Partial regularity for solutions to subelliptic eikonal equations

On a bounded domain Ω in the Euclidean space ${\mathbb{R}}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.     

Remarks on the regularity of weak solutions to the Navier-Stokes equations near the boundary

We give a simple proof of the so-called -regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. Bibliography: 7 titles.Dedicated to Olga Aleksandrovna Ladyzhenskaya__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 168–179.     

On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

We consider suitable weak solutions to an incompressible viscous Newtonian fluid governed by the Navier-Stokes equations in the half space \({\mathbb {R}^3_+}\). Our main result is a direct proof of the partial regularity up to the flat boundary based on a new decay estimate, which implies the regularity in the cylinder \({Q_\rho ^+(x_0, t_0)}\) provided

$\limsup_{R\to 0}\frac {1} {R}\int\limits_{Q_R^+(x_0, t_0)} |{\rm rot}\,\mathbf u|^2 dxdt \,\leq\, \varepsilon _0$

with ε ₀ sufficiently small. In addition, we get a new condition for the local regularity beyond Serrin’s class which involves the L ²-norm of ?u and the L ^3/2-norm of the pressure.

     

Local regularity of the Navier-Stokes equations near the curved boundary

We present some regularity conditions for suitable weak solutions of the Navier-Stokes equations near the curved boundary of a sufficiently smooth domain. Our extend the work that was results established near a flat boundary by Gustafson, Kang and Tsai (2006) [6].     

Global viscosity solutions for eikonal equations on class A Lorentzian 2-tori

On the Abelian cover \(({\mathbb {R}}^{2},g)\) of a class A Lorentzian 2-torus \(({\mathbb {T}}^{2},g)\), we showed the existence of global viscosity solutions to the eikonal equation

$$\begin{aligned} g(\nabla u,\nabla u)=-1 \end{aligned}$$

associated to those homologies in the interior of the stable time cone. Some other related dynamical properties are also considered. As an application of the main results, we study the differentiability of the unit sphere of the stable time separation associated to the class A Lorentzian 2-torus.

     

On the regularity of solutions of hyperbolic nonlinear equations

Summary Some regularity results for the solutions of certain hyperbolic nonlinear equations are given. This paper was written while the author was visiting Department of Mathematics, Sir GeorgeWilliams University, Montreal (Canada). Entrata in Redazione il 22 dicembre 1971.     

On viscosity solutions of irregular Hamilton-Jacobi equations

It is proved that the initial-value problem for admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x, p) is discontinuous in x. Particular attention is devoted to the linear transport equation , where a may be discontinuous. Received: 21 October 2002     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

Smoothness near the boundary of the solutions of the elliptic Bellman equations

One considers Bellman's elliptic equation with constant coefficients and zero boundary values on a plane part of the boundary. In this case one gives a simplified proof of N. V. Krylov's result regarding the boundary estimates of the Hölder constants of the second derivatives of the solutions of the Bellman equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 150–154, 1985.     

On the definition of viscosity solutions for parabolic equations

In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.

     

Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions

It is shown how one can get upper bounds for when and are the (viscosity) solutions of

respectively, in with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form.

     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

On the regularity of the axisymmetric solutions of the Navier-Stokes equations

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations. Received: 11 April 2000; in final firm: 26 November 2000 / Published online: 28 February 2002     

On the boundary regularity of weak solutions to the MHD system

The partial regularity of boundary suitable weak solutions to the MHD system near the plane part of the boundary is proved. Bibliography: 16 titles.     

Behavior near the boundary of positive solutions of second order parabolic equations

Journal of Fourier Analysis and Applications -