Related papers: An Onsager-type theorem for SQG
We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ with a pressure law $p\in C^{1,\gamma-1}$, where $1\le \gamma <2$. This includes all physically relevant cases, e.g.\ the monoatomic gas. We investigate…
Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler…
A procedure to construct $K$-matrices from the generalized $q$-Onsager algebra $\cO_{q}(\hat{g})$ is proposed. This procedure extends the intertwiner techniques used to obtain scalar (c-number) solutions of the reflection equation to…
We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-i\theta}u_t&=&u_{xx}-|u|^2u-\alpha g(v)u& v_t+(f(v))_x&=&\alpha (g'(v)|u|^2)_x&…
The paper aims on the construction of weak solutions to equations of a model of compressible viscous fluids, being a simplification of the classical compressible Navier-Stokes system. We present a novel scheme for approximating systems that…
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the…
Deformed sine-Gordon (DSG) models $\partial_\xi \partial_\eta \, w + \frac{d}{dw}V(w) = 0$, with $V(w)$ being the deformed potential, are considered in the context of the Riccati-type pseudopotential approach. A compatibility condition of…
We establish the well/ill-posedness theories for the inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in H\"older spaces, where $\alpha = 0$ and $\alpha = 1$ correspond to the two-dimensional Euler equation in the…
We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} d\theta_t +…
We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t \theta - \nabla^\perp \log(10+(-\Delta)^{\frac12})\theta \cdot \nabla \theta = 0 ,$$ and…
We prove the existence of global $L^2$ weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field…
We consider a vanishing viscosity sequence of weak solutions of the three-dimensional Navier--Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit…
We prove non-uniqueness of weak solutions to the forced $\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \alpha + \frac{2}{p}$, covering the 2D Euler equation ($\alpha = 0$), the Surface…
This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $\alpha\in[1,2)$ in the whole plane, where $\alpha=1$ corresponds…
We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }\theta+ u\cdot\nabla\theta = -\nu(-\Delta)^{\gamma/2}\theta+ \zeta,\qquad u=\nabla^{\perp}(-\Delta)^{-1}\theta, $$ on…
We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov…
We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the…
We consider local balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent…
This paper concerns the existence of global weak solutions \`a la Leray for compressible Navier-Stokes equations with a pressure law that depends on the density and on time and space variables $t$ and $x$. The assumptions on the pressure…
In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the…