arXiv subjects

Michele Gorini

Publications and source records attributed to Michele Gorini.

3 recordsLinked to original sources

arXiv2026

Weak and dissipative solutions for the Hasegawa-Mima equation

We consider the Hasegawa-Mima equation in its ``Euler-like'' velocity form: \[\partial_t(u-\Delta^{-1}u)+(u\cdot\nabla)u-u^\perp\log n_0=0,\] $n_0$ being the time-independent function appearing in the particle count $n=n_0e^{\frac{e\varphi}{T}}$, and $u$ being the drift velocity $\nabla^\perp\varphi=-\nabla\varphi\times\hat z$. Adapting the notion from Lions' book on the Euler equations, we prove the existence of dissipative solutions for this equation for any $L^2$ divergence free initial condition $w\in L^2(D)$, for $D=\mathbb T^2$ and $D\subset\mathbb R^2$ a bounded $\mathcal{C}^1$ domain.

math.AP↗

arXiv2023

Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

In this paper, we consider the fractional Navier-Stokes equations. We extend a previous non-uniqueness result due to Cheskidov and Luo, found in [5], from Navier-Stokes to the fractional case, and from $L^1$-in-time, $W^{1,q}$-in-space solutions for every $q > 1$ to $L^s$-in-time, $W^{1,q}$-in-space solutions for appropriate ranges of $s,q$.

math.AP↗

arXiv2022

$L^2$-Density of Wild Initial Data for the Hypodissipative Navier-Stokes Equations

In this paper we deal with the Cauchy problem for the hypodissipative Navier-Stokes equations in the three-dimensional periodic setting. For all Laplacian exponents $θ<\frac13$, we prove non-uniqueness of dissipative $L^2_tH^θ_x$ weak solutions for an $L^2$-dense set of $\mathcal C^β$ Hölder continuous wild initial data with $θ<β<\frac13$. This improves previous results of non-uniqueness for infinitely many wild initial data ([8,20]) and generalizes previous results on density of wild initial data obtained for the Euler equations ([14, 13]).

math.AP↗