Vanishing viscosity limit of the 3D incompressible micropolar equations in a bounded domain

In this paper, we study the vanishing viscosity limit for the 3D incompressible micropolar equations in a flat domain with boundary conditions. We prove the existence of the global weak solution of the micropolar equations and obtain the uniform estimate of the strong solution. Furthermore, we establish the convergence rate from the solution of the micropolar equations to that of the ideal micropolar equations as all viscosities tend to zero (i.e., (ε,χ,γ,κ) → 0).

Geodesic stretch, pressure metric and marked length spectrum rigidity

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

Local and global behavior of solutions to 2 D-elliptic equation with exponentially-dominated nonlinearities

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Eisenstein series and an asymptotic for the K-Bessel function

We produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

Rate of convergence in the transfer theorem for min-scheme

In this paper non-uniform estimate of convergence rate in the min-scheme is obtained. Presented results make the estimates, given in [1] and [2], more precise.

On approximations of the function |x|s by the Vallee Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions

The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

Jackson’s rational singular integral on the cut

The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases. Next, we introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated. The obtained estimates are asymptotically exact as n → ∞; an estimate of deviation of Jackson’s rational singular integral from the function |x|s, 0 < s < 2 depending on the position of the point on the segment, a uniform estimate of the deviation on the segment [–1, 1] and its asymptotics are found. The optimal value of the parameter is obtained, for which the deviation error of the studied approximation apparatus from the function |x|s, 0 < s < 2 on the interval [–1, 1] has the highest rate of zero; the order of approximation of the function |x| on the interval [–1, 1] byJackson’s considered singular integral is found. It is shown that with a special choice of the parameter, the velocity of the approximation error tending to zero is higher in comparison with the polynomial case. All results of this article are new. The work is both theoretical and applied. It is possible to apply the results to solve specific problems of computational mathematics and to read special courses at mathematical faculties.

Precipitation Estimation Methods in Continuous, Distributed Urban Hydrologic Modeling

Quantitative precipitation estimation (QPE) remains a key area of uncertainty in hydrological modeling and prediction, particularly in small, urban watersheds, which respond rapidly to precipitation and can experience significant spatial variability in rainfall fields. Few studies have compared QPE methods in small, urban watersheds, and studies that have examined this topic only compared model results on an event basis using a small number of storms. This study sought to compare the efficacy of multiple QPE methods when simulating discharge in a small, urban watershed on a continuous basis using an operational hydrologic model and QPE forcings. The research distributed hydrologic model (RDHM) was used to model a basin in Roanoke, Virginia, USA, forced with QPEs from four methods: mean field bias (MFB) correction of radar data, kriging of rain gauge data, uncorrected radar data, and a basin-uniform estimate from a single gauge inside the watershed. Based on comparisons between simulated and observed discharge at the basin outlet for a six-month period in 2018, simulations forced with the uncorrected radar QPE had the highest accuracy, as measured by root mean squared error (RMSE) and peak flow relative error, despite systematic underprediction of the mean areal precipitation (MAP). Simulations forced with MFB-corrected radar data consistently and significantly overpredicted discharge, but had the highest accuracy in predicting the timing of peak flows.