Entropy condition for quasi-linear evolution equations

Question

Let's consider the problem: $$ \partial_t u + \partial_x(f(u)) = 0, (x,t)\in \mathbb R \times \mathbb R^+.\\ u|_{t=0}=u_0. $$ I have seen three formulations for the entropy condition of this equation. The first two can be found in Evan's PDE book.

1. (Page 142.) Consider the characteristics of the above PDE, assuming that it is given by $(x(t),t).$ Then we have $x'(t) =f(u(x(t),t)).$ Since $u$ is constant along this curve, we actually have $x'(t)=\text{const.}$ So, we have straight line characteristics. In the case where we have a solution which is smooth expect for one single curve of discontinuity, the entropy condition says that we cannot have crossing of characteristics when we go backward in time. More precisely this means $f'(u_-) <\sigma < f'(u_+)$, where $\sigma$ is the speed of propagation of discontinuity, and $u_{\pm}$ are the left and right limit of $u$ at a point on the curve of discontinuity.

2. (On page 148 of the book) The entropy condition is that $u(x+z,t) -u(x,t) \leq \frac{Cz}{t}$ for all $x,z,t.$ It is explained in the book how this implies the first entropic condition stated above.

3. Here is a third definition: a solution $u\in L^\infty$ is entropic if for all $\varphi \in C_c^\infty$ and and $\eta$ convex and piecewise smooth, and $\psi\in C^1$ satisfying $\psi'=f'\eta ',$ $$ \int_0^\infty \int_{\mathbb R} \eta(u) \partial_t \varphi +\psi u \partial_x \psi dtdx + \int_\mathbb R \varphi(x,t=0) \eta(u_0(x))dx \geq 0.$$ This is the definition used for an important uniqueness result in the following paper: Kruzkov, S. N. 1970. First order quasilinear equations with several independent variables.

These definitions are certainly not equivalent, and some of them assume more things than others.

I therefore find it a bit hard to grasp the link/differences between those entropy inequalities. Are they just assumptions cooked up to make certain results work, so that we need invent new "entropy conditions" when we intend to prove something new?

We know there are plenty of technical bounds in PDE theory. How can I know whether a certain bound is considered an "entropy condition"? Are there any good ways to understand what "entropy" means here in general? (The connection to thermodynamics seems not so clear.)

Answer 1

This is an extended comment. Dennis Serre is on this site and can weigh in on this better than I can, if he is up to it.

"These definitions are certainly not equivalent, and some of them assume more things than others." I somewhat disagree. They are equivalent in the sense that each condition guarantees uniqueness within a certain class, and when you are in a setting when all the above notions make sense, the solutions will agree.

To my mind, only (3) is an "entropy condition". Each $\eta$ is an "entropy", and $\psi$ is the corresponding "entropy flux". For smooth and decaying solutions, $\int \eta(x,t) \, dx < +\infty$ and is conserved. You would like your weak solutions to acknowledge this somehow, either by conserving entropy or the next best thing, keeping it monotone. It makes sense to consider also the local version of the entropy conservation, which is captured by the "local entropy inequality" in (3). This is an equality for smooth solutions, but for non-smooth solutions, the inequality allows for the entropy to be dissipated. However, it turns out that this condition is enough to restore uniqueness!

One might add a condition (4), which is that the solution is obtained by vanishing viscosity approximation: $u$ is a suitable limit of solutions $u^\nu$ to the PDE $\partial_t u^\nu + \partial_x f(u^\nu) = \nu \partial_{xx}u^\nu$, where $\nu > 0$. The fact that (3) is allowed to be an inequality rather than an equality is related to this fact: The limiting solution "remembers" the dissipation, at least where it is non-smooth.