Convex Integration for a Class of Active Scalar Equations
نویسنده
چکیده
T̂ [θ](ξ) = m(ξ)θ̂(ξ), ξ ∈ Z. We make the following assumptions about m. We assume that m : R\{0} → R is even, 0-homogeneous, and m(ξ) · ξ = 0 a.e. Note that m is not necessarily smooth or even bounded, so the operator T is not assumed to be of Calderón-Zygmund type. We say that the pair (θ, u) ∈ Lloc(R × T) is a weak solution of the system (1)–(4) on R if for every φ ∈ C∞ 0 (R× T) the following holds: ∫
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