Weak solution

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

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {\partial u}{\partial x}}=0}$

(1)

where u = u(t, x) is a function of two real variables. To indirectly probe the properties of a possible solution u, one integrates it against an arbitrary smooth function ${\displaystyle \varphi \,\!}$ of compact support, known as a test function, taking

${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(t,x)\,\varphi (t,x)\,dx\,dt}$

For example, if ${\displaystyle \varphi }$ is a smooth probability distribution concentrated near a point ${\displaystyle (t,x)=(t_{\circ },x_{\circ })}$, the integral is approximately ${\displaystyle u(t_{\circ },x_{\circ })}$. Notice that while the integrals go from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$, they are essentially over a finite box where ${\displaystyle \varphi }$ is non-zero.

Thus, assume a solution u is continuously differentiable on the Euclidean space R², multiply the equation (1) by a test function ${\displaystyle \varphi }$ (smooth of compact support), and integrate:

${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\frac {\partial u(t,x)}{\partial t}}\varphi (t,x)\,\mathrm {d} t\,\mathrm {d} x+\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\frac {\partial u(t,x)}{\partial x}}\varphi (t,x)\,\mathrm {d} t\,\mathrm {d} x=0.}$

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:

${\displaystyle -\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(t,x){\frac {\partial \varphi (t,x)}{\partial t}}\,\mathrm {d} t\,\mathrm {d} x-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(t,x){\frac {\partial \varphi (t,x)}{\partial x}}\,\mathrm {d} t\,\mathrm {d} x=0.}$

(2)

(Boundary terms vanish since ${\displaystyle \varphi }$ is zero outside a finite box.) 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 ${\displaystyle \varphi }$, but such u may not be differentiable and so cannot satisfy equation (1). An example is u(t, x) = |t − x|, as one may check by splitting the integrals over regions x ≥ t and x ≤ t where u is smooth, and reversing the above computation using integration by parts. A weak solution of equation (1) means any solution u of equation (2) over all test functions ${\displaystyle \varphi }$.

The general idea which follows from this example is that, when solving a differential equation in u, one can rewrite it using a test function ${\displaystyle \varphi }$, such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to ${\displaystyle \varphi }$, resulting in an equation without derivatives of u. This new equation generalizes the original equation to include solutions which are not necessarily differentiable.

The approach illustrated above works in great generality. Indeed, consider a linear differential operator in an open set W in Rⁿ:

${\displaystyle P(x,\partial )u(x)=\sum a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}(x)\,\partial ^{\alpha _{1}}\partial ^{\alpha _{2}}\cdots \partial ^{\alpha _{n}}u(x),}$

where the multi-index (α₁, α₂, …, α_n) varies over some finite set in Nⁿ and the coefficients ${\displaystyle a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}}$ are smooth enough functions of x in Rⁿ.

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

${\displaystyle \int _{W}u(x)Q(x,\partial )\varphi (x)\,\mathrm {d} x=0}$

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

${\displaystyle Q(x,\partial )\varphi (x)=\sum (-1)^{|\alpha |}\partial ^{\alpha _{1}}\partial ^{\alpha _{2}}\cdots \partial ^{\alpha _{n}}\left[a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}(x)\varphi (x)\right].}$

The number

${\displaystyle (-1)^{|\alpha |}=(-1)^{\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}}$

shows up because one needs α₁ + α₂ + ⋯ + α_n integrations by parts to transfer all the partial derivatives from u to ${\displaystyle \varphi }$ 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, ∂) (cf adjoint of an operator).

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

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

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

${\displaystyle \int _{W}u(x)\,Q(x,\partial )\varphi (x)\,\mathrm {d} x=0}$

for every smooth function ${\displaystyle \varphi }$ with compact support in W.

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.

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