Weak solutions and strong friendships
When I was finishing my undergraduate studies at the Universidad Distrital (UD) by 2017 an atmosphere of uncertainty was damping my days. I had already decided that I wanted to go over postgraduate studies but I looked to the future with trepidation. I had made up my mind to study a master degree in mathematical sciences and I had applied a total of three times to this master: One to Los Andes University (from which I was not admitted) another one to the National University of Colombia (I wasn’t admitted again) and the third one, to the National University one more time, being admitted this time to begin my studies in the second semester of 2018 (the third time’s the charm).
I was frightened about the complexity of the advanced courses that I was going to take, but there was something else that was haunting my mind. I don’t know exactly why, but there was a hearsay that mathematics students at the National University of Colombia (UNAL from now) were too competitive academically and that sometimes this competitiveness reached the point of selfishness and even haughtiness. Fortunately, it took me very little time to debunk these spurious assertions. When my courses began, I realized that actually, the students in the UNAL were as friendly and warm as they were in my beloved UD. In fact one of my very first courses in the UNAL (Measure Theory) brought J. Andrés to my life, one of my most valuable friendships.
J. Andrés started the master degree the same semester as me and in a few weeks we began to share a lot of time together. I always have believed that he is a sort of nomadic; J. Andrés was born in Riohacha, La Guajira, then he moved to Bucaramanga, Santander where he did his undergraduate studies in mathematics in the Industrial University of Santander (UIS) and after that, he traveled to Bogotá to began his master degree (No doubt he is flying soon to continue his journey along the world). By the beginning of 2019 we already were close friends and we were sharing many courses together. As we got to know each other better I got astonished by his cleverness. Undoubtedly, he is one of the most brilliant people I know, he is an excellent mathematician and he fairly goes beyond that; he knows a ream of things about history, chemistry, general culture and of course, physics. His insightful point of view is remarkable and the best part of all of this, is that he has been always willing to share these things with me and to help me during my studies.
Besides sharing many interesting topics in mathematics and physics with me, J. Andrés helped me out a lot in some mandatory courses of the master degree. I remember long nights of sleep deprivation working on troublesome assignments of Differential Geometry (a subject that has always been especially challenging to me) that could have not been accomplished without his help. And that brings us to the mathematical side of this entry (please do not leave). When I was working on my master thesis I was addressing a problem regarding the existence of solutions for a non-linear system of partial differential equations called “the p-system”; a toy model to study and understand some phenomena related to gas dynamics.
This system is a specific case of a wider kind of equations that arise in many physical scenarios where conserved quantities such as mass, momentum and energy are present and due to that, they receive the name of “systems of conservation laws”:
One very special feature of these kinds of equations is that in many cases global classical solutions can not exist; it is possible to show that for classical solutions, singularities develop in finite time and this is an enormous obstacle to study the underlying theory behind these problems. Roughly speaking, if we look for a solution to the above problem, that is, a differentiable function u such that its time and spatial derivatives satisfy the equation presented in (1), then for a finite time “t”, the value of the function u goes to infinity ! In that points the function is no longer differentiable and therefore it can not satisfy equation (1). These problems forced mathematicians to change the notion of solution for equations like in (1). They came up with a new concept of solution that somehow circumvented the problems related to the differentiability of the possible solution.
A weak solution of equation (1) is a function u that satisfies the integral relation
where “phi” is any test function: a infinitely differentiable function with compact support on the region of interest. Essentially, the function “phi” is a infinitely times differentiable function that is zero outside a closed and bounded set.
J. Andrés and I used to study friday’s mornings when few people were in the university and it was easy to find available whiteboards to write on. One morning, I was discussing with J. Andrés about this specific subject and he asked me about the idea behind introducing this new concept of solution. Gathering all the things that I had read about the topic I tried to answer to him.
In the first place, this new idea of solution actually tackles the problem of differentiability of the function u. A priori, we don’t need u to be differentiable to fulfill equation (2), if it is nice enough to be integrable then we won’t have problem dealing with undetermined objects. Secondly, it is expected that all the classical solutions remain valid for this new formulation of solution, and this is the case: If u is a differentiable function that is a classical solution (satsifies equation (1)), then by integrating by parts the left hand side of equation (2) and since “phi” is zero outside a closed and bounded set, we recover the expression
which is zero for u fulfilling equation (1). So it turns out that this new concept embraces possible solutions that may fail being differentiable but also allows classical solutions to remain being solutions for the problem: it is an authentic generalization of the old concept of solution.
At this moment I was rather satisfied with the ideas I had exposed to J. Andrés. However, he kept gazing at the whiteboard as if something was wrong with my argument. He suddenly told me “Yeah, but that was not what I was asking for”. At that moment, I understood that I have misinterpreted his question about the nature of this new concept. What J. Andrés was asking was much more deeper. He was not asking about how this new concept fitted with the previous idea of solution, but why this new concept should be taken as the new definition of solution for these kinds of problems. This was a rougher question, and it was the typical kind of questions that J. Andrés usually bears in mind; the really meaningful ones. Although I tried to sketch some ideas about this in the whiteboard, I could not find a compelling reason of why this was the “right” way to extend the concept of solution.
It took me long to find out why this formulation was actually the more natural way to extend the concept of solution. Actually, I understood it with the help of many posts on MathematisStackExchange and an excellent book called: Numerical Methods for Conservation Laws written by Randal J. LeVeque, professor at Washington University. One way to understand the core of the ideas involving weak solutions has a close connection with the sceneries from which these equations come from and to set forth this in a simple way, I will draw out a very simple context in which conservation laws arise.
Imagine that you want to study the flow of gas in a thin tube. We suppose that the tube is so thin that the velocity and the density of the gas can be assumed to be constant along cross sections of the tube. We also assume that the walls of the tube are impermeable and that mass is neither created nor destroyed inside the tube. We denote by u the velocity of the gas and by p its density. If we take a little section of the tube and we study the total mass of gas within this section, then according to our assumptions, the mass inside can change only because of gas flowing across the endpoints of the section. Since the total mass of gas is given by the integral of the density in the section, and the flux of gas is given by the product of the density and the velocity of the gas, the previous considerations show that the following relation holds:
If you integrate the above expression with respect to t in a time interval, using the fundamental theorem of calculus you will end up with the equation
Observe that via the fundamental theorem of calculus, if u and p are differentiable this is equivalent to say that
and from this it is deduced that we must have
which is in the form of (1). With this in mind, a natural way to define a generalized solution that does not require differentiability is to go back to the integral formulation in (3). We then will say that u is a generalized solution if relation (3) holds for all space and time intervals. However this approach has a serious disadvantage; we have to guarantee that the relation is satisfied for any time and space interval and it is then desired to find a way to fix the domain of integration. This can be achieved by means of test functions.
The idea is to take the original equation in (1), multiply it by a test function “phi” defined on the domain of interest and then integrate over the whole region
Since the test function will vanish outside its support (we may take a test function whose support is precisely the time and space interval appearing in (4) ), integrating by parts we obtain the expression
It turns out that working with expressions like in (5) is much easier than working with expressions as in (3) where the domain of integration varies. The equation in (5) becomes a convenient way to define a new concept of solution for the problem that doesn’t require differentiability and has the same structure as our initial definition of weak solution in (2) . Now we have a slightly clearer point of view about its nature (I hope).
All this story seeks mainly two things. The first one, to give a little more detailed explanation about the idea of weak solutions in the context of conservation laws. A thorough discussion about this topic can be found in this beautiful post, where people from diverse areas of mathematics gather to discuss their own point of view about the concept of weak solutions in their respective fields. The second point is to illustrate one of the many situations in which my very favorite quotation applies in a wide sense:
“I get by with a little help from my friends” John Lennon — Paul McCartney
J. Andrés has always been not only a support in my academic and personal development, but also a constant source of motivation and inspiration. He is a great friend and I owe him a lot. I really hope we can continue sharing many things inside and outside the academy.
Postcript : I apologize for the poor quality of the mathematical environment of the equations. Medium does not support LATEX.