Bulletin of the Brazilian Mathematical Society, New Series最新文献

In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem of the drifting Laplacian in several cases, and establish some universal inequalities that are different from those obtained previously in (Du et al. in Z Angew Math Phys 66(3):703–726, 2015).

Abstract

In this work, we are concerned with the main mechanism for possible blow-up criteria of smooth solutions to the 3D incompressible Boussinesq equations. The main results state that the finite-time blowup/global existence of smooth solutions to the Boussinesq equation is controlled by either of the criteria $$begin{aligned} u_{h}in L^{2}left( 0,T;dot{B}_{infty ,infty }^{0}({mathbb {R}} ^{3})right) quad text {or}quad nabla _{h}u_{h}in L^{1}left( 0,T;dot{B} _{infty ,infty }^{0}left( {mathbb {R}}^{3}right) right) , end{aligned}$$ where (u_{h}) and (nabla _{h}) denote the horizontal components of the velocity field and partial derivative with respect to the horizontal variables, respectively. We present a new simple proof for the regularity of this system without using the higher-order energy law and without any assumptions on the temperature (theta .) Our results extend the Navier–Stokes equations results in Dong and Zhang (Nonlinear Anal Real World Appl 11:2415–2421, 2010), Dong and Chen (J Math Anal Appl 338:1–10, 2008) and Gala and Ragusa (Electron J Qual Theory Differ Equ, 2016a) to Boussinesq equations.

In this paper, we investigate the analytical solutions to the cylindrically symmetric compressible Navier–Stokes equations with density-dependent viscosity and vacuum free boundary. The shear and bulk viscosity coefficients are assumed to be a power function of the density and a positive constant, respectively, and the free boundary is assumed to move in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. We obtain a global analytical solution by using some ansatzs and reducing the original partial differential equations into a nonlinear ordinary differential equation about the free boundary. The free boundary is shown to grow at least sub-linearly in time and not more than linearly in time for the analytical solution by using a new averaged quantity.

In this paper, we study results of well-posedness and regularity of higher order in time abstract non-autonomous semilinear Cauchy problems associated with Newton’s binomial theorem and the theory of sectorial operators. Our approach to parabolic problems of arbitrarily order n apparently has never been addressed earlier in the existing literature. Also, we present applications to evolutionary equations involving the fractional Laplacian in bounded smooth domains of ({mathbb {R}}^N).

We consider a complete noncompact minimal hypersurface (Sigma ^n) in a product manifold ({{mathbb {S}}}^{n}(sqrt{2(n-1)})times {{mathbb {R}}}) ((nge 3)). We get that there admits no nontrivial (L^2) harmonic 1-forms on (Sigma ) if the square of (L^n)-norm of the second fundamental form is less than (frac{alpha ^2n}{2C_0(n-1)}) or the square of the length of the second fundamental form is less than (frac{nalpha ^2}{2(n-1)}). Here (alpha ) is an angle function and (C_0) is the Sobolev constant depending only on n.

We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups.

Kwapień’s theorem asserts that every continuous linear operator from (ell _{1}) to (ell _{p}) is absolutely (left( r,1right) )-summing for (1/r=1-left| 1/p-1/2right| .) When (p=2) it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators.

Given a strictly increasing continuous function (phi :mathbb {R}_{ge 0} longrightarrow mathbb {R}cup {-infty }) with (lim _{trightarrow infty }phi (t) = infty ), a function (f:[a,b] longrightarrow mathbb {R}_{ge 0}) is said to be (phi )-concave if (phi circ f) is concave. When (phi (t) = t^p), (p>0), this notion is that of p-concavity whereas for (phi (t) = log (t)) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set (Ksubset mathbb {R}^n) (of positive volume) w.r.t. some hyperplane H passing through its centroid is (phi )-concave, then one can find a sharp lower bound for the ratio (textrm{vol}(K^{-})/textrm{vol}(K)), where (K^{-}) denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for (phi )-concave functions (and involving the centroid) are shown.

It is established existence and multiplicity of solutions for semilinear elliptic problems defined in the whole space (mathbb {R}^N) considering subcritical nonlinearities with some parameters. Here we emphasize that our nonlinearities can be sign-changing functions. The main difficulty is proving the existence of nontrivial solutions by using the Nehari method, taking into account that the Lagrange multipliers theorem cannot be directly applied in our setting. In fact, we consider the case where the fibering map admits inflection points. In other words, we consider the case where the Nehari set admits degenerate critical points. Hence our main contribution is to consider a huge class of semilinear elliptic problems where the standard Nehari method cannot be applied. Using some fine estimates and recovering some compactness results together with the nonlinear Rayleigh quotient, we prove that our main problem admits at least three nontrivial solutions depending on the parameters.