Conservation laws describe quantities transported through space without being created or destroyed. Even smooth initial data can form discontinuities, so weak solutions and an additional entropy principle are essential.
Weak formulation and jump condition
Definition. A bounded function \(u\) is a weak solution of \[ u_t+\partial_x f(u)=0,\qquad u(x,0)=u_0(x), \] if for every \(\varphi\in C_c^\infty(\mathbb R\times[0,\infty))\), \[ \int_0^\infty\!\!\int_{\mathbb R} \bigl(u\varphi_t+f(u)\varphi_x\bigr)\,\mathrm{d}x\,\mathrm{d}t +\int_{\mathbb R}u_0(x)\varphi(x,0)\,\mathrm{d}x=0. \]
Proposition (Rankine-Hugoniot). If a piecewise smooth weak solution jumps from \(u_-\) to \(u_+\) across \(x=s(t)\), then \[ s'(t)[u]=[f(u)],\qquad s'(t)=\frac{f(u_+)-f(u_-)}{u_+-u_-}. \]
Proof. Integrate the conservation law over a thin spacetime control volume crossing the curve. Flux through the two vertical sides converges to \(f(u_-)-f(u_+)\), while transport through the moving interface contributes \(s'(t)(u_+-u_-)\). Vanishing of the total boundary flux gives the identity.
Characteristics and shock formation
Where \(u\) is smooth, \[ u_t+f'(u)u_x=0. \] Along \(x'(t)=f'(u)\), one has \(u'=0\). Thus the characteristic starting at \(\xi\) is \[ x(t)=\xi+t f'(u_0(\xi)),\qquad u(x(t),t)=u_0(\xi). \] Differentiation gives \[ u_x(x(t),t)= \frac{u_0'(\xi)} {1+t f''(u_0(\xi))u_0'(\xi)}. \] For convex \(f\), negative initial slope can make the denominator vanish in finite time: characteristics cross and a shock forms.
Entropy solutions
Definition (Kruzhkov). A weak solution is an entropy solution if, for every \(k\in\mathbb R\), \[ \partial_t|u-k| +\partial_x\!\left[ \operatorname{sgn}(u-k)(f(u)-f(k)) \right]\le0 \] in distributions.
For a convex flux, this condition excludes expansion shocks. A shock from \(u_-\) to \(u_+\) is admissible precisely when characteristics enter it: \[ f'(u_+)\le \frac{f(u_+)-f(u_-)}{u_+-u_-} \le f'(u_-). \] This is the Lax condition.
Theorem (Kruzhkov contraction). For \(u_0,v_0\in L^1\cap L^\infty\), scalar entropy solutions satisfy \[ \|u(t)-v(t)\|_{L^1(\mathbb R)} \le \|u_0-v_0\|_{L^1(\mathbb R)}. \] Hence the entropy solution is unique and depends continuously on its data.
Proof outline. Apply the entropy inequality for \(u(x,t)\) with \(k=v(y,s)\), and the corresponding inequality for \(v(y,s)\) with \(k=u(x,t)\). Add them and test against a nonnegative approximation to \(\delta_{x=y}\delta_{t=s}\psi((t+s)/2)\). Passing to the diagonal gives \[ \partial_t|u-v|+ \partial_x\bigl(\operatorname{sgn}(u-v)(f(u)-f(v))\bigr)\le0. \] Integration in \(x\) removes the flux term and yields the contraction.
The Riemann problem
Example. For a strictly convex flux and data \(u_0=u_L\) on \(x<0\), \(u_0=u_R\) on \(x>0\): if \(u_L>u_R\), the entropy solution is a shock moving with \[ \sigma=\frac{f(u_L)-f(u_R)}{u_L-u_R}. \] If \(u_L<u_R\), it is the rarefaction \[ u(x,t)= \begin{cases} u_L,&x/t\le f'(u_L),\\ (f')^{-1}(x/t),&f'(u_L)<x/t<f'(u_R),\\ u_R,&x/t\ge f'(u_R). \end{cases} \] The Burgers flux \(f(u)=u^2/2\) makes these formulas explicit.
For systems \(U_t+F(U)_x=0\), each eigenvalue of \(DF(U)\) defines a characteristic family. Genuine nonlinearity, linear degeneracy, interacting waves, and loss of a universal entropy theory make the system case substantially harder.