Vladimir Sverak

University of Minnesota Duluth, Duluth, Minnesota, United States

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Publications (14)14.17 Total impact

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    ABSTRACT: In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.
    07/2014;
  • H. Jia, G. Seregin, V. Sverak
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    ABSTRACT: In this note, all the nontrivial bounded ancient solutions to the Stokes system in a half-space with nonslip boundary conditions are described. Bibliography: 5 titles.
    Journal of Mathematical Sciences 11/2013; 195(1).
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    Alexander Kiselev, Vladimir Sverak
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    ABSTRACT: We construct an initial data for two-dimensional 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.
    10/2013;
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    Nathan Glatt-Holtz, Vladimir Sverak, Vlad Vicol
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    ABSTRACT: We study inviscid limits of invariant measures for the 2D Stochastic Navier-Stokes equations. As shown in \cite{Kuksin2004} the noise scaling $\sqrt{{\nu}}$ is the only one which leads to non-trivial 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 Batchelor-Krainchnan 2D turbulence theory, we also consider inviscid limits for the weakly damped stochastic Navier-Stokes 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.
    02/2013;
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    G. Seregin, V. Sverak
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    ABSTRACT: In the paper, we have introduced the notion of mild bounded ancient solutions to the Navier-Stokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the Navier-Stokes system with non-slip boundary conditions have blowups of Type I.
    02/2013;
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    Lu Li, Vladimir Sverak
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    ABSTRACT: It is known that a bounded solution of the heat equation in a half-space 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 half-space 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 lower-order terms with bounded measurable coefficients.
    Communications in Partial Differential Equations 11/2010; · 1.03 Impact Factor
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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 divergence-free 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 divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm $\|b\|_{L_1}$. In three dimensions, on the other hand, bounded solutions with $L_1$ drifts may be discontinuous.
    10/2010;
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    G. Seregin, V. Sverak
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    ABSTRACT: Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.
    Communications in Partial Differential Equations 05/2008; · 1.03 Impact Factor
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    G. Seregin, V. Sverak
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    ABSTRACT: We prove two sufficient conditions for local regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. One of these conditions implies the smoothness of L3,∞-solutions as a particular case. Bibliography: 12 titles.
    Journal of Mathematical Sciences 09/2005; 130(4):4884-4892.
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    Petr Plechác, Vladimír Sverák
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    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 blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding the existence of self-similar singular solutions to a semi-linear heat equation.
    Nonlinearity 09/2003; 16(6):2083. · 1.60 Impact Factor
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    L. Escauriaza, G. A. Seregin, Vladimir Sverak
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    ABSTRACT: It is shown that the L3,∞-solutions of the Cauchy problem for the three-dimensional Navier-Stokes equations are smooth.
    Russian Mathematical Surveys 01/2003; 58(2):211-250. · 0.78 Impact Factor
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    Vladimír Sverák, Xiaodong Yan
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    ABSTRACT: We construct non-Lipschitz 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.
    Proceedings of the National Academy of Sciences 12/2002; 99(24):15269-76. · 9.81 Impact Factor
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    Petr Plechac, Vladimir Sverak
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    ABSTRACT: We address the open problem of existence of singularities for the complex Ginzburg-Landau equation. Using a combination of rigourous results and numerical computations, we describe a countable family of self-similar singularities. Our analysis includes the super-critical non-linear 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.
    08/2000;
  • G. Seregin, V. Sverak
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    ABSTRACT: In this paper, a simple shear flow in a half-space, 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.
    Journal of Mathematical Sciences 178(3).

Publication Stats

238 Citations
14.17 Total Impact Points

Institutions

  • 2003–2008
    • University of Minnesota Duluth
      • Department of Mathematics & Statistics
      Duluth, Minnesota, United States
  • 2002–2003
    • University of Minnesota Twin Cities
      • Department of Mathematics
      Minneapolis, MN, United States