254A, Notes 2: Weak solutions of the Navier-Stokes equations

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In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time are not high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit $latex {nu rightarrow 0}&fg=000000$.

However, it is possible to construct “weak” solutions…

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