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This paper concerns the validity of the Prandtl boundary layer theory for steady, incompressible Navier-Stokes flows over a rotating disk. We prove that the Navier Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid…
In this article, I would like to express some of my views on the nature of turbulence. These views are mainly drawn from the author's recent results on chaos in partial differential equations \cite{Li04}. Fluid dynamicists believe that…
We study a three-dimensional fluid-structure interaction problem describing the motion of an incompressible, viscous fluid coupled with a deformable elastic shell of Koiter type that forms part of the fluid boundary. The fluid motion is…
We show the existence of weak solutions to the fluid-structure interaction problem of a largely deforming viscoelastic bulk solid with a viscous fluid governed by the incompressible Navier-Stokes equations. In contrast to previous works,…
The paper proves existence of a large class of smooth solutions to the incompressible Navier-Stokes equations in the three dimensional space. The viscosity coefficient is put to be $1$. Our result points a new class of regular solutions…
In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts,…
This paper provides a rigorous mathematical analysis of the global regularity problem for the 3D incompressible Navier-Stokes (NS) equations, specifically addressing the conditions under which smooth initial data may lead to a loss of…
In the theory of the Navier-Stokes equations, the viscous fluid in incompressible flow is modelled as a homogeneous and dense assemblage of constituent "fluid particles" with viscous stress proportional to rate of strain. The crucial…
This paper constructs a rigorous mathematical framework for investigating laminar-turbulent transition induced by weak singularities of incompressible Navier-Stokes (NS) equations. By integrating the energy identity of Leray weak solutions…
In this study, we propose a computational method for solving the turbulence problem of incompressible viscous Newtonian fluids based on the extended Navier-Stokes (N-S) equations. With some phenomenological observations and H. J. Kreuer's…
We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the Kelvin-Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the…
We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the…
This article attempts to use the ideas from the field of complexity sciences to revisit the classical field of fluid mechanics. For almost a century, the mathematical self-consistency of Navier-Stokes equations has remained elusive to the…
We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $\rho_0 u_0$ belongs to the set of…
We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $\epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1,…
We consider any cover $\mathscr{C}$ of $\mathbb{R}^3$ by balls of radius bigger or equal $1$ satisfying two conditions: (i) any ball intersects at most $\sigma>0$ other balls, and (ii) intersecting balls have comparable sizes. We consider a…
We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear…
In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied…
This paper extends the resolvent formalism for wall turbulence proposed by McKeon and Sharma(2010) to account for the effect of streamwise-constant riblets. Under the resolvent formulation, the Navier-Stokes equations are interpreted as a…
We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…