Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn? 4 the L_∞-norm of weak solutions of the Navier-Stokes equations on ?ⁿ with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε L_p p >n.     

Von Kármán equations in L p spaces

We show the existence of solution in L ^p spaces for a generalized form of the classical Von Kármán equations where the coefficients of nonlinear terms are variable. We use Campanato's near operators theory.     

Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

On the nonexistence of solutions for random differential equations in banach spaces ∗

Random differential equations in Banach spaces without solutions are considered. Some density results are established.     

Unconditional structures of translates for L p (ℝ d )

We prove that a sequence (f _i ) _i=1 ^∞ of translates of a fixed f ∈ L _p (?) cannot be an unconditional basis of L _p (?) for any 1 ≤ p < ∞. In contrast to this, for every 2 < p < ∞, d ∈ ? and unbounded sequence (λ _n ) _n∈? ? ? ^d we establish the existence of a function f ∈ L _p (? ^d ) and sequence (g _n *) _n∈? ? L _p *(? ^d ) such that \({({T_{{\lambda _n}}}f,g_n^*)_{n \in {\Bbb N}}}\) forms an unconditional Schauder frame for L _p (? ^d ). In particular, there exists a Schauder frame of integer translates for L _p (?) if (and only if) 2 < p < ∞.     

Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise

In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also. This work was supported by National Basic Research Program of China (Grant No. 2006CB8059000), Science Fund for Creative Research Groups (Grant No. 10721101), National Natural Science Foundation of China (Grant Nos. 10671197, 10671168), Science Foundation of Jiangsu Province (Grant Nos. BK2006032, 06-A-038, 07-333) and Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences     

Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev spaceH ^s=H^s(R ⁿ) of fractional orders>n/2 under the following assumptions. (1) Concerning the Cauchy data ?∈H ^s: ‖?;L ²‖ is relatively small with respect to ‖?;H ^σ‖ for any fixed σ withn/2<σ≤s. (2) Concerning the nonlinearityf: f(u) behaves as a conformal poweru ¹⁺⁴/n near zero and has an arbitrary growth rate at infinity.     

Navier-Stokes equations under slip boundary conditions: Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in L∞(L3)

Science China Mathematics - The main purpose of this paper is to extend the result obtained by Beirão da Veiga (2000) from the whole-space case to slip boundary cases. Denote by ū two...     

Stability of solutions for nonlinear Schrdinger equations in critical spaces

We consider the Cauchy problem for nonlinear Schrdinger equation iut + Δu = ±|u|pu,4/d< p <4 /d-2 in high dimensions d 6. We prove the stability of solutions in the critical space H˙xsp , where sp = d/2-p/2 .     

A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces

For the incompressible Navier–Stokes equations in R³${\mathbb{R}}^3$, a regularity criterion for weak solutions is proved under the assumption that the pressure belongs to the scaling invariant Lorentz space with small norm, while corresponding results for the velocity field were proved by Sohr. The main theorem continues and extends a previous result given by the author.     

Time-periodic solutions of dissipative ε-approximations for modified Navier-Stokes equations

In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:

Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the local solutions can be extended globally in time. Our solutions also belong to the Serrin class in the usual Lebesgue space. The method is based on the maximal Lorentz regularity theorem of the Stokes equations in the homogeneous Besov spaces. As an application, we may handle such singular data as the Dirac measure and the single layer potential supported on the sphere.     

On monochromatic solutions of some nonlinear equations in ℤ/pℤ

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ? such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ?/p?, where p is a prime, and obtain an even stronger density result.     

ℋ p stability of solutions of stochastic differential equations

Summary Solutions of systems of stochastic differential equations are shown to be stable in ^p under ^p perturbations of semimartingale differentials. Analogous results are obtained inp ^p when the solutions are not semimartingales but are only cadlag, adapted processes. Also, the solutions are shown to be stable under almost sure perturbations. These results are contrasted with the lack of stability under non- ^p perturbations, a result originally obtained by Wong and Zakai.This research was supported in part by NSF Grant No. MCS77-00095