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The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called…
We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation…
This paper studies the local existence of strong solutions to the Cauchy problem of the 2D fluid-particle interaction model with vacuum as far field density. Notice that the technique used by Ding et al.\cite{SBH} for the corresponding 3D…
The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann…
We consider here time-dependent three-dimensional stratified geophysical water flows of finite depth over a variable bottom with a free surface and an interface (separating two layers of constant and different densities). Under the…
We consider a new nonlocal and nonlinear one-dimensional evolution model arising in the study of oceanic flows in equatorial regions, recently derived in [A. Constantin and L. Molinet, Global Existence and Finite-Time Blow-Up for a…
We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for…
In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone…
We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic…
We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the…
We investigate an evolutive system of non-linear partial differential equations derived from Oldroyd models on Non-Newtonian flows. We prove global existence of weak solutions, in the case of a smooth bounded domain, for general initial…
In this paper, we are concerned with the global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on the product space $\mathbb{R}^2\times\mathbb{T}$. These quasilinear wave systems include 3D…
In the present work, we investigate stochastic third grade fluids equations in a $d$-dimensional setting, for $d = 2, 3$. More precisely, on a bounded and simply connected domain $\mathcal{D}$ of $\mathbb{R}^d$, $d = 2,3$, with a…
We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We…
We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by $ (-\Delta)^{s/2} $ with $ s \in (1,2)$ and $s = 1$,…
In this article, we introduce a moving-frame approach to the geophysical equation of two-dimensional uniformly stratified rotational fluid in oceans and find a family of exact solutions containing ten arbitrary parameter functions.
In this work, we study a phase transition model in atmospheric dynamics, inspired by the works [6,14,15], which analyze the primitive equations governing the evolution of velocity, temperature, and specific humidity. The main difficulty…
We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable…
The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such…
We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with…