An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains

The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a noncoercive homogenization problem.     

On the Leray–Hopf extension condition for the steady-state Navier–Stokes problem in multiply-connected bounded domains

Employing the approach of Takeshita (Pacific J Math 157:151–158, 1993), we give an elementary proof of the invalidity of the Leray–Hopf Extension Condition for certain multiply connected bounded domains of \({\mathbb {R}}^{n}\) , \(n=2,3\) , whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by Heywood (J Math Fluid Mech 13:449–457, 2011).     

The error term in the Sato–Tate conjecture

Let \({f(z) = \sum_{n=1}^\infty a(n)e^{2\pi i nz} \in S_k^{\mathrm{new}}(\Gamma_0(N))}\) be a newform of even weight \({k \geq 2}\) that does not have complex multiplication. Then \({a(n) \in \mathbb{R}}\) for all n; so for any prime p, there exists \({\theta_p \in [0, \pi]}\) such that \({a(p) = 2p^{(k-1)/2} {\rm cos} (\theta_p)}\) . Let \({\pi(x) = \#\{p \leq x\}}\) . For a given subinterval \({[\alpha, \beta]\subset[0, \pi]}\) , the now-proven Sato–Tate conjecture tells us that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} \sim \mu_{ST} ([\alpha, \beta])\pi(x),\quad \mu_{ST} ([\alpha, \beta]) = \int\limits_{\alpha}^\beta \frac{2}{\pi}{\rm sin}^2(\theta) d\theta. $$ Let \({\epsilon > 0}\) . Assuming that the symmetric power L-functions of f are automorphic, we prove that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} = \mu_{ST} ([\alpha, \beta])\pi(x) + O\left(\frac{x}{(\log x)^{9/8-\epsilon}} \right), $$ where the implied constant is effectively computable and depends only on k,N, and \({\epsilon}\) .     

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

Mathematical Notes - This paper deals with the existence of a solution of a problem involving Leray–Lions type operators in Sobolev spaces with variable exponent. The proofs of our main...     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

The lower estimate for the linear combinations of Bernstein–Kantorovich operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein–Kantorovich operators, which gives the optimal approximation rate. On the basis of this inequality, we further obtain the lower estimate for these operators.     

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.     

Well-posedness in energy space for the periodic modified Benjamin–Ono equation

We prove that the periodic modified Benjamin–Ono equation is locally well-posed in the energy space H^1/2$H^{1/2}$. This ensures the global well-posedness in the defocusing case. The proof is based on an X^s,b$X^{s,b}$ analysis of the system after a gauge transform.     

The limitations of the Poincaré inequality for Grušin type operators

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\) . We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H _δ is a generalized Gru?in operator, $$H_\delta=-\nabla_{x_1}\,|x_1|^{\left(2\delta_1,2\delta_1'\right)} \,\nabla_{x_1}-|x_1|^{\left(2\delta_2,2\delta_2'\right)} \,\nabla_{x_2}^2.$$ Here \({x_1 \in \mathbf{R}^n,\; x_2 \in \mathbf{R}^m,\;\delta_1,\delta_1'\in[0,1\rangle,\;\delta_2,\delta_2'\geq0}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta}}\) if \({|x_1|\leq 1}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta'}}\) if \({|x_1|\geq 1}\) . We prove that the Poincaré inequality, formulated in terms of the geometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and \({\delta_1\vee\delta_1'\in[0,1/2\rangle}\) but it fails if n = 1 and \({\delta_1\vee\delta_1'\in[1/2,1\rangle}\) . The failure is caused by the leading term. If \({\delta_1\in[1/2, 1\rangle}\) , it is an effect of the local degeneracy \({|x_1|^{2\delta_1}}\) , but if \({\delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , it is an effect of the growth at infinity of \({|x_1|^{2\delta_1'}}\) . If n = 1 and \({\delta_1\in[1/2, 1\rangle}\) , then the semigroup S generated by the Friedrichs’ extension of H is not ergodic. The subspaces \({x_1\geq 0}\) and \({x_1\leq 0}\) are S-invariant, and the Poincaré inequality is valid on each of these subspaces. If, however, \({n=1,\; \delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , then the semigroup S is ergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these results for the Gaussian and non-Gaussian behaviour of the semigroup S.     

The unbounded solution of a periodic mixed Sturm–Liouville problem in an infinite strip for the Laplacian

The unbounded solution, at the points where the boundary conditions change, for a mixed Sturm–Liouville problem of the Dirichlet–Neumann type can be obtained using the method of the integral equation formulation. Since this formulation is usually reduced to an infinite algebraic system in which the unknowns are the Fourier coefficients of the unknown unbounded entity, a study of ?_p-solutions imposes itself concerning the influence of the truncation on such systems. This study is achieved and the well-known theorem on the ?₂-solutions of the infinite algebraic systems is generalized.     

Unilateral problem for the Navier–Stokes operators with variable viscosity in noncylindrical domain

We study a unilateral problem for the operator L perturbed of Navier–Stokes operator in a noncylindrical case, where

Lu=u^′-(_ν0+_ν1∥u(t)∥²)Δu+(u.∇)u-f+∇p.$\mathit{Lu}=u^{\prime }-({\nu }_0+{\nu }_1\parallel u(t){\parallel }^2)\mathrm{\Delta }u+(u.\nabla )u-f+\nabla p\text{.}$

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

The incompressible Navier–Stokes flows in a 2D exterior domain are considered, for which the associated total net force to the boundary may not vanish. The decay properties are shown for the first and second derivatives of the Navier–Stokes flows in L ¹ and weighted spaces, respectively, which improve Theorem 1.2 in Bae and Jin (J Funct Anal 240:508–529, 2006).     

On the homogenization principle in a time-periodic problem for the Navier–Stokes equations with rapidly oscillating mass force

We study the behavior of the set of time-periodic solutions of the three-dimensional system of Navier–Stokes equations in a bounded domain as the frequency of the oscillations of the right-hand side tends to infinity. It is established that the set of periodic solutions tends to the solution set of the homogenized stationary equation.     

The bifurcation and exact peakons,solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation

In this paper, the Kudryashov–Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, the existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some new exact travelling wave solutions are presented through some special phase orbits.