We give a characterization of Sobolev spaces of bivariate periodic functions with dominating smoothness properties in terms of Sobolev spaces of univariate functions. The mixed Sobolev norm is proved to be a uniform crossnorm. This property can be used as a powerful tool in approximation theory. x1. Introduction Beside the approximation of functions from the usual isotropic periodic Sobo-lev sp...

Significance The Sobolev spaces, introduced in the 1930s, have become ubiquitous analysis and applied mathematics. They involve L p norms of gradient a function u . We present an alternative point view where deri...

We consider the Cauchy problem of 2 dimensional incompressible Euler equations. We study the problem of growth rate of high Sobolev norms of the solutions. A well-known result that is implicit in [J. T. Beals ,T. Kato and A. Majda, Comm. Math. Phys. 94 (1984), 61-66] is the double exponential rate. An open question that has been asked by Tao is wether we can get an exponential rate. In this pap...

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, the theory electrorheological fluids as well image processing regions where p ( x ) reaches value 1.

We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit ε→ 0, with ε being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a...

We consider the biharmonic Dirichlet problem on a polygonal domain. Regularity estimates in terms of Sobolev norms of fractional order are proved. The analysis is based on new interpolation results which generalizes Kellogg’s method for solving subspace interpolation problems. The Fourier transform and the construction of extension operators to Sobolev spaces on R are used in the proof of the i...

We survey recent results on the approximation of functions from Sobolev spaces by stochastic linear algorithms based on function values. The error is measured in various Sobolev norms, including positive and negative degree of smoothness. We also prove some new, related results concerning integration over Lipschitz domains, integration with variable weights, and study tractability of generalize...

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t →∞ to t...