Entropy condition
An entropy condition is an additional admissibility criterion imposed on weak solutions of a nonlinear conservation law. Its principal function is to distinguish the physically relevant solution from other distributional solutions that satisfy the same initial data and conservation equation. The condition expresses irreversibility by requiring a suitable entropy quantity to dissipate, or at least not to increase, as discontinuities develop.
For a scalar conservation law in one spatial dimension,
[ \partial_t u+\partial_x f(u)=0, \qquad u(x,0)=u_0(x), ]
a classical differentiable solution generally exists only until characteristics intersect. Beyond that time, the solution is represented in the weak sense and may contain shock waves. Weakness alone does not provide uniqueness because discontinuities corresponding to compressive shocks and discontinuities corresponding to nonphysical expansion shocks can both satisfy the conservation law. Entropy conditions restore uniqueness by excluding the latter class.
Weak solutions and shock discontinuities
A locally integrable function (u) is a weak solution when
[ \int_0^\infty\int_{\mathbb R} \left( u,\partial_t\varphi+f(u),\partial_x\varphi \right),dx,dt + \int_{\mathbb R}u_0(x)\varphi(x,0),dx =0 ]
for every smooth, compactly supported test function (\varphi). This formulation remains meaningful when (u) is discontinuous, since derivatives are transferred from (u) to the test function.
Suppose that a discontinuity separates constant states (u_L) and (u_R) and propagates with speed (s). The weak formulation then yields the Rankine–Hugoniot condition,
[ s(u_R-u_L)=f(u_R)-f(u_L). ]
This relation enforces conservation across the discontinuity but does not determine whether the discontinuity is admissible. For a strictly convex flux, the same algebraic relation permits both a compressive shock with (u_L>u_R) and an expansion shock with (u_L<u_R). The second configuration conflicts with the characteristic evolution because its characteristics move away from the discontinuity rather than enter it.
The ambiguity is visible in the inviscid Burgers equation,
[ \partial_tu+\partial_x\left(\frac{u^2}{2}\right)=0. ]
For Riemann initial data with (u_L>u_R), the admissible solution is a shock moving at
[ s=\frac{u_L+u_R}{2}. ]
When (u_L<u_R), the admissible solution is instead a continuous rarefaction fan. A discontinuous expansion shock still satisfies the Rankine–Hugoniot relation, but it violates every standard entropy criterion.
Entropy pairs
An entropy pair for a scalar conservation law consists of a convex function (\eta(u)) and an associated entropy flux (q(u)) satisfying
[ q'(u)=\eta'(u)f'(u). ]
Every smooth solution obeys the additional conservation identity
[ \partial_t\eta(u)+\partial_xq(u)=0. ]
For discontinuous solutions, the corresponding entropy condition is the distributional inequality
[ \partial_t\eta(u)+\partial_xq(u)\leq 0. ]
Equivalently, for every nonnegative smooth test function (\varphi) with compact support,
[ \int_0^\infty\int_{\mathbb R} \left( \eta(u),\partial_t\varphi + q(u),\partial_x\varphi \right),dx,dt + \int_{\mathbb R}\eta(u_0(x))\varphi(x,0),dx \geq 0. ]
Across a shock, this distributional inequality reduces to
[ q(u_R)-q(u_L)-s\bigl(\eta(u_R)-\eta(u_L)\bigr)\leq 0. ]
The inequality states that the entropy production concentrated on the shock is nonpositive under the mathematical sign convention. In thermodynamic notation, entropy is commonly assigned the opposite sign, so the corresponding physical entropy is nondecreasing. Mathematical entropy is therefore a structural generalization of thermodynamic entropy, rather than an assertion that every conservation law describes thermal processes.
Convexity is essential because it makes entropy sensitive to the direction in which characteristics are compressed. For scalar equations, requiring the inequality for a sufficiently rich family of convex entropies eliminates expansion shocks and supplies a contraction principle for the resulting solution operator.
Historical development
The modern entropy framework developed from the study of nonlinear wave propagation during the mid-twentieth century. Eberhard Hopf obtained an explicit representation of solutions to viscous Burgers equations and analyzed the zero-viscosity limit, providing an early connection between parabolic regularization and shock admissibility.
During the 1950s, Olga Oleinik and You Watanabe established one-sided estimates for scalar conservation laws with convex flux. Their analysis related the decay of positive spatial increments to the exclusion of expansion discontinuities. In a common normalized form, the resulting estimate for Burgers-type evolution is
[ \frac{u(x+a,t)-u(x,t)}{a}\leq \frac{1}{t}, \qquad a>0,\quad t>0. ]
For a general uniformly convex flux satisfying (f''(u)\geq c>0), the corresponding upper bound is proportional to ((ct)^{-1}). This condition permits downward jumps while preventing upward jumps, in agreement with the characteristic geometry of compressive shocks and rarefaction waves.
The one-sided estimate is closely connected with Oleinik’s (E)-condition, which compares the slope of a flux chord across a shock with slopes of intermediate chords. For a discontinuity from (u_L) to (u_R), the condition requires the shock speed to be positioned so that characteristics on both sides enter the discontinuity. Under strict convexity, this formulation is equivalent to the scalar entropy inequality.
Kružkov entropy solutions
For general scalar conservation laws, the standard uniqueness theory is based on the entropy family introduced by S. N. Kružkov. For every constant (k\in\mathbb R), the Kružkov entropy and entropy flux are
[ \eta_k(u)=|u-k| ]
and
[ q_k(u)=\operatorname{sgn}(u-k)\bigl(f(u)-f(k)\bigr). ]
An entropy solution satisfies
[ \partial_t|u-k| + \partial_x \left[ \operatorname{sgn}(u-k)\bigl(f(u)-f(k)\bigr) \right] \leq 0 ]
in the sense of distributions for every (k). Although (\eta_k) is not differentiable at (u=k), it arises as a limit of smooth convex entropies and is well suited to comparison arguments.
The Kružkov inequalities imply the (L^1)-contraction property
[ |u(\cdot,t)-v(\cdot,t)|{L^1(\mathbb R)} \leq |u_0-v_0|{L^1(\mathbb R)} ]
for entropy solutions (u) and (v) whose initial difference is integrable. Contraction gives uniqueness, continuous dependence on the initial data, and compatibility with the semigroup structure of the evolution. The same framework extends to scalar laws in several spatial dimensions, where the flux is vector-valued and the entropy flux has one component for each spatial coordinate.
Vanishing viscosity
The entropy condition is also selected by vanishing viscosity. The hyperbolic equation is replaced by the parabolic regularization
[ \partial_tu^\varepsilon+\partial_xf(u^\varepsilon)
\varepsilon,\partial_{xx}u^\varepsilon, \qquad \varepsilon>0. ]
For a convex entropy (\eta), multiplication by (\eta'(u^\varepsilon)) gives
[ \partial_t\eta(u^\varepsilon) + \partial_xq(u^\varepsilon)
\varepsilon,\partial_x \left( \eta'(u^\varepsilon)\partial_xu^\varepsilon \right)
\varepsilon,\eta''(u^\varepsilon) \left|\partial_xu^\varepsilon\right|^2. ]
The final term is nonpositive because (\eta) is convex. When (u^\varepsilon) converges as (\varepsilon\to0^+), the limit consequently satisfies the entropy inequality. For scalar conservation laws under standard boundedness assumptions, this limit agrees with the unique Kružkov entropy solution.
The viscous traveling-wave equation also explains the directionality of shock admissibility. A compressive shock can be resolved into a smooth transition layer whose thickness is proportional to (\varepsilon). An expansion shock has no corresponding stable viscous profile with the required end states, so it does not survive the zero-viscosity limit.
Lax condition for systems
For a system of conservation laws,
[ \partial_tU+\partial_xF(U)=0, ]
the state (U) is vector-valued, and the Jacobian matrix (DF(U)) determines several characteristic families. The scalar ordering of characteristic speeds is replaced by the eigenstructure of this matrix, making admissibility more dependent on the geometry of the system.
Peter Lax formulated a characteristic condition for shocks in strictly hyperbolic systems. A shock belonging to the (k)-th characteristic family satisfies a compressivity relation in which the (k)-characteristics enter the shock from both sides. With the characteristic speeds ordered as
[ \lambda_1(U)<\lambda_2(U)<\cdots<\lambda_n(U), ]
a (k)-shock with speed (s) has the central inequalities
[ \lambda_k(U_R)<s<\lambda_k(U_L), ]
together with the ordering relations that keep the remaining characteristic families on their appropriate sides of the shock. This is the system-level analogue of the compressive condition for a convex scalar flux.
A system entropy is a scalar function (\eta(U)) with an entropy flux (q(U)) satisfying
[ \nabla q(U)=\nabla\eta(U),DF(U). ]
When (\eta) is strictly convex, it also provides a symmetrizing structure for the system. Across an admissible shock, the inequality
[ q(U_R)-q(U_L)-s\bigl(\eta(U_R)-\eta(U_L)\bigr)\leq0 ]
expresses entropy dissipation. Unlike the scalar case, a single convex entropy does not automatically yield a complete uniqueness theory for arbitrary large solutions of a system.
Nonclassical shocks
Some regularizations produce admissible discontinuities that do not satisfy the classical Lax compressivity condition. This occurs in models containing competing diffusion and dispersion, for which the small-scale equation can take the form
[ \partial_tu+\partial_xf(u)
\varepsilon,\partial_{xx}u + \delta,\partial_{xxx}u. ]
The limiting shock set depends on the relative scaling of (\delta) and (\varepsilon). Entropy dissipation alone can then leave several weak solutions admissible. A kinetic relation, derived from the traveling-wave structure of the regularized equation, supplies the additional constitutive information needed to determine the selected shock.
This phenomenon does not contradict the scalar Kružkov theory because the limiting mechanism differs from purely positive viscosity. It instead shows that an entropy condition encodes a particular class of unresolved small-scale dynamics. Classical entropy solutions correspond to dissipative regularizations dominated by viscosity, while nonclassical solutions retain information from a more complicated balance within the internal shock layer.
Relation to irreversible evolution
Entropy conditions convert a formally time-reversible differential equation into a forward evolution with irreversible loss at shocks. Before singularities form, classical solutions satisfy entropy equalities and retain their local information. After characteristic crossing, the entropy solution merges incoming states into discontinuities, and the associated contraction property prevents reconstruction of the earlier configuration from later data.
The entropy inequality therefore has three mutually connected roles. It defines admissibility at discontinuities through local entropy production, identifies limits of dissipative regularizations, and produces the stability estimates required for a well-defined solution semigroup. For scalar conservation laws these roles lead to equivalent formulations under broad hypotheses, whereas systems and nonclassical models require additional structural conditions.