformly convex $C^2$, $L=F^{\ast }$, $w_0={\int }_0^{\cdot }{u}_0$, and $u(x,t)=G((x-y(x,t))/t)$ with $y$ the minimizer and $G=(F^{\prime }{)}^{-1}$. Then $u$ is a weak solution of $u_t+F(u{)}_x=0$ with data $u_0$.

Proof. Let $w(x,t)={\min }_y\{tL((x-y)/t)+w_0(y)\}$ be the Hopf-Lax solution of $w_t+F(w_x)=0$, $w(\cdot ,0)=w_0$, which is Lipschitz and satisfies the Hamilton-Jacobi equation a.e. and in the viscosity sense 02.18.05. At a.e. $(x,t)$, $w$ is differentiable with $w_t+F(w_x)=0$ and $w_x=G((x-y)/t)=u$ (the minimizer condition $F^{\prime }(w_x)=(x-y)/t$ inverts to $w_x=G((x-y)/t)$). Differentiating the Hamilton-Jacobi equation in $x$ where $w\in C^2$ gives $w_{xt}+F^{\prime }(w_x)w_{xx}=0$, i.e. $u_t+F(u{)}_x=0$ classically off the shock set. Across shock curves $w$ is continuous with a corner, so $u=w_x$ jumps; testing the conservation law against $\phi \in C_c^1$ and integrating by parts, the corner contributions of the Lipschitz $w$ telescope (since $w$ has no jump, only $u=w_x$ does), leaving precisely the weak identity. The boundary term recovers $u_0$ because $w(\cdot ,t)\to w_0$ uniformly and $w_x\to u_0$ in $L_{\mathrm{loc}}^1$ as $t\downarrow 0$. Hence $u$ is a weak solution. $\square$

Proposition 4 (Lax-Oleinik is the entropy solution). Under the hypotheses of Proposition 3, $u$ satisfies the Kruzhkov entropy inequalities, hence is the unique entropy solution.

Proof. The Hopf-Lax $w$ is the viscosity solution of $w_t+F(w_x)=0$, characterized by the one-sided semiconcavity estimate $w(x+z,t)+w(x-z,t)-2w(x,t)\le C|z{|}^2/t$ (semiconcavity of value functions of convex problems) 02.18.05. Differentiating, this is the Oleinik one-sided bound $u(x+z,t)-u(x,t)\le z/(\theta t)$ (Exercise 7), which forbids upward jumps: every discontinuity of $u$ has $u_{\ell }\ge u_r$. By Proposition 1 each such jump dissipates every convex entropy, so $\eta (u{)}_t+q(u{)}_x\le 0$ holds across shocks; off shocks $u$ is classical and the entropy identity holds with equality. Summing the distributional contributions (the only singular part lives on the shock curves, where the inequality is $\le 0$), $u$ satisfies the Kruzhkov inequalities for all convex $\eta$, in particular for ${\eta }_k=|u-k|$. Uniqueness (Proposition 2) identifies $u$ as the entropy solution. $\square$

Proposition 5 (vanishing-viscosity limits satisfy the entropy inequality). Let $u^{\epsilon }$ solve $u_t^{\epsilon }+F(u^{\epsilon }{)}_x=\epsilon u_{xx}^{\epsilon }$ classically with uniform $L^{\infty }$ and $BV$ bounds, and $u^{\epsilon }\to u$ in $L_{\mathrm{loc}}^1$. Then $u$ is an entropy solution of $u_t+F(u{)}_x=0$.

Proof. Fix a convex $\eta \in C^2$ with flux $q^{\prime }={\eta }^{\prime }{F}^{\prime }$. Multiply the regularized equation by ${\eta }^{\prime }(u^{\epsilon })$: $$\eta (u^{\epsilon }{)}_t+q(u^{\epsilon }{)}_x=\epsilon {\eta }^{\prime }(u^{\epsilon })u_{xx}^{\epsilon }=\epsilon (\eta (u^{\epsilon }{)}_{xx}-{\eta }^{\prime \prime }(u^{\epsilon })(u_x^{\epsilon }{)}^2)\le \epsilon \eta (u^{\epsilon }{)}_{xx},$$ the inequality because ${\eta }^{\prime \prime }\ge 0$ makes the dropped term $-\epsilon {\eta }^{\prime \prime }(u^{\epsilon })(u_x^{\epsilon }{)}^2\le 0$. Test against any nonnegative $\phi \in C_c^1$: $\iint (\eta (u^{\epsilon }){\phi }_t+q(u^{\epsilon }){\phi }_x)\ge -\epsilon \iint \eta (u^{\epsilon }){\phi }_{xx}$. As $\epsilon \downarrow 0$, the uniform bounds and $u^{\epsilon }\to u$ in $L_{\mathrm{loc}}^1$ give $\eta (u^{\epsilon })\to \eta (u)$, $q(u^{\epsilon })\to q(u)$ in $L_{\mathrm{loc}}^1$ (continuity of $\eta ,q$ on the common $L^{\infty }$ range), the left side converges to $\iint (\eta (u){\phi }_t+q(u){\phi }_x)$, and the right side $\to 0$ because $\epsilon \iint \eta (u^{\epsilon }){\phi }_{xx}\to 0$. Hence $\iint (\eta (u){\phi }_t+q(u){\phi }_x)\ge 0$ for all $\phi \ge 0$, i.e. $\eta (u{)}_t+q(u{)}_x\le 0$ distributionally. This holds for every convex $\eta$, so $u$ is an entropy solution. $\square$

Connections Master

• The entropy solution is the differentiated twin of the viscosity solution of Hamilton-Jacobi 02.18.05: in one space dimension, if $w$ solves $w_t+F(w_x)=0$ then $u=w_x$ solves $u_t+F(u{)}_x=0$, a corner of $w$ becomes a downward jump of $u$, the touching-from-below test becomes the entropy inequality, and the very same doubling-of-variables argument proves comparison there and $L^1$ contraction here. The Lax-Oleinik formula for $u$ is literally the spatial derivative of the Hopf-Lax formula for $w$, so the two units share their entire convex-duality backbone.

• The vanishing-viscosity construction (Theorem 3) and the uniform $BV$ bounds run on the parabolic-smoothing and finite-speed machinery connected to the wave equation 02.13.04: the regularized equation $u_t^{\epsilon }+F(u^{\epsilon }{)}_x=\epsilon u_{xx}^{\epsilon }$ is a nonlinear perturbation of the heat equation, its maximum principle and total-variation diminishing structure give the compactness, and the hyperbolic finite-speed-of-propagation that bounds shock speeds is the nonlinear analogue of the light-cone causality there. The clean linear hyperbolic theory of 02.13.04 is the $F(u)=cu$ degenerate case in which no shock ever forms.

• The functional setting for weak solutions and the entropy dissipation rests on the Sobolev and $BV$ trace theory of 24.01.01: the test-function pairing that defines weak solutions, the one-sided traces $u_{\ell },u_r$ at a shock, and the $L^1$-$BV$ compactness used in the existence proof are all instances of the function-space machinery established there, specialized from the linear $H^s$/$W^{k,p}$ theory to the low-regularity $L^{\infty }\cap BV$ class that conservation laws force.

Historical & philosophical context Master

The recognition that nonlinear hyperbolic equations require discontinuous solutions predates the modern theory by a century: the jump relations across a shock were written by Rankine in 1870 and Hugoniot in 1887 for the gas-dynamics equations of compressible flow, derived from conservation of mass, momentum, and energy across a moving front. The mathematical difficulty — that the conservation form admits many weak solutions, only one of which is physical — was isolated in the 1950s. Eberhard Hopf's 1950 study of the viscous Burgers equation $u_t+u{u}_x=\mu u_{xx}$ ^{[Hopf 1950]} gave the explicit linearizing transformation and the vanishing-viscosity limit, showing concretely how diffusion selects the admissible jump. Peter Lax, in his 1954 and 1957 papers on weak solutions and hyperbolic systems ^{[Lax 1954] [Lax 1957]}, formulated the entropy condition $F^{\prime }(u_{\ell })\ge s\ge F^{\prime }(u_r)$ as a characteristic-geometry criterion and gave the Lax-Oleinik representation for convex flux.

Olga Oleinik's 1957 paper ^{[Oleinik 1957]} supplied the one-sided inequality $u(x+z,t)-u(x,t)\le z/(\theta t)$ and proved uniqueness for convex flux in one dimension through it. The decisive generalization came in Stanislav Kruzhkov's 1970 Matematicheskii Sbornik paper ^{[Kruzhkov 1970]}, which introduced the entropy pairs $|u-k|$ indexed by all constants $k$, defined entropy solutions for general (non-convex) flux in several space dimensions, and proved uniqueness and $L^1$ contraction by the doubling-of-variables technique — the same device Crandall and Lions would later adapt for viscosity solutions 02.18.05. The structure theory of $BV$ solutions, the Riemann problem, and the extension to systems were developed by Glimm, Lax, Dafermos, and Serre, recorded in the monographs of Dafermos ^{[Dafermos 2016]} and Serre ^{[Serre 1999]}.

Bibliography Master

@article{Kruzhkov1970,
  author  = {Kru{\v{z}}kov, Stanislav N.},
  title   = {First order quasilinear equations in several independent variables},
  journal = {Matematicheskii Sbornik (N.S.)},
  volume  = {81(123)},
  number  = {2},
  year    = {1970},
  pages   = {228--255}
}

@article{Oleinik1957,
  author  = {Oleinik, Olga A.},
  title   = {Discontinuous solutions of non-linear differential equations},
  journal = {Uspekhi Matematicheskikh Nauk},
  volume  = {12},
  number  = {3},
  year    = {1957},
  pages   = {3--73}
}

@article{Lax1957,
  author  = {Lax, Peter D.},
  title   = {Hyperbolic systems of conservation laws II},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {10},
  number  = {4},
  year    = {1957},
  pages   = {537--566}
}

@article{Lax1954,
  author  = {Lax, Peter D.},
  title   = {Weak solutions of nonlinear hyperbolic equations and their numerical computation},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {7},
  number  = {1},
  year    = {1954},
  pages   = {159--193}
}

@article{Hopf1950,
  author  = {Hopf, Eberhard},
  title   = {The partial differential equation $u_t + u u_x = \mu u_{xx}$},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {3},
  number  = {3},
  year    = {1950},
  pages   = {201--230}
}

@book{Dafermos2016,
  author    = {Dafermos, Constantine M.},
  title     = {Hyperbolic Conservation Laws in Continuum Physics},
  edition   = {4},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {325},
  publisher = {Springer},
  year      = {2016}
}

@book{Serre1999,
  author    = {Serre, Denis},
  title     = {Systems of Conservation Laws I: Hyperbolicity, Entropies, Shock Waves},
  publisher = {Cambridge University Press},
  year      = {1999}
}

@book{Evans2010,
  author    = {Evans, Lawrence C.},
  title     = {Partial Differential Equations},
  edition   = {2},
  series    = {Graduate Studies in Mathematics},
  volume    = {19},
  publisher = {American Mathematical Society},
  year      = {2010}
}