In this paper, we solve the obstacle problems on metric measure spaces with generalized Ricci lower bounds. We show the existence and Lipschitz continuity of the solutions, and then we establish some regularities of the free boundaries.

In this paper we study the hydrostatic limit of the Navier-Stokes-alpha model in a very thin strip domain. We derive some Prandtl-type limit equations for this model and we prove the global well-posedness of the limit system for small initial conditions in an appropriate analytic function space.

This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integrals $\mu_{\Omega, \beta}$ and the commutators $\mu_{\Omega, \beta}^b$ generated by $\mu_{\Omega, \beta}$ with $b\in L_{\rm loc}(\mathbb{R}^n)$ on weighted Hardy spaces. Under the assumption of that the homogeneous kernel $\Omega$ satisfies certain regularities, the authors obtain the boundedness of $\mu_{\Omega, \beta}$ from the weighted Hardy spaces $H^p_{\omega^p}(\mathbb{R}^n)$ to the weighted Lebesgue spaces $L^q_{\omega^q}(\mathbb{R}^n)$ for $n/(n+\beta)\le p\le 1$ with $1/q=1/p-\beta/n$, as well as the same $(H^p_{\omega^p}, L^q_{\omega^q})$-boudedness of $\mu_{\Omega, \beta}^b$ when $b$ belongs to $\mathcal{BMO}_{\omega^p, p}(\mathbb{R}^n)$, which is a non-trivial subspace of ${\rm BMO}(\mathbb{R}^n)$.

Let $\tau$ be a generalized Thue-Morse substitution on a two-letter alphabet $\{a, b\}$:$\tau(a)=a^mb^m$, $\tau(b)=b^ma^ms$ for the integer $m\ge 2$. Let $\xi$ be a sequence in $\{a, b\}^{\mathbb{Z}}$ that is generated by $\tau$. We study the one-dimensional Schrödinger operator $H_{m, \lambda}$ on $l^2(\mathbb{Z})$ with a potential given by $$v(n)=\lambda V_{\xi}(n), $$ where $\lambda>0$ is the coupling and $V_\xi(n)=1$ ($V_\xi(n)=-1$) if $\xi(n)=a$ ($\xi(n)=b$). Let $\Lambda_2=2$, and for $m>2$, let $\Lambda_m=m$ if $m\equiv0\mod 4$; let $\Lambda_m=m-3$ if $m\equiv1\mod 4$; let $\Lambda_m=m-2$ if $m\equiv2\mod 4$; let $\Lambda_m=m-1$ if $m\equiv3\mod 4$. We show that the Hausdorff dimension of the spectrum $\sigma(H_{m, \lambda})$ satisfies that $$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$$ It is interesting to see that $\dim_H \sigma(H_{m, \lambda})$ tends to $1$ as $m$ tends to infinity.

This paper is concerned with inverse acoustic scattering in an inhomogeneous medium with a conductive boundary condition and the unknown buried impenetrable objects inside. Using a variational approach, we establish the well-posedness of the direct problem. For the inverse problem, we shall numerically reconstruct the inhomogeneous medium from the far-field data for different kinds of cases. For the case when a Dirichlet boundary condition is imposed on the buried object, the classical factorization method proposed in [2] is justified as valid for reconstructing the inhomogeneous medium from the far-field data. For the case when a Neumann boundary condition is imposed on the buried object, the classical factorization method of [1] cannot be applied directly, since the middle operator of the factorization of the far-field operator is only compact. In this case, we develop a modified factorization method to locate the inhomogeneous medium with a conductive boundary condition and the unknown buried objects. Some numerical experiments are provided to demonstrate the practicability of the inversion algorithms developed.

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem. The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order. The concentration equation is solved by the method of mixed finite element, where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method. The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. In actual computations the characteristics adopts a large time step without any loss of accuracy. The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux. In order to derive the optimal $3/2$-order error estimate in $L^2$ norm, a post-processing technique is included in the approximation to the scalar unknowns. Numerical experiments are illustrated finally to validate theoretical analysis and efficiency. This method can be used to solve such an important problem.

We investigate the global classical solutions of the non-relativistic Vlasov-Darwin system with generalized variables (VDG) in three dimensions. We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials. Then, we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system (VP) at the asymptotic rate of $\frac{1}{c^2}$ as the speed of light $c$ tends to infinity for all time. Moreover, we obtain rigorously an asymptotic estimate of the difference between the two systems.

In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents $ \left\{\begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u +\frac{ \vert u \vert ^{2^{*}(s)-2}u}{ \vert x \vert ^{s}}+ a(x) \vert u \vert ^{q-2} u & \;{\rm in} \; \Omega, \\ u=0 & \;{\rm on} \; \partial \Omega, \end{array}\right.$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $ \partial \Omega$ at 0 are negative, $a \in \mathcal{C}^{1}(\bar{\Omega}, \mathbb{R^{\ast}}^{+}), $ $ \mu> 0, $ $0<s<2, $ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s):=\frac{2 (N-s)}{N-2}$ we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.

Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function $f(\xi)=\xi+\sum\limits_{m=2}^\infty a_m\xi^m$ in the class $\mathcal{C}$, $$ \max\limits_{f\in \mathcal{C}}|a_3-\lambda a_2^2|\leq \left\{\begin{array}{ll} 3-4\lambda, \quad & \lambda\in[0, \frac{1}{3}], \\[3mm] \frac{1}{3}+\frac{4}{9\lambda}, \quad & \lambda\in[\frac{1}{3}, \frac{2}{3}], \\[3mm] 1, \quad & \lambda\in[\frac{2}{3}, 1]. \end{array}\right.$$ By applying this inequality, it can be proven that $||a_3|-|a_2||\leq 1$ for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

In this paper, we study a quantum kinetic-fluid model in a three-dimensional torus. This model is a coupling of the Vlasov-Fokker-Planck equation and the compressible quantum Navier-Stokes equations with degenerate viscosity. We establish a global weak solution to this model for arbitrarily large initial data when the pressure takes the form $ p(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$.

This paper deals with the asymptotic behavior of solutions of the stochastic $g$-Navier-Stokes equation driven by nonlinear noise. The existence and uniqueness of weak pullback mean random attractors for the equation in Bochner space is proven for when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we also establish the existence of invariant measures for the equation.

In this paper, we study the complex symmetric $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on the weighted Hardy spaces $H_\gamma$ of the unit disk $\mathbb{D}$. It is well-known that there are only two classes of weighted composition conjugations $\mathcal{A}_{u, v}$ on $H_\gamma(\mathbb{D})$: either $\mathcal{C}_1$ or $\mathcal{C}_2$. We completely characterize $\mathcal{C}_1$-symmetric ($\mathcal{C}_2$-symmetric) $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on $H_\gamma(\mathbb{D})$.}

This paper is concerned with a diffuse interface model called Navier-Stokes/Cahn-Hilliard system. This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface. For the periodic boundary value problem of this system in torus $\mathbb{T}^3$, we prove that there exists a global unique strong solution near the phase separation state, which means that no vacuum, shock wave, mass concentration, interface collision or rupture will be developed in finite time. Furthermore, we establish the large time behavior of the global strong solution of this system. In particular, we find that the phase field decays algebraically to the phase separation state.

The purpose of this work is to implement a discontinuous Galerkin (DG) method with a one-sided flux for a singularly perturbed Volterra integro-differential equation (VIDE) with a smooth kernel. First, the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided. Then the existence and uniqueness of the DG solution are proven. Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established. Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants, the DG method achieves the uniform convergence in the $L^2$ norm with respect to the singular perturbation parameter $\epsilon$ when the space of polynomials with degree $p$ is used. A numerical experiment validates the theoretical results. Furthermore, an ultra-convergence order $2p+1$ at the nodes for the one-sided flux, uniform with respect to the singular perturbation parameter $\epsilon$, is observed numerically.

In this paper, we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only, it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves, while the initial perturbation and the strength of rarefaction waves are suitably small.

We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman, weighted Dirichlet, and weighted Besov spaces on the upper half-plane by dilatations and eventually by analytic polynomials.

In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations (FBSDEs), whose coefficients not only depend on the solution but also on the law of the solution. The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations (BSDEs) under Lipschitz conditions, and for the one-dimensional case a comparison theorem is studied. With the help of this comparison result, we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions. It should be mentioned that, under appropriate assumptions, we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.

This paper focusses on a peeling phenomenon governed by a nonlinear wave equation with a free boundary. Under the hypotheses that the total variation of the intial data and the boundary data are small, the global existence of a weak solution to the nonlinear problem (1.1)-(1.3) is proven by a modified Glimm scheme. The regularity of the peeling front is established, and the asymptotic behaviour of the obtained solution and the peeling front at infinity is also studied.

In this paper, for a bounded $C^2$ domain, we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive $C^2$ boundary data. We have a non-existence result, which is the justification for taking into account the restricted boundary data. There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

In this paper, we study the following Schrödinger-Poisson system with critical growth: \begin{equation*} \begin{cases}-\Delta u+V(x)u+ \phi(x)u =f(u)+|u|^4u, \ & x\in\mathbb{R}^3, \\ -\Delta \phi=u^2, \ & x\in\mathbb{R}^3. \end{cases} \end{equation*} We establish the existence of a positive ground state solution and a least energy sign-changing solution, providing that the nonlinearity $f$ is super-cubic, subcritical and that the potential $V(x)$ has a potential well.

Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.

The optimization problem to minimize the weighted sum of $\alpha$-z Bures-Wasserstein quantum divergences to given positive definite Hermitian matrices has been solved. We call the unique minimizer the $\alpha$-z weighted right mean, which provides a new non-commutative version of generalized mean (Hölder mean). We investigate its fundamental properties, and give many interesting operator inequalities with the matrix power mean including the Cartan mean. Moreover, we verify the trace inequality with the Wasserstein mean and provide bounds for the Hadamard product of two right means.