Usbrth

After taking a course of ODEs, I began reading about the theory of weak solutions. Without any examples the author claimed that i.e. the function being differentiable twice in the interior of the interval is very unrealistic and in practice are often not satisfied.

Is there a easy real world example (i.e. from physics or engineering) I can understand as maths student without physics or engineering background which illustrates where and why we would need weak solutions.

Intuitively, all physical processes are continuous and smooth enough and even taking the example of a shock wave one could i.e. model the pressure right after the shock, since before the show the pressure will likely be constant and therefore not interesting. On the other hand, according to the linked wikipedia article, the change in pressure only happens "almost instantaneously", and can therefore be modelled with a continuous function, which just has a very steep increase at the point of the shock.

Intuitively, all physical processes are continuous and smooth enough and even taking the example of a shock wave one could i.e. model the pressure right after the shock, since before the show the pressure will likely be constant and therefore not interesting. On the other hand, according to the linked wikipedia article, the change in pressure only happens "almost instantaneously", and can therefore be modelled with a continuous function, which just has a very steep increase at the point of the shock.

This is quite a different matter altogether, as you are trying to change how you model things here. In order to do this you need to understand what actually happens during shock formations in this small time frame, at which point you're considering a different problem entirely.

Note the PDE model is always a mathematical approximation to the physical system you are studying, typically because more accurate models are either not known, or too difficult to consider. In the case of shocks, allowing the notion of a weak solution gives us a way to mathematically give meaning to this phenomenon, and hence provides a way to model their behavior.

This naturally leads us to a really good source of examples: conservation laws. In one-dimension these are equations of the form (solved on $(x,t) in mathbb R times (0,infty)$),
$$ u_t + f(u)_x = 0. $$
Here the subscript denotes partial differentiation. It is known that for many choices of $f$ (e.g. $f(u) = u^2/2$ which gives Burgers' equation), one finds solutions will develop singularities in finite time (in the form of shock waves). However it is still possible to continue these solutions after these singularities form, which is why we want a notion of a weak solutions.

A detailed discussion of this can be found in many PDE texts, see for example section 3.4 of the book by Evans. This setting is nice because you can often work out solutions explicitly and see discontinuities forming, so it's a good way to develop your understanding of how these things work.

All this aside however, it's also important to keep in mind that a key motivation for weak solutions comes from the technical reason that they are much easier to study. In PDE theory it's usually very hard to show the existence of classical solutions directly, and most equations cannot be solved explicitly. Showing the existence of weak solutions on the other hand is manageable in many cases (the key analytical reason is because we can work in spaces with better compactness properties), and from there one can ask whether the weak solutions we obtain are actually suitably regular. An example of this are the Navier-Stokes equations, where weak existence can be shown but proving these are regular would answer one of the millennium problems.

Added later: As requested in the comments, an explicit example of a PDE which develops singularities in finite time. This is example 1 in section 3.4 of Evans.

Consider the initial value problem for the 1-dimensional Burgers' equation,
$$ u_t + uu_x = 0 $$
for $(x,t) in mathbb R times (0,infty)$ subject to the initial condition $u(0,t) = g(t)$ where,
$$ g(t) =
begincases
1, & textif x leq 0 \
1-x, & textif 0 leq x leq 1 \
0, & textif x geq 1.
endcases
$$
Note the initial data is continuous. Using the method of characteristics, we obtain that the unique solution for $0 leq t leq 1$ must be given by,
$$ u(x,t) =
begincases
1, & textif x leq t, 0 leq t leq 1 \
frac1-x1-t, & textif t leq x leq 1, 0 leq t leq 1 \
0, & textif x geq 1, 0 leq t leq 1.
endcases
$$
We see that $u$ is continuous in this interval $mathbb R times (0,1),$ but it is discontinuous on $mathbb R times (0,1).$ Indeed we have $u(x_n,1) rightarrow 0$ as $x_n rightarrow 1,$ while $u(1,t_n) rightarrow 0$ as $t_n rightarrow 1.$

The problem is that we get characteristic lines that cross at $(1,1),$ which prevents us from getting a uniquely defined continuous solution.