Related papers: The Derivative Structure for a Quadratic Nonlinear…
We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}L_{x}^{p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the…
Consider the three-dimensional Navier--Stokes flow past a moving rigid body $\mathscr{O} \subset \mathbb{R}^3$ with prescribed translational and angular velocities, where $\mathscr{O}$ stands for a bounded Lipschitz domain. We prove that…
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient…
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…
The Navier-Stokes motions in cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions are proved. The solutions are bounded by the same constant for all time.…
We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field.…
We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport.…
We study the global and local existence and uniqueness of solutions to the Navier-Stokes equations with anisotropic viscosity in a bounded cylindrical domain $Q=\Omega\times (0,1)$, where $\Omega$ is a star-shaped domain in $R^2$. In this…
We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type…
We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the $N$-dimensional Euclidean space for $N\geq 2$ when the gravity is not taken into account. The aim of this paper is…
In this article, we consider non-smooth time-dependent domains and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic…
A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of…
This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between…
We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the…
Via probabilistic convex integration, we prove non-uniqueness in law of the two-dimensional surface quasi-geostrophic equations forced by random noise of additive type. In its proof we work on the equation of the momentum rather than the…
This paper is a continuation of our previous work (arXiv:2507.03505), where the global existence and incompressible limit of weak solutions to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and…
We integrate in closed implicit form the Navier-Stokes equations for an incompressible fluid and the kinematical dynamo equation, in smooth manifolds and Euclidean space. This integration is carried out by applying Stochastic Differential…
We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We…
This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and…
In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is ill-posed for…