The theory of weak solutions is a mathematical framework used to analyze partial differential equations (PDEs) when traditional solutions may not exist. In this approach, solutions are defined in a broader sense, allowing for functions that may not be smooth but still satisfy the equation in an integral form. This is particularly useful in cases where solutions exhibit discontinuities or singularities. Weak solutions are essential in various fields, including fluid dynamics and material science, where complex behaviors arise. By employing tools from functional analysis, such as Sobolev spaces, mathematicians can rigorously study these solutions, ensuring that they retain essential properties of classical solutions while accommodating more general scenarios.