Newtonian motivations for general relativity

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.

Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions.

Weak solutions are important because a great many differential equations encountered in modelling real world phenomena do not admit sufficiently smooth solutions and then the only way of solving such equations is using the weak formulation. Even in situations where an equation does have differentiable solutions, it is often convenient to first prove the existence of weak solutions and only later show that those solutions are in fact smooth enough.

A concrete example

As an illustration of the concept, consider the first-order wave equation

$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0(1)$

(see partial derivative for the notation) where u = u(t, x) is a function of two real variables. Assume that u is continuously differentiable on the Euclidean space R², multiply this equation (1) by a smooth function $\phi$ of compact support, and integrate. One obtains

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\partial u(t,x)}{\partial t}\phi (t,x)\mathrm{d}t\mathrm{d}x+\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\partial u(t,x)}{\partial x}\phi (t,x)\mathrm{d}t\mathrm{d}x=0.$

Using Fubini's theorem which allows one to interchange the order of integration, as well as integration by parts (in t for the first term and in x for the second term) this equation becomes

$-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}u(t,x)\frac{\partial \phi (t,x)}{\partial t}\mathrm{d}t\mathrm{d}x-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}u(t,x)\frac{\partial \phi (t,x)}{\partial x}\mathrm{d}t\mathrm{d}x=0.(2)$

(Notice that while the integrals go from −∞ to ∞, the integrals are essentially over a finite box because $\phi$ has compact support, and it is this observation which also allows for integration by parts without the introduction of boundary terms.)

We have shown that equation (1) implies equation (2) as long as u is continuously differentiable. The key to the concept of weak solution is that there exist functions u which satisfy equation (2) for any $\phi$, and such u may not be differentiable and thus, they do not satisfy equation (1). A simple example of such function is u(t, x) = |t − x| for all t and x. (That u defined in this way satisfies equation (2) is easy enough to check, one needs to integrate separately on the regions above and below the line x = t and use integration by parts.) A solution u of equation (2) is called a weak solution of equation (1).

General case

The general idea which follows from this example is that, when solving a differential equation in u, one can rewrite it using a so-called test function $\phi$, such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to $\phi$. In this way one obtains solutions to the original equation which are not necessarily differentiable.

The approach illustrated above works for equations more general than the wave equation. Indeed, consider a linear differential operator in an open set W in Rⁿ

$P(x,\partial )u(x)=\sum a_{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}(x){\partial }^{{\alpha }_1}{\partial }^{{\alpha }_2}\cdots {\partial }^{{\alpha }_n}u(x)$

where the multi-index (α₁, α₂, ..., α_n) varies over some finite set in Nⁿ and the coefficients $a_{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}$ are smooth enough functions of x.

The differential equation P(x, ∂)u(x) = 0 can, after being multiplied by a smooth test function $\phi$ with compact support in W and integrated by parts, be written as

$\underset{W}{\int }u(x)Q(x,\partial )\phi (x)\mathrm{d}x=0$

where the differential operator Q(x, ∂) is given by the formula

$Q(x,\partial )\phi (x)=\sum (-1{)}^{\left|\alpha \right|}{\partial }^{{\alpha }_1}{\partial }^{{\alpha }_2}\cdots {\partial }^{{\alpha }_n}[a_{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}(x)\phi (x)].$

The number

$(-1{)}^{\left|\alpha \right|}=(-1{)}^{{\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n}$

shows up because one needs α₁ + α₂ + ... + α_n integrations by parts to transfer all the partial derivatives from u to $\phi$ in each term of the differential equation, and each integration by parts entails a multiplication by −1.

The differential operator Q(x, ∂) is the formal adjoint of P(x, ∂) (see also adjoint of an operator for the concept of adjoint).

In summary, if the original (strong) problem was to find a |α|-times differentiable function u defined on the open set W such that

$P(x,\partial )u(x)=0\text{ for all }x\in W$

(a so-called strong solution), then an integrable function u would be said to be a weak solution if

$\underset{W}{\int }u(x)Q(x,\partial )\phi (x)\mathrm{d}x=0$

for every smooth function $\phi$ with compact support in W.

Other kinds of weak solution

The notion of weak solution based on distributions is sometimes inadequate. In the case of hyperbolic systems, the notion of weak solution based on distributions does not guarantee uniqueness, and it is necessary to supplement it with entropy conditions or some other selection criterion. In fully nonlinear PDE such as the Hamilton-Jacobi equation, there is a very different definition of weak solution called viscosity solution.

References

• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2