Transport Equations and Conservation Laws

1Introduction

This lecture explains how a transport equation describes a quantity carried by a flow and how this description leads to a conservation law. Having completed the construction of solutions for second-order linear PDEs, we return to first-order PDEs and extend the method of characteristics to nonlinear problems with discontinuities.

2Constant-Coefficient Transport

The equation

transports a profile at constant velocity $c$c. It is also a conservation law with flux $f\left(u\right)=cu$f(u)=cu. The value of $u$u remains constant along characteristics $x-ct=\text{constant}$x-ct=\text{constant}. For initial data $u(x,0)=g\left(x\right)$u(x,0)=g(x), the solution is

If $c>0$c>0, the profile moves to the right with velocity $c$c. Unlike heat diffusion, this motion translates the initial profile without smoothing it.

3Conservation Laws

A conservation lawConservation law relates a density $u$u and flux $f\left(u\right)$f(u) by

Here $u(x,t)$u(x,t) is the conserved quantity per unit length, while $f\left(u\right(x,t\left)\right)$f(u(x,t)) is the amount crossing $x$x to the right per unit time.

4From the Integral Balance to the PDE

The amount in an interval $[a,b]$[a,b] is ${\int }_a^{b}u(x,t)dx$\int_a^b u(x,t)\,dx. Its rate of change is determined by flux entering at $a$a and leaving at $b$b:

For sufficiently smooth $u$u, differentiation under the integral and the fundamental theorem of calculus give

Since this holds for every interval, localization yields $u_t+f(u{)}_x=0$u_t+f(u)_x=0. Conversely, integrating the PDE over $[a,b]$[a,b] recovers the integral balance.

On the real line, suppose $u(·,t)\in L^1\left(\mathbb{R}\right)$u(\cdot,t)\in L^1(\mathbb R), $f\left(u\right(x,t\left)\right)\to 0$f(u(x,t))\to0 as $x\to \pm \mathrm{\infty }$x\to\pm\infty, and the solution has enough regularity to interchange differentiation and the relevant limits. Letting $a\to -\mathrm{\infty }$a\to-\infty and $b\to \mathrm{\infty }$b\to\infty gives

Thus local flux can redistribute the quantity, but the total amount remains constant when there is no flux at infinity.

5Weak Solutions

Nonlinear conservation laws can produce shocks even from smooth initial data. At a shock, $u_x$u_x need not exist as an ordinary function, so one replaces pointwise classical solutions with weak solutions defined by integration. Assume

meaning that their absolute values are integrable on every bounded spacetime region. Let the initial data be $g\in L^{1_{\mathrm{l}\mathrm{o}\mathrm{c}}}\left(\mathbb{R}\right)$g\in L^1_{\mathrm{loc}}(\mathbb R). A function $u$u is a weak solution with $u(x,0)=g\left(x\right)$u(x,0)=g(x) if, for every test function $\phi \in C_{c^{\mathrm{\infty }}}(\mathbb{R}\times [0,\mathrm{\infty }\left)\right)$\varphi\in C_c^\infty(\mathbb R\times[0,\infty)) that is smooth up to $t=0$t=0,

For a smooth solution, integration by parts recovers the PDE and initial condition. The definition differentiates the test function rather than $u$u, and therefore permits discontinuities.

6Finite Propagation Speed

For a transport equation, information in the initial profile moves along characteristics. A discontinuity in the initial data propagates along them as well. This finite-speed behavior contrasts with heat diffusion. For nonlinear conservation laws, selecting the physically relevant discontinuous solution additionally requires an entropy condition.

7Nonlinear Conservation Laws and Intersecting Characteristics

For a smooth solution,

can be written as $u_t+f^{\prime }\left(u\right)u_x=0$u_t+f'(u)u_x=0. The characteristic velocity $f^{\prime }\left(u\right)$f'(u) depends on the value of $u$u, so different states travel at different speeds. When characteristics intersect, a classical solution breaks down and a weak solution containing a shock becomes necessary.

Suppose a discontinuity moves with speed $s$s, with left and right states $u_L$u_L and $u_R$u_R. The Rankine--Hugoniot condition is

Applying the integral balance to a thin spacetime region crossing the discontinuity gives

This condition enforces conservation across the shock, but by itself does not make a weak solution unique.

8Entropy Conditions and the Burgers Equation

For the one-dimensional scalar conservation laws considered here, an entropy conditionEntropy condition is an admissibility criterion that selects a physically relevant weak solution. Under standard hypotheses, the vanishing-viscosity limit of

as $\epsilon \downarrow 0$\varepsilon\downarrow0 agrees with the Kruzhkov entropy solution. This lecture uses vanishing viscosity as a selection principle; proofs of existence, convergence, and uniqueness lie outside its scope. For a convex flux with $f^{\prime }\text{'}\left(u\right)>0$f''(u)>0 and a shock satisfying $u_L>u_R$u_L>u_R, the Lax entropy condition

requires characteristics from both sides to enter the shock.

Let initial data $g_1,g_2$g_1,g_2 and their corresponding entropy solutions $u_1,u_2$u_1,u_2 take values in a common interval $I$I, and assume

The standard local $L^1$L^1 contraction estimate states that, for every $a<b$a<b,

Consequently, if $g_1=g_2$g_1=g_2 on $[a-Mt,b+Mt]$[a-Mt,b+Mt], then $u_1=u_2$u_1=u_2 almost everywhere on $[a,b]$[a,b] at time $t$t. This estimate is the precise finite-propagation statement for entropy solutions. Its proof is part of the standard local contraction theory and lies outside the scope of this lecture.

The standard example is the Burgers equation with $f\left(u\right)=u^2/2$f(u)=u^2/2:

Consider Riemann data equal to $u_L$u_L for $x<0$x<0 and $u_R$u_R for $x>0$x>0. If $u_L=1$u_L=1 and $u_R=0$u_R=0, the Rankine--Hugoniot speed is $s=1/2$s=1/2, and $1>s>0$1>s>0 verifies the entropy condition. If instead $u_L=0$u_L=0 and $u_R=1$u_R=1, the characteristics spread into a rarefaction fan, and for $t>0$t>0 the entropy solution is

9Characteristic Geometry

For constant-coefficient transport, the characteristics $x-ct=\text{constant}$x-ct=\text{constant} are parallel lines. For a nonlinear conservation law, their slopes depend on $u$u. Diverging characteristics create a rarefaction, whereas intersecting characteristics require a shock. This geometric picture connects the method of characteristics with conservation laws.

10Scope of Validity

The theory of nonlinear conservation laws must address not only existence of weak solutions but also entropy-based uniqueness, stability with respect to initial data, and interactions among shocks. This lecture has introduced only one-dimensional scalar laws with convex flux.

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