Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u₀ ∈ H^s( R ²) for s > 0 in 2-D, or u₀ ∈ H¹( R ³) satisfying ‖u₀‖_L² ‖?u₀‖_L² being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10 Danchin , R. , Mucha , P.B. ( 2013 ). Incompressible flows with piecewise constant density . Arch. Rat. Mech. Anal. 207 : 991 – 1023 .[Crossref], [Web of Science ®] , [Google Scholar]], which requires the initial velocity u₀ ∈ H²( R ^d) for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.     

A NONCLASSICAL LAW OF THE ITERATED LOGARITHM FOR I.I.D. SQUARE INTEGRABLE RANDOM VARIABLES. II

Let {X _n , n≥1} be a sequence of i.i.d. random variable with EX ₁=0 and E X ₁ ²=1 and let {b _n , n≥1} be a sequence of positive constants monotonically approaching infinity such that lim inf _n→∞ b _n /log log n=1. It is proved that lim sup _n→∞ ∑ _i=1 ⁿ X _i /√2nb _n =1 almost certainly thereby extending the work of Klesov and Rosalsky[4] Klesov, O. and Rosalsky, A. 2001. A Nonclassical Law of the Iterated Logarithm for I.I.D. Square Integrable Random Variables. Stoch. Anal. Appl., 19: 627–641. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] to a much larger class of sequences {b _n , n≥1}. The tools employed in the argument are results of Bulinskii[1] Bulinskii, A.V. 1977. On Normalization in the Law of the Iterated Logarithm. Teor. Veroyatnost. i Primen., 22: 407–409. In Russian, English translation in Theory Probab. Appl., 22 (1977), 398–399 [Google Scholar] and Feller[2] Feller, W. 1943. The General Form of the So-Called Law of the Iterated Logarithm. Trans. Am. Math. Soc., 54: 373–402.  [Google Scholar] and the Strassen[5] Strassen, V. 1964. An Invariance Principle for the Law of the Iterated Logarithm. Z. Wahrsch. Verw. Gebiete, 3: 211–226. [Crossref] , [Google Scholar] strong invariance principle.     

Factorisable Semigroups of Generalized Transformations

ABSTRACT

If X and Y are sets, we let P(X, Y ) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). We define an operation * on P(X, Y ) by choosing θ ∈ P(Y, X) and writing: α*β = α °θ°β, for each α, β ∈ P(X, Y ). Then (P(X, Y ), *) is a semigroup, and some authors have determined when this is regular (Magill and Subbiah, 1975 Magill , K. D. , Jr. Subbiah , S. ( 1975 ). Green's relations for regular elements of sandwich semigroups. I. General results . Proc. London Math. Soc. 31 : 194 – 210 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), when it contains a “proper dense subsemigroup” (Wasanawichit and Kemprasit, 2002 Wasanawichit , A. , Kemprasit , Y. ( 2002 ). Dense subsemigroups of generalized transformation semigroups . J. Austral. Math. Soc. 73 ( 3 ): 433 – 445 . [CSA] [Crossref] , [Google Scholar]) and when it is factorisable (Saengsura, 2001 Saengsura , K. ( 2001 ). Factorizable on (P(X, Y ), θ) , MSc thesis, 23 pp (in Thai, with English summary), Department of Mathematics, Khon Kaen University, Khon Kaen, Thailand, 2001.  [Google Scholar]). In this paper, we extend the latter work to certain subsemigroups of (P(X, Y ), *). We also consider the corresponding idea for partial linear transformations from one vector space into another. In this way, we generalise known results for total transformations and for injective partial transformations between sets, and we establish new results for linear transformations between vector spaces.     

A New Proof for the Existence of an Equivariant Entire Solution Connecting the Minima of the Potential for the System Δu − W u (u) = 0

Recently, Giorgio Fusco and the author in [2 Alikakos , N.D. , Fusco , G. ( 2011 ). Entire solutions to equivariant elliptic systems with variational structure . Arch. Rat. Mech. Anal. 202 : 567 – 597 .[Crossref], [Web of Science ®] , [Google Scholar]] studied the system Δu ? W _u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of equivariant solutions connecting the minima in certain directions at infinity, together with an estimate. In this paper a new proof is given which, in particular, avoids both the introduction of a pointwise constraint in the minimization process and the equivariant extensions of the various test functions.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

For N = 1,2, we consider singularly perturbed elliptic equations ?²Δ u ? V(x) u + f(u)= 0, u(x)> 0 on R ^N , lim_|x|→∞ u(x)= 0. For small ? > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007 Byeon , J. , Oshita , Y. ( 2004 ). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations . Comm. PDE 29 : 1877 – 1904 . [Google Scholar]).     

Semiclassical L p Estimates of Quasimodes on Submanifolds

Let P = P(h) be a semiclassical pseudodifferential operator on a Riemannian manifold M. Suppose that u(h) is a localized, L ² normalized family of functions such that P(h)u(h) is O(h) in L ², as h → 0. Then, for any submanifold Y ? M, we obtain estimates on the L ^p norm of u(h) restricted to Y, with exponents that are sharp for h → 0. These results generalize those of Burq et al. [4 Burq , N. , Gérard , P. , Tzvetkov , N. ( 2007 ). Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds . Duke Math. J. 138 : 445 – 486 .[Crossref], [Web of Science ®] , [Google Scholar]] on L ^p norms for restriction of Laplacian eigenfunctions. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao [8 Keel , M. , Tao , T. ( 1998 ). Endpoint Strichartz estimates . Amer. J. Math. 120 : 955 – 980 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The Convexity Estimates for the Solutions of Two Elliptic Equations

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17 Makar-Limanov , L.G. ( 1971 ). Solution of Dirichlet's problem for the equation Δu = ?1 on a convex region . Math. Notes Acad. Sci. USSR 9 : 52 – 53 .[Crossref] , [Google Scholar]] for the torsion equation and Acker et al. [1 Acker , A. , Payne , L.E. , Philippin , G. ( 1981 ). On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem . Z. Angew. Math. Phys. 32 : 683 – 694 .[Crossref], [Web of Science ®] , [Google Scholar]] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation vΔv = ? (1 + |?v|²) in a smooth, bounded and strictly convex domain Ω in ? ⁿ with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.     

Droplet Spreading Under Weak Slippage—Existence for the Cauchy Problem

Abstract

In this paper, we consider the thin film equation u _t  + div(|u| ⁿ ?Δu) = 0 in the multi-dimensional setting and solve the Cauchy problem in the parameter regime n ∈ [2, 3). New interpolation inequalities applied to the energy estimate enable us to control third order derivatives of appropriate powers of solutions. In such a way, a natural solution concept – reminiscent of that one used by Bernis and Friedman [Bernis, F., Friedman, A., (1990 Bernis, F. and Friedman, A. 1990. Higher order nonlinear degenerate parabolic equations. J. Differential Equations, 83: 179–206. [Crossref], [Web of Science ®] , [Google Scholar]). Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83:179–206] in space dimension N = 1 ? becomes available for the first time in the multi-dimensional setting. In addition, we provide the key integral estimate to establish results on the qualitative behavior of solutions like finite speed of propagation or occurrence of a waiting time phenomenon.     

Prophet Regions for Discounted,Uniformly Bounded Random Variables

Abstract

Let X ₁, X ₂,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max _j≤n Y _j ) and the stop rule supremum sup _t E Y _t for two types of discounted sequences: (i) Y _j  = b _j X _j , where {b _j } is a nonincreasing sequence of positive numbers with b ₁ = 1; and (ii) Y _j  = B ₁… B _j?1 X _j , where B ₁, B ₂,… are independent [0,1]-valued random variables that are independent of the X _j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup _t E Y _{{(x, y): x=sup t E Y} and y = E(max _j≤n Y _j ), for some sequence X ₁,…,X _n and Y _j  = b _j X _j }, is precisely the convex closure of the union of the sets {(b _j x, b _j y): (x, y) ∈ C _j }, j = 1,…,n, where C _j  = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j ? 1)(1 ? x ^1/(j?1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8 Hill , T.P. , and R.P. Kertz . 1983 . Stop rule inequalities for uniformly bounded sequences of random variables . Trans. Amer. Math. Soc. 278 : 197 – 207 . [CSA]  [Google Scholar]]. As a special case, it is shown that the maximum possible difference E(max _j≤n β ^j?1 X _j ) ? sup _t E(β ^t?1 X _t ) is attained by independent random variables when β ≤ 27/32, but by a martingale-like sequence when β > 27/32. Prophet regions for infinite sequences are given also.     

Addendum to “On the Sectional Geometric Genus of Quasi-Polarized Varieties,I” by Yoshiaki Fukuma

Let (X, L) be a polarized manifold defined over the complex number field with dim X = n such that L is very ample. In this article, we will improve the classification of (X, L) with g ₂(X, L) = h ²(𝒪 _X ) + 1 which was obtained in Fukuma (Fukuma 2004 Fukuma , Y. ( 2004 ). On the sectional geometric genus of quasi-polarized varieties, I . Comm. Alg. 32 : 1069 – 1100 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 3.6), where g ₂(X, L) denotes the second sectional geometric genus of (X, L).     

Power series over strongly Hopfian bounded rings

Let R be a commutative ring with identity. We show that R[[X]] is strongly Hopfian bounded if and only if R has a strongly Hopfian bounded extension T such that I_c(T) contains a regular element of T. We deduce that if R[[X]] is strongly Hopfian bounded, then so is R[[X,Y]] where X,Y are two indeterminates over R. Also we show that if R is embeddable in a zero-dimensional strongly Hopfian bounded ring, then so is R[[X]] (this generalizes most results of Hizem [11 Hizem, S. (2011). Formal power series over strongly Hopfian rings. Commun. Algebra 39(1):279–291.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]). For a chained ring R, we show that R[[X]] is strongly Hopfian if and only if R is strongly Hopfian.     

GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT

The algebras M _{a, b} (E) ? E and M _a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991 Kemer , A. ( 1991 ). Ideals of identities of associative algebras . Translations Math. Monographs 87 . Providence , RI : Am. Math. Soc. .[Crossref] , [Google Scholar]); other proofs of it were given by Regev (1990 Regev , A. ( 1990 ). Tensor products of matrix algebras over the Grassmann algebra . J. Alg. 133 ( 2 ): 512 – 526 . [CROSSREF]  [Google Scholar]), and in several particular cases, by Di Vincenzo (1992 Di Vincenzo , O. M. ( 1992 ). On the graded identities of M_{1, 1}( E) . Israel J. Math. 80 ( 3 ): 323 – 335 .[Crossref], [Web of Science ®] , [Google Scholar]), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M _{1, 1}(E) ? E and M ₂(E) when the base field is infinite and of characteristic p > 2. The algebra M _{a, a} (E) ? E satisfies certain graded identities that are not satisfied by M _2a (E). In another paper we proved that the algebras M _{1, 1}(E) and E ? E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.     

Analytic Wave Front Set for Solutions to Schrödinger Equations II – Long Range Perturbations

This paper is a continuation of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]], where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]] to long-range perturbations (in particular, we can allow potentials growing like ?x?^2?? at infinity). More precisely, we construct a modified quantum free evolution G ₀(?s, hD _z ) acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution e ^?itH u ₀ of the Schrödinger equation, in terms of the semiclassical exponential decay of G ₀(?th ^?1, hD _z )T u ₀, where T stands for the Bargmann-transform. The result is valid for t < 0 near the forward non trapping points, and for t > 0 near the backward non trapping points. It is an extension of [12 Nakamura , S. ( 2009 ). Semiclassical singularities propagation properties for the Schrödinger equations . J. Math. Soc. Japan 61 : 177 – 211 . [Google Scholar]] to the analytic framework.     

Sharp estimates for solutions of mean field equations with collapsing singularity

The pioneering work of Brezis-Merle [7 Brezis, H., Merle, F. (1991). Uniform estimates and blow-up behavior for solutions of ?Δu = V(x)e^u in two dimensions. Commun. Partial Differential Equation 16:1223–1254.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Li-Shafrir [27 Li, Y.Y., Shafrir, I. (1994). Blow-up analysis for solutions of ?Δu = V(x)e^u in dimension two. Indiana Univ. Math. J. 43:1255–1270.[Crossref], [Web of Science ®] , [Google Scholar]], Li [26 Li, Y.Y. (1999). Harnack inequality: the method of moving planes. Commun. Math. Phys. 200:421–444.[Crossref], [Web of Science ®] , [Google Scholar]], and Bartolucci-Tarantello [3 Bartolucci, D., Tarantello, G. (2002). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229:3–47.[Crossref], [Web of Science ®] , [Google Scholar]] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30 Lin, C.S., Tarantello, G. (2016). When “blow-up” does not imply “concentration”: A detour from Brezis-Merle’s result. C. R. Math. Acad. Sci. Paris 354:493–498.[Crossref], [Web of Science ®] , [Google Scholar]] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.