Publications (20)27.72 Total impact
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ABSTRACT: The aim of the note is to discuss different definitions of solutions to the Cauchy problem for the NavierStokes equations with the initial data belonging to the Lebesgue space $L_3(\mathbb R^3)$  [Show abstract] [Hide abstract]
ABSTRACT: In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axisymmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finitetime blowup from smooth data.  [Show abstract] [Hide abstract]
ABSTRACT: In this note, all the nontrivial bounded ancient solutions to the Stokes system in a halfspace with nonslip boundary conditions are described. Bibliography: 5 titles.  [Show abstract] [Hide abstract]
ABSTRACT: We construct an initial data for twodimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp  the double exponential growth is the fastest possible growth rate.  [Show abstract] [Hide abstract]
ABSTRACT: We study inviscid limits of invariant measures for the 2D Stochastic NavierStokes equations. As shown in \cite{Kuksin2004} the noise scaling $\sqrt{{\nu}}$ is the only one which leads to nontrivial limiting measures, which are invariant for the 2D Euler equations. We show that any limiting measure $\mu_{0}$ is in fact supported on bounded vorticities. Relationships of $\mu_{0}$ to the long term dynamics of Euler in the $L^{\infty}$ with the weak$^{*}$ topology are discussed. In view of the BatchelorKrainchnan 2D turbulence theory, we also consider inviscid limits for the weakly damped stochastic NavierStokes equation. In this setting we show that only an order zero noise (i.e. the noise scaling $\nu^0$) leads to a nontrivial limiting measure in the inviscid limit.  [Show abstract] [Hide abstract]
ABSTRACT: In the paper, we have introduced the notion of mild bounded ancient solutions to the NavierStokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the NavierStokes system with nonslip boundary conditions have blowups of Type I.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, a simple shear flow in a halfspace, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier–Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles.  [Show abstract] [Hide abstract]
ABSTRACT: It is known that a bounded solution of the heat equation in a halfspace which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the halfspace is replaced by cones with opening angle smaller than 90 degrees. Here we show the result remains true for cones with opening angle larger than 110 degrees. The proof covers heat equations having lowerorder terms with bounded measurable coefficients. 
Article: On Divergencefree Drifts
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ABSTRACT: We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t  \Delta +b\cdot\nabla$ and $\Delta +b\cdot\nabla$ with a divergencefree drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{1})$ resp. $b\in BMO^{1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergenceform operators with an elliptic symmetric part and a BMO skewsymmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the nonscaleinvariant norm $\b\_{L_1}$. In three dimensions, on the other hand, bounded solutions with $L_1$ drifts may be discontinuous. 
Article: On Type I Singularities of the Local AxiSymmetric Solutions of the Navier–Stokes Equations
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ABSTRACT: Local regularity of axially symmetric solutions to the NavierStokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the 1d viscous Burgers equation considered for complex valued functions develops finitetime singularities from compactly supported smooth data. By means of the ColeHopf transformation, the singularities of the solutions are related to zeros of complexvalued solutions of the heat equation. We prove that such zeros are isolated if they are not present in the initial data.  [Show abstract] [Hide abstract]
ABSTRACT: We prove two sufficient conditions for local regularity of suitable weak solutions to the threedimensional NavierStokes equations. One of these conditions implies the smoothness of L3,∞solutions as a particular case. Bibliography: 12 titles.  [Show abstract] [Hide abstract]
ABSTRACT: We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vectorvalued function in an open subset of $R^n$. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counterexamples to regularity of solutions of EulerLagrange systems satisfying classical ellipticity conditions.  [Show abstract] [Hide abstract]
ABSTRACT: We study a dissipative nonlinear equation modelling certain features of the Navier–Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions n≤4. For dimensions n>4, we present strong numerical evidence supporting the existence of blowup solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding the existence of selfsimilar singular solutions to a semilinear heat equation.  [Show abstract] [Hide abstract]
ABSTRACT: It is shown that the L3,∞solutions of the Cauchy problem for the threedimensional NavierStokes equations are smooth.  [Show abstract] [Hide abstract]
ABSTRACT: Abstract We outline an approach to study the properties of nonlinear partial differential equations through the geometric properties of a set in the space of m × n matrices which is naturally associated to the equation. In particular, different notions of convex hulls play a crucial role. This work draws heavily on Tartar’s work on oscillations in nonlinear pde and compensated compactness and on Gromov’s work on partial dif ferential relations and convex integration. We point out some recent successes of this approach and outline a number of open problems, most of which seem to require a better geometric understanding of the different convexity notions. Contents  [Show abstract] [Hide abstract]
ABSTRACT: We construct nonLipschitz minimizers of smooth, uniformly convex functionals of type I(u) = integral (Omega) f(Du(x))dx. Our method is based on the use of null Lagrangians.  [Show abstract] [Hide abstract]
ABSTRACT: We address the open problem of existence of singularities for the complex GinzburgLandau equation. Using a combination of rigourous results and numerical computations, we describe a countable family of selfsimilar singularities. Our analysis includes the supercritical nonlinear Schroedinger equation as a special case, and most of the described singularities are new even in that situation. We also consider the problem of stability of these singularities. 
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ABSTRACT: We consider variational integrals\( I(u) = \int_\Omega {f(Du(x))dx}\) defined for (sufficiently regular) functionsu: Ω →R m . Here Ω is a bounded open subset ofR n ,Du(x) denotes the gradient matrix ofu atx, andf:M m×n →R is given,M m×n denoting the space of real (m ×n)matrices. We are interested in the casem, n ≥ 2.
Publication Stats
622  Citations  
27.72  Total Impact Points  
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19972013

University of Minnesota Duluth
 Department of Mathematics & Statistics
Duluth, Minnesota, United States
