Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG
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DOI: 10.1088/1361-6544/adf9b1
Abstract: The general surface quasi-geostrophic equation is the scalar transport equation defined by {∂θ∂t+v1γ∂θ∂x1+v2γ∂θ∂x2=0,vγ=∇⊥ψγ=(∂2ψγ,−∂1ψγ), ψγ=−Λ−1+γθ,θ(⋅,0)=θ0(⋅), for γ∈(−1,1), where the non-local operator Λα=(−Δ)α2 is defined on the Fourier side by Λαf^(ξ)=|ξ|αf^(ξ). The PDE is well-posed in the Sobolev… read more here.
Keywords: ill posedness; sobolev spaces; non existence; strong ill ... See more keywords
Strong Ill-Posedness of the 3D Incompressible Euler Equation in Borderline Spaces
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DOI: 10.1093/imrn/rnz158
Abstract: For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we… read more here.
Keywords: incompressible euler; strong ill; euler equation; borderline ... See more keywords