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We prove that the inviscid surface quasigeostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data H 2 ‫ޔ(‬ 2 ) without any solutions in L ∞ t H 2 .Moreover, we prove strong critical norm inflation for C ∞ -smooth data.Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with the two-dimensional incompressible Euler equations.

<p style='text-indent:20px;'>We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form <inline-formula><tex-math id="M1">\begin{document}$ -1/r - 1/r^2 $\end{document}</tex-math></inline-formula> instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either

We investigate the well-posedness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -surface quasi-geostrophic ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:ann

We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady state $ω^*$ that is orbitally stable up to a translation and has a saddle point, we construct ${\tildeω}_0 \in C^\infty(\mathbb{T}^2)$ that is arbitrarily close to $ω^*$ in $L^2$, such that superlinear growth of the vorticity gradient occurs for an open set of s

We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by \begin{document}$U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$ \end{document} can be well approximated by the linear Schrödinger equation,

Abstract We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\begin{aligned} \begin{aligned} \partial _t \theta - \nabla ^\perp \log (10+(-\Delta )^{\frac{1}{2}})\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t

We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in $\mathbb{R}^{3}$. Namely, for any $s\in (0,3/2)$ and $\varepsilon &gt;0$, we construct a divergence-free initial vorticity $ω_0$ defined in $\mathbb{R}^{3}$ satisfying $\| ω_0 \|_{H^s}\leq \varepsilon$, as well as $T&gt;0$, $c&gt;0$ and a corresponding local-in-time solution $ω$ such that, for each $t\in [0,T]$, $ω(\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$ and $ ω(\cd