Weak Solution · aiwiki.page

A weak solution is a solution of a differential equation defined through identities against auxiliary functions rather than solely through pointwise differentiation. The concept is especially important for partial differential equations: it allows equations to be studied when solutions lack classical derivatives, when coefficients are rough, or when discontinuities develop. “Weak” describes the formulation and regularity requirements, not an approximate satisfaction of the equation. The precise definition depends on the equation and the chosen function spaces. (math.ucdavis.edu)

Test functions and weak derivatives

The central device is integration by parts, which transfers derivatives from the unknown function to a smooth test function. Let $\Omega$ be an open subset of $\mathbb R^n$ . A locally integrable function $g$ is the $i$ -th weak derivative of $u$ if

\int_\Omega u\,\partial_i\varphi\,dx = -\int_\Omega g\,\varphi\,dx \qquad \text{for every }\varphi\in C_c^\infty(\Omega).

Here $C_c^\infty(\Omega)$ consists of smooth functions with compact support inside $\Omega$ , so no boundary term occurs. For continuously differentiable functions, this agrees with the ordinary derivative. Weak derivatives are determined only up to equality almost everywhere. (math.ucdavis.edu)

In distribution theory, derivatives are defined by the same transfer of differentiation to test functions. Every distribution has distributional derivatives, but these need not be represented by locally integrable functions. For example, the derivative of a step function is a Dirac delta distribution. Thus, being differentiable almost everywhere is not sufficient to possess a weak derivative represented by a function. Sobolev spaces organize functions according to the integrability of their weak derivatives. (math.ucdavis.edu)

Example: Poisson’s equation

Consider the homogeneous Dirichlet boundary value problem

-\Delta u=f\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega,

where $\Omega$ is bounded. Multiplication by a test function and integration by parts give the weak formulation of Poisson’s equation:

\boxed{ \int_\Omega \nabla u\cdot\nabla v\,dx = \langle f,v\rangle \quad\text{for every }v\in H_0^1(\Omega). }

A weak solution is sought in $H_0^1(\Omega)$ , the closure of $C_c^\infty(\Omega)$ in the $H^1$ norm. This incorporates the zero boundary condition; on sufficiently regular domains it is equivalent to zero boundary trace. If $f\in L^2(\Omega)$ , the right-hand side is $\int_\Omega fv\,dx$ ; more generally, $f$ may belong to the dual space $H^{-1}(\Omega)$ . Only first weak derivatives of $u$ are required, rather than classical second derivatives. (math.ucdavis.edu)

The Lax–Milgram theorem, together with the Poincaré inequality, gives existence and uniqueness for this problem. Elliptic regularity can subsequently establish additional differentiability when the data and domain permit it. A sufficiently regular weak solution then satisfies the original equation classically. (math.ucdavis.edu)

Evolution equations

For time-dependent equations, the function spaces must also specify temporal regularity and the interpretation of initial data. For the heat equation

u_t-\Delta u=f

with homogeneous Dirichlet boundary conditions, a standard energy-space formulation requires

u\in L^2(0,T;H_0^1(\Omega)), \qquad u_t\in L^2(0,T;H^{-1}(\Omega)),

and

\langle u_t,v\rangle+ \int_\Omega\nabla u\cdot\nabla v\,dx = \langle f,v\rangle

for every spatial test function $v\in H_0^1(\Omega)$ , for almost every time. These conditions give an $L^2(\Omega)$ -continuous representative of $u$ , making the initial condition meaningful. Galerkin approximations and energy estimates provide a route to existence. (math.ucdavis.edu)

Conservation laws and discontinuities

For a one-dimensional scalar conservation law,

u_t+\partial_x F(u)=0,

smooth solutions may develop discontinuities. Assuming sufficient local integrability, the weak formulation with initial data $u_0$ is

\int_0^\infty\!\!\int_{\mathbb R} \bigl(u\varphi_t+F(u)\varphi_x\bigr)\,dx\,dt + \int_{\mathbb R}u_0(x)\varphi(x,0)\,dx=0.

This identity remains meaningful across shocks. A jump moving with speed $s$ , between constant states $u_L$ and $u_R$ , must satisfy the Rankine–Hugoniot condition

s(u_R-u_L)=F(u_R)-F(u_L).

(web.stanford.edu)

The integral identity alone generally does not guarantee uniqueness. An entropy condition supplies additional admissibility requirements. For Burgers’ flux $F(u)=u^2/2$ , both increasing and decreasing jumps satisfy the jump relation with the appropriate speed, but the standard entropy condition admits decreasing shocks; increasing Riemann data produce a rarefaction wave instead. For scalar conservation laws under standard hypotheses, entropy conditions select a unique solution. (web.stanford.edu)

Numerical applications

Weak formulations underpin the finite element method. A conforming Galerkin discretization replaces the infinite-dimensional trial and test spaces by finite-dimensional subspaces. For Poisson’s equation, one seeks $u_h\in V_h\subset H_0^1(\Omega)$ satisfying

\int_\Omega\nabla u_h\cdot\nabla v_h\,dx = \int_\Omega fv_h\,dx \qquad\text{for all }v_h\in V_h.

Expanding $u_h$ in a basis produces a system of linear equations. Piecewise polynomial functions can serve as trial functions despite lacking classical second derivatives across element boundaries. The discrete solution is an approximation to the weak solution; the weak solution itself is an exact solution of the continuous integral formulation. (jsdokken.com)

Related notions and limitations

“Distributional solution” and “weak solution” are sometimes used interchangeably, but a variational weak solution usually includes specified function-space and boundary requirements. A distributional solution need not satisfy those additional conditions. Likewise, “strong solution” has equation-dependent meanings, commonly involving enough weak derivatives to satisfy the equation almost everywhere. (math.ucdavis.edu)

Viscosity solutions provide a different generalized formulation, particularly for fully nonlinear equations: smooth functions touching the solution from above or below impose differential inequalities. They are not simply the integral weak formulation under another name. More generally, weakening regularity does not automatically establish existence, uniqueness, or admissibility; each requires hypotheses appropriate to the equation. (math.stanford.edu)

References

Sobolev Spaces — Notes on Partial Differential Equations, Chapter 3math.ucdavis.edu
Elliptic PDEs — Notes on Partial Differential Equations, Chapter 4math.ucdavis.edu
Parabolic Equations — Notes on Partial Differential Equations, Chapter 6math.ucdavis.edu
Conservation Laws — Stanford Math 220A Lecture Notesweb.stanford.edu
Solving the Poisson Equation — FEniCSx Tutorialjsdokken.com
Stanford Lecture Notes on Hamilton–Jacobi Equations and Viscosity Solutionsmath.stanford.edu