Related papers: Remarks on singular solutions of the Euler equatio…
This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which…
We consider Euler flows on two-dimensional (2D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2D Taylor-Green vortex. As the first main result, numerical evidence is…
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in…
Smooth solutions of the forced incompressible Euler equations satisfy an energy balance, where the rate-of-change in time of the kinetic energy equals the work done by the force per unit time. Interesting phenomena such as turbulence are…
Using extrapolation theory, we develop a new framework to prove the uniqueness of solutions for transport equations. We apply our methodology to unify and extend the classical results of Yudovich and Vishik for 2D Euler equations. In…
We present a formal, approximate model for singularity formation in classical fluid dynamics in three dimensions. The construction utilizes an approximation of local two-dimensionality to study an anti-parallel hairpin vortex structure with…
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop…
In this paper, we study the full regularity and well-posedness of classical solutions to the nonlinear unsteady Prandtl equations with Robin or Dirichlet boundary condition in half space. Under Oleinik's monotonicity assumption, we prove…
We introduce many families of explicit solutions to the three dimensional incompressible Euler equations for nonviscous fluid flows using the Lagrangian framework. Almost no exact Lagrangian solutions exist in the literature prior to this…
In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the…
Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can…
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^\alpha$…
In this paper, we study the existence and uniqueness of three dimensional steady Euler flows in rectangular nozzles when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the…
In this paper, we study the well-posedness of classical solutions to a two-phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier-Stokes equations via a drag forcing term. We consider…
In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow…
In this paper, we study the stability two-dimensional (2D) steady Euler flows with sharply concentrated vorticity in a simply-connected bounded domain. These flows are obtained as maximizers of the kinetic energy subject to the constraint…
Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its…
We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time,…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the $L^s$…