Weak Formulation
The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L2(Ω))3 has an orthogonal decomposition:
where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl,Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.
For a slightly smoother vector field u ∈ H(curl,Ω), a similar decomposition holds:
where φ ∈ H1(Ω) and v ∈ (H1(Ω))d.
Read more about this topic: Helmholtz Decomposition
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