Scalar conservation laws $\partial_t u + \partial_x f(u) = 0$ model the transport of conserved quantities — mass, momentum, energy — in fluid dynamics, traffic flow, and gas dynamics. Unlike elliptic or parabolic equations, hyperbolic conservation laws develop singularities (shocks) in finite time from smooth initial data, forcing the theory into the framework of weak solutions. The central difficulty is that weak solutions are non-unique: one must impose additional *entropy conditions* to select the physically correct solution. [motivation] ## Motivation ### Characteristics and Shock Formation The [method of characteristics](/page/Method%20of%20Characteristics) transforms the PDE $\partial_t u + f'(u)\partial_x u = 0$ into a family of ODEs: along the curve $\dot{x}(t) = f'(u_0(x(0)))$, the solution $u$ is constant. When $f$ is nonlinear, different characteristics travel at different speeds. If the initial data $u_0$ is decreasing in a region (so that faster characteristics are behind slower ones), the characteristic lines converge and cross in finite time. At the crossing point, the solution would need to take two values simultaneously — a contradiction that signals the formation of a **shock**, a propagating discontinuity. ### The Failure of Classical Solutions The crossing of characteristics is not an artifact of the method — it reflects a genuine breakdown of regularity. For Burgers' equation $\partial_t u + u \, \partial_x u = 0$ with smooth decreasing initial data, one can show that $\partial_x u \to -\infty$ in finite time, and no $C^1$ solution can exist beyond that point. The physical quantity $u$ remains bounded (it is merely redistributed), but its spatial gradient blows up. To continue the solution past this time, we must enlarge the solution class to allow discontinuities. ### Weak Solutions and Non-Uniqueness The weak formulation — requiring $\int\!\!\int (u \, \partial_t\varphi + f(u)\,\partial_x\varphi) \, d\mathcal{L}^2 = 0$ for all [test functions](/page/Test%20Function) $\varphi$ — accommodates shocks, but it admits too many solutions. Given the same initial data, there may be infinitely many weak solutions, most of which are unphysical. The prototypical failure is the **expansion shock**: a discontinuity that, while satisfying the weak formulation and the [Rankine-Hugoniot condition](/theorems/578), violates the second law of thermodynamics by decreasing entropy. The purpose of the entropy condition is to rule out such solutions and restore uniqueness. [/motivation] ## Scalar Conservation Laws To make the discussion precise, we need to specify what equation we are solving, what regularity the flux has, and what data we prescribe. The following definition [sets](/page/Set) the framework for the entire article. [definition: Scalar Conservation Law] Let $f: \mathbb{R} \to \mathbb{R}$ be a Lipschitz [continuous](/page/Continuity) [function](/page/Function) (the **flux**). The **scalar conservation law** is the first-order PDE \begin{align*} \partial_t u + \partial_x f(u) = 0 \quad \text{in } \mathbb{R} \times (0, T), \end{align*} where $u: \mathbb{R} \times [0, T) \to \mathbb{R}$ is the unknown (the conserved density) and $T > 0$. The **Cauchy problem** supplements the PDE with initial data $u(\cdot, 0) = u_0$ for a given $u_0 \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$. [/definition] The quantity $f'(u)$ is the **characteristic speed** — the speed at which information propagates through the solution. When $f$ is linear ($f(u) = au$), the equation is the transport equation with constant speed $a$, and characteristics never cross. All the difficulties of the theory arise from the nonlinearity of $f$. ## Weak Solutions Once shocks form, no $C^1$ solution exists. We relax the notion of solution by testing against smooth functions and integrating by parts, transferring all [derivatives](/page/Derivative) to the test function. This allows $u$ to have jump discontinuities while still satisfying the equation in an [integral](/page/Integral) sense. [definition: Weak Solution] A bounded measurable function $u \in L^\infty(\mathbb{R} \times (0, T))$ is a **weak solution** of the Cauchy problem if \begin{align*} \int_0^T \!\!\int_{\mathbb{R}} \left(u \, \partial_t\varphi + f(u) \, \partial_x\varphi\right) d\mathcal{L}^2 + \int_{\mathbb{R}} u_0(x) \, \varphi(x, 0) \, d\mathcal{L}^1(x) = 0 \end{align*} for every test function $\varphi \in C_c^\infty(\mathbb{R} \times [0, T))$. [/definition] A piecewise smooth weak solution with a jump discontinuity must satisfy a compatibility condition along the shock curve. Without such a condition, the jump could be arbitrary — any two states $u_L, u_R$ at any speed $\sigma$ would be acceptable. The Rankine-Hugoniot condition constrains $\sigma$ as a function of $u_L$, $u_R$, and $f$, reflecting the underlying conservation law: the total flux entering the shock from the left must equal the total flux leaving to the right. [quotetheorem:578] The Rankine-Hugoniot condition is necessary for a discontinuity to be a weak solution, but it says nothing about which discontinuities are *admissible*. Both compression shocks (where characteristics converge into the shock from both sides) and expansion shocks (where characteristics diverge away) satisfy R-H with the same speed formula. [example: Non Uniqueness For Burgers Equation] Consider Burgers' equation $\partial_t u + \partial_x(u^2/2) = 0$ with initial data $u_0(x) = 0$ for $x < 0$ and $u_0(x) = 1$ for $x > 0$ (so $u_L = 0$, $u_R = 1$). Two weak solutions exist: The **rarefaction wave** $u(x, t) = 0$ for $x < 0$, $u(x, t) = x/t$ for $0 \leq x \leq t$, and $u(x, t) = 1$ for $x > t$. This is continuous and self-similar. The **expansion shock** $u(x, t) = 0$ for $x < t/2$ and $u(x, t) = 1$ for $x > t/2$, traveling at speed $\sigma = (f(1) - f(0))/(1 - 0) = 1/2$. This satisfies R-H but is unphysical: characteristics on the left travel at speed $f'(0) = 0 < 1/2 = \sigma$ and on the right at speed $f'(1) = 1 > 1/2 = \sigma$, so information flows *away* from the shock on both sides. The shock has no mechanism to sustain itself. The entropy condition selects the rarefaction as the unique physical solution. [/example] ## Entropy Conditions The mathematical device that selects the correct weak solution comes from thermodynamics: the physical entropy of a system can only increase across a shock. This principle is formalised through entropy-flux pairs. ### Vanishing Viscosity Motivation The physically correct solution is the one obtained as the [limit](/page/Limit) of viscous approximations $\partial_t u^\varepsilon + \partial_x f(u^\varepsilon) = \varepsilon \, \partial_x^2 u^\varepsilon$ as $\varepsilon \to 0$. The viscous term models diffusion (heat conduction, viscosity), which is always present in real systems at small scales. The entropy condition is the mathematical consequence of this limiting process: multiplying the viscous equation by $\eta'(u^\varepsilon)$ for a convex function $\eta$ produces $\partial_t \eta(u^\varepsilon) + \partial_x q(u^\varepsilon) = \varepsilon \, \partial_x(\eta' \partial_x u^\varepsilon) - \varepsilon \eta''(u^\varepsilon)(\partial_x u^\varepsilon)^2 \leq \varepsilon \, \partial_x(\eta' \partial_x u^\varepsilon)$, using $\eta'' \geq 0$. In the limit $\varepsilon \to 0$, the right side vanishes but the inequality persists, producing the entropy inequality. ### Entropy-Flux Pairs To formalise the vanishing viscosity argument, we need a class of test inequalities parametrised by convex functions. [definition: Entropy Flux Pair] A pair of functions $(\eta, q)$ with $\eta, q: \mathbb{R} \to \mathbb{R}$ is an **entropy-flux pair** for the conservation law $\partial_t u + \partial_x f(u) = 0$ if $\eta \in C^2(\mathbb{R})$ is convex and $q$ satisfies the compatibility relation $q'(u) = \eta'(u) f'(u)$. The function $\eta$ is called an **entropy** and $q$ the associated **entropy flux**. [/definition] For a smooth solution, the chain rule gives $\partial_t \eta(u) + \partial_x q(u) = 0$ — entropy is conserved along smooth flows. Across shocks, this equality breaks down and is replaced by an inequality: entropy is *produced* (in the mathematical sign convention, where the inequality is $\leq 0$, entropy is *dissipated* in the sense that the entropy functional decreases). ### Kružkov Entropy Solutions Kružkov's insight was that the family of entropies $\eta_k(u) = |u - k|$ (parametrised by $k \in \mathbb{R}$) suffices to characterise the physically correct solution. These are convex but not smooth; the associated fluxes are $q_k(u) = \operatorname{sgn}(u - k)(f(u) - f(k))$. The entropy condition requires the inequality $\partial_t |u - k| + \partial_x[\operatorname{sgn}(u-k)(f(u)-f(k))] \leq 0$ to hold distributionally for *every* $k$. [definition: Entropy Solution] A weak solution $u \in L^\infty(\mathbb{R} \times (0, T))$ is an **entropy solution** (in the sense of Kružkov) if for every $k \in \mathbb{R}$ and every non-negative test function $\varphi \in C_c^\infty(\mathbb{R} \times [0, T))$: \begin{align*} \int_0^T \!\!\int_{\mathbb{R}} \left(|u - k| \, \partial_t\varphi + \operatorname{sgn}(u - k)(f(u) - f(k)) \, \partial_x\varphi\right) d\mathcal{L}^2 + \int_{\mathbb{R}} |u_0 - k| \, \varphi(x, 0) \, d\mathcal{L}^1 \geq 0. \end{align*} [/definition] This definition is equivalent to requiring $\partial_t \eta(u) + \partial_x q(u) \leq 0$ distributionally for *all* convex entropies $(\eta, q)$. The Kružkov formulation is preferred because it works with the specific family $|u - k|$, which is amenable to the doubling-of-variables technique in the uniqueness proof. For a more detailed treatment of the various entropy conditions (Lax, Oleinik) and their equivalences, see the [entropy condition page](/page/Entropy%20Condition%20(Scalar%20Conservation%20Laws)). ## Uniqueness The raison d'être of the entropy condition is the following theorem: entropy solutions are unique and depend continuously on the initial data in $L^1$. [quotetheorem:579] The proof is Kružkov's celebrated **doubling of variables** argument. The entropy inequality for $u$ is written with $k = v(y, s)$ and for $v$ with $k = u(x, t)$. Testing both against a four-variable kernel $\psi((x+y)/2, (t+s)/2) \, \rho_\varepsilon(x-y) \rho_\varepsilon(t-s)$ and letting $\varepsilon \to 0$ collapses the four-dimensional inequality to a two-dimensional one for $|u - v|$, from which the contraction follows. The $L^1$ contraction is far stronger than mere uniqueness: it says the solution operator $u_0 \mapsto u(\cdot, t)$ is a contraction on $L^1$. This is the nonlinear analogue of the energy dissipation for the [heat equation](/page/Heat%20Equation) — in both cases, the equation irons out differences between solutions over time. ## Existence Uniqueness without existence is vacuous. The entropy condition does not merely filter weak solutions — it also characterises the limit of physically motivated approximations. [quotetheorem:580] The proof adds a viscous regularisation $\varepsilon \, \partial_x^2 u^\varepsilon$ to the conservation law, solves the resulting parabolic problem (which has smooth solutions by the theory of the [heat equation](/page/Heat%20Equation)), derives uniform $L^\infty$ and $BV$ bounds, extracts a convergent subsequence, and verifies that the limit satisfies the Kružkov inequality. The convexity of $\eta$ is essential in Step 4: the dissipative term $-\varepsilon \eta''(u^\varepsilon)(\partial_x u^\varepsilon)^2 \leq 0$ is what produces the entropy inequality in the limit. ## The Riemann Problem General initial data can produce multiple interacting shocks and rarefactions, making the solution structure extremely complex. To build intuition and develop the theory, we start with the simplest non-trivial initial data: a single jump discontinuity at $x = 0$. The resulting Riemann problem is solvable by self-similar methods, and its solutions — shocks and rarefaction fans — are the building blocks from which solutions with general data are constructed (via Glimm's scheme or front-tracking algorithms). [definition: Riemann Problem] The **Riemann problem** for $\partial_t u + \partial_x f(u) = 0$ is the Cauchy problem with initial data \begin{align*} u_0(x) = \begin{cases} u_L & \text{if } x < 0, \\ u_R & \text{if } x > 0, \end{cases} \end{align*} where $u_L, u_R \in \mathbb{R}$ are given constants. [/definition] The solution is self-similar ($u(x, t) = v(x/t)$) and takes one of two forms depending on the relationship between $u_L$, $u_R$, and the convexity of $f$. ### Shocks and Rarefactions For a **convex flux** ($f'' > 0$), the characteristic speed $f'(u)$ is increasing. The Riemann problem has a sharp dichotomy. When $u_L > u_R$: characteristics on the left are faster than those on the right, they collide, and the entropy solution is a **shock** traveling at the Rankine-Hugoniot speed $\sigma = (f(u_L) - f(u_R))/(u_L - u_R)$. When $u_L < u_R$: characteristics on the left are slower, they fan out, and the entropy solution is a **rarefaction wave** $u(x, t) = (f')^{-1}(x/t)$ for $f'(u_L) \leq x/t \leq f'(u_R)$. ### The Lax Entropy Condition The Kružkov inequality is a global (integral) condition. For piecewise smooth solutions of conservation laws with convex flux, it reduces to a simple pointwise inequality at each shock — a condition on how the characteristic speeds on either side relate to the shock speed. This local criterion is often easier to verify in practice, especially for the Riemann problem. [definition: Lax Entropy Condition] A shock with speed $\sigma$ separating states $u_L$ (left) and $u_R$ (right) satisfies the **Lax entropy condition** if \begin{align*} f'(u_L) > \sigma > f'(u_R). \end{align*} [/definition] This means characteristics on the left impinge on the shock from behind ($f'(u_L) > \sigma$), and characteristics on the right are overtaken by it ($f'(u_R) < \sigma$). Information flows *into* the shock from both sides, making it stable. An expansion shock has the opposite inequality — characteristics flow away from it — and is ruled out. [example: Burgers Shock] For Burgers' equation ($f(u) = u^2/2$, so $f'(u) = u$) with $u_L = 2$ and $u_R = 1$: the Rankine-Hugoniot speed is $\sigma = (4/2 - 1/2)/(2 - 1) = 3/2$. The Lax condition requires $f'(u_L) = 2 > 3/2 > 1 = f'(u_R)$, which holds. The entropy solution is the shock $u(x, t) = 2$ for $x < 3t/2$ and $u(x, t) = 1$ for $x > 3t/2$. [/example] [example: Burgers Rarefaction] For Burgers' equation with $u_L = 1$ and $u_R = 2$: a shock would have speed $\sigma = 3/2$, but $f'(u_L) = 1 < 3/2$, violating Lax. Instead, characteristics fan out from $x = 0$, and the entropy solution is \begin{align*} u(x, t) = \begin{cases} 1 & \text{if } x < t, \\ x/t & \text{if } t \leq x \leq 2t, \\ 2 & \text{if } x > 2t. \end{cases} \end{align*} This continuous rarefaction wave satisfies the conservation law classically in each region and connects $u_L$ to $u_R$ through the self-similar profile $u = x/t$. [/example] ## References - Evans, L. C. (2010). *Partial Differential Equations* (2nd ed.). American Mathematical Society. - Kružkov, S. N. (1970). First order quasilinear equations in several independent variables. *Mathematics of the USSR-Sbornik*, 10(2), 217–243. - Dafermos, C. M. (2016). *Hyperbolic Conservation Laws in Continuum Physics* (4th ed.). Springer. - LeVeque, R. J. (2002). *Finite Volume Methods for Hyperbolic Problems*. Cambridge University Press.