Journal of the London Mathematical Society-Second Series最新文献

Alt, Caffarelli, and Friedman [Arch. Ration. Mech. Anal. 81(1983), 97–149] established the well-posedness of the axially symmetric incompressible jet flow without vorticity. In this paper, we mainly extend their work to the rotational case. Precisely, our main results show that for a given incoming horizontal velocity and atmosphere pressure at the outlet, there exists a unique solution to incompressible rotational jet flow issuing from a three-dimensional axisymmetric nozzle, and the free boundary initiates smoothly at the endpoint of the nozzle wall. Moreover, it is proved that there is no singularity of the velocity field near the axis. Motivated by Serrin's work [J. Ration. Mech. Anal. 2(1953), 563–575] on irrotational free streamline problems, we establish the under-over theorem and the single intersection property of the free boundary for the rotational case. Then the convexity of the free boundary is also obtained. Finally, we prove that the free boundary is monotonic with respect to the nozzle wall.

Let $�Q$ denote the unique radial solution to the equation

Furthermore, by employing the co-compact method, we derive a global stability result from the above local one when $��d�=�1�$ or when $�u$ is non-negative. In the complex-valued case, the analysis becomes more intricate, and may depend on the phase difference. Finally, we give some applications to the nonlinear Schrödinger equation.

We prove that the Flat Cover Conjecture holds for the category of (right) acts over any right-reversible monoid $�S$, provided that the flat $�S$-acts are closed under stable Rees extensions. The argument shows that the class $�F$-Mono ($�S$-act monomorphisms with flat Rees quotient) is cofibrantly generated in such categories, answering a question of Bailey and Renshaw. But cofibrant generation of $�\mathrm{SF}$-Mono ($�S$-act monomorphisms with strongly flat Rees quotient) appears much stronger, since we show it implies that there is a bound on the size of the indecomposable strongly flat acts. Similarly, cofibrant generation of $�U_F$ (unitary monomorphisms with flat complement) implies a bound on the size of indecomposable flat acts. The key tool is a new characterization of cofibrant generation of a class of monomorphisms in terms of “almost everywhere” effectiveness of the class.

We prove that the distribution of the number of steps of the Euclidean algorithm of rationals in imaginary quadratic fields with denominators bounded by $�N$ is asymptotically Gaussian as $�N$ goes to infinity, extending a result by Baladi and Vallée for the real case. The proof is based on the spectral analysis of the transfer operator associated to the nearest integer complex continued fraction map, which is piecewise analytic and expanding but not a full branch map. By observing a finite Markov partition with a regular CW-structure, which enables us to associate the transfer operator acting on a direct sum of spaces of $�C^1$-functions, we obtain the limit Gaussian distribution as well as residual equidistribution.

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a potential term. Our results reveal the interplay between the manifold's geometry, the power nonlinearity, and the potential's behavior at infinity. Using a test function argument, we identify explicit parameter ranges where nonexistence holds.

We give some criteria for the Lie algebra $��{\mathrm{HH}}^1��\left(�B�\right)��$ to be solvable, where $�B$ is a $�p$-block of a finite group algebra, in terms of the action of an inertial quotient of $�B$ on a defect group of $�B$.

Let $�M_g$ be the moduli space of hyperbolic surfaces of genus $�g$ endowed with the Weil–Petersson metric. We view the regularized determinant $��\log �det�\left(�{\Delta }_X�\right)�$ of Laplacian as a function on $�M_g$ and show that there exists a universal constant $��E�>�0�$ such that as $��g�\to �\infty �$,