In this note, we consider the homogenization of compressible Navier-Stokes equations in a periodically perforated domain $\mathbb{R}^3$. Assuming that particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and Mach number decreases fast enough, show limit $\varepsilon\to 0$, velocity density converge to solution incompressible with Brinkman term. We strongly ...

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain optimal rate convergence $\mathcal{O}(\varepsilon)$ $u^\varepsilon \rightarrow u$ as $\varepsilon 0^+$ for large class convex Hamiltonians $H(x,y,p)$ in one dimension. This includes from cl...

Abstract In this article, we study some results on the value distribution of differential polynomials and mainly prove following theorem: suppose that P P is a polynomial with deg ≥ 3 {\rm{\deg }}\hspace{0.33em}P\ge 3 f...

We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u \phi = f(u)\\ \varepsilon^{4} \Delta^{2}\phi 4\pi\varepsilon u^{2}\\ \end{array} \right.$$ where $\varepsilon > 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R}, f:\mathbb{R} \mathbb{R}$ satisfy suitable assumptions. By using variational methods, we pr...

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ $ {\mathbb {R}}^{\alpha },$ study Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ one its hyperbolic sectors. We prove (theorem 1.1) that derivative $\partial _...

In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*} subject non-local boundary conditions x(0)=h_1+\varepsilon\eta_1(x)\text{ and } x(\pi)=h_2+\varepsilon\eta_2(x). this paper, we study existence solutions above pr...

Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and let $F\subseteq TM$ an integrable subbundle $TM$. $g^F=g^{TM}|_{F}$ the restricted metric on $F$ $k^F$ associated leafwise scalar curvature. $f:M\to S^n(1)$ smooth area decreasing map along $F$, which is locally constant near infinity non-zero degree. We show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ support ${\...

For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b :\, a,b\in A\}$. Combining structural theorem of Freiman on sets with small doubling constants together discrete analogue Pr\'ekopa--Leindler inequality we prove lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate conjecture about the v...

We study the asymptotic behavior, when $$\varepsilon \rightarrow 0$$ , of minimizers $$\{u_\varepsilon \}_{\varepsilon >0}$$ for energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$ over class maps $$u\in H^1(\Omega ,{{\mathbb {R}}}^2)$$ satisfying boundary condition $$u=g$$ on $$\partial \Omega ...