We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces. More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous Markov-Feller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior, then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.

In order to measure the uncertainty of financial asset returns in the stock market, this paper presents a new model, called SV-dtC model, a stochastic volatility (SV) model assuming that the stock return has a doubly truncated Cauchy distribution, which takes into account the high peak and fat tail of the empirical distribution simultaneously. Under the Bayesian framework, a prior and posterior analysis for the parameters is made and Markov Chain Monte Carlo (MCMC) is used for computing the posterior estimates of the model parameters and forecasting in the empirical application of Shanghai Stock Exchange Composite Index (SSE-CI) with respect to the proposed SV-dtC model and two classic SV-N (SV model with Normal distribution) and SV-T (SV model with Student-t distribution) models. The empirical analysis shows that the proposed SV-dtC model has better performance by model checking, including independence test (Projection correlation test), Kolmogorov-Smirnov test(K-S test) and Q-Q plot. Additionally, deviance information criterion (DIC) also shows that the proposed model has a significant improvement in model fit over the others.

This article proves the existence and uniqueness conditions of the solution of two-dimensional time-space tempered fractional diffusion-wave equation. We find analytical solution of the equation via the two-step Adomian decomposition method (TSADM). The existence result is obtained with the help of some fixed point theorems, while the uniqueness of the solution is a consequence of the Banach contraction principle. Additionally, we study the stability via the Ulam-Hyers stability for the considered problem. The existing techniques use numerical algorithms for solving the two-dimensional time-space tempered fractional diffusion-wave equation, and thus, the results obtained from them are the approximate solution of the problem with high computational and time complexity. In comparison, our proposed method eliminates all the difficulties arising from numerical methods and gives an analytical solution with a straightforward process in just one iteration.

In this paper, we investigate the global solution and the structures of interaction between two dimensional non-selfsimilar shock wave and rarefaction wave of general two-dimensional scalar conservation law in which flux functions f(u) and g(u) do not need to be convex, and the initial value contains three constant states which are respectively separated by two general initial discontinuities. When initial value contains three constant states, the cases of selfsimilar shock wave and rarefaction wave had been studied before, but no results of the cases of neither non-selfsimilar shock wave or non-selfsimilar rarefaction wave. Under the assumption that Condition H which is generalization of one dimensional convex condition, and some weak conditions of initial discontinuity, according to all the kinds of combination of elementary waves respectively staring from two initial discontinuities, we get four cases of wave interactions as S + S, S + R, R + S and R + R. By studying these interactions between non-selfsimilar elementary waves, we obtain and prove all structures of non-selfsimilar global solutions for all cases.

A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number ${R}_{0}$. Specifically, it shows that the disease-free equilibrium ${E}^{0}$ is globally asymptotically stable (GAS) for ${R}_{0}<1$, and globally attractive (GA) for ${R}_{0}=1$, while the endemic equilibrium $E^{\ast}$ is GAS and ${E}^{0}$ is unstable for ${R}_{0}>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast}$ for $R_{0}>1$, the weak persistence of the model is proved by some analysis techniques.

In present work studied the new boundary value problem for semi linear (Power-type nonlinearities) system equations of mixed hyperbolic -elliptic Keldysh type in the multivariate dimension with the changing time direction. Considered problem and equation belongs to the modern level partial differential equations. Applying methods of functional analysis, topological methods, “ε” -regularizing. and continuation by the parameter at the same time with aid of a prior estimates, under assumptions conditions on coefficients of equations of system, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev’s space. In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations, moreover in this case constructed new weithged spaces, the identity between of strong and weak solutions is established.

This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form $\dot{x}=-y+xf(x,y),\ \dot{y}=x+yf(x,y)$, where $f(x,y)=a_{1}x+a_{2}xy+a_{3}xy^{2}+a_{4}xy^{3}+a_{5}xy^4=x\sigma(y)$, and any zero of $1+a_{1}y+a_{2}y^2+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}$, $y=\overline{y}$ is an invariant straight line. At last, all global phase portraits are drawn on the Poincarédisk.

In this paper, we consider a modification of the well-known logistic family using a family of fuzzy numbers. The dynamics of this modified logistic map is studied by computing its topological entropy with a given accuracy. This computation allows us to characterize when the dynamics of the modified family is chaotic. Besides, some attractors that appear in bifurcation diagrams are explained. Finally, we will show that the dynamics induced by the logistic family on the fuzzy numbers need not be complicated at all.

This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

In the current paper, the best linear unbiased estimators (BLUEs) of location and scale parameters from location-scale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling (MERSS). Explicit mathematical expressions of these estimators and their variances are derived. Their relative efficiencies with respect to the minimum variance unbiased estimators (MVUEs) under simple random sampling (SRS) are compared for the cases of some usual distributions. The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\cdots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N, N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $1\leq i\leq r$. We show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum\limits^{j+1}_{i=1}\alpha_i$ for $j=1,2,\cdots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$

In this paper, we study the irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.

In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in L_x,v² ∩L_x,v^∞ in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.

Let $\mathbf{G}=\{G_i: i\in[n]\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$, where $\mathbf{G}$ can be seen as an edge-colored (multi)graph and each $G_i$ is the set of edges with color $i$. A graph $F$ on $V$ is called rainbow if any two edges of $F$ come from different $G_i$s'. We say that $\mathbf{G}$ is rainbow pancyclic if there is a rainbow cycle $C_{\ell}$ of length $\ell$ in $\mathbf{G}$ for each integer $\ell\in [3,n]$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If $\delta(G_i)\geq\frac{n}{2}$ for each $i\in[n]$, then there is a rainbow Hamiltonian cycle in $\mathbf{G}$. In this paper, under the same condition, we show that $\mathbf{G}$ is rainbow pancyclic except that $n$ is even and $\mathbf{G}$ consists of $n$ copies of $K_{\frac{n}{2},\frac{n}{2}}$. This result supports the famous meta-conjecture posed by Bondy.

For a revised model of Caldentey and Stacchetti (Econometrica, 2010) in continuous-time insider trading with a random deadline which allows market makers to observe some information on a risky asset, a closed form of its market equilibrium consisting of optimal insider trading intensity and market liquidity is obtained by maximum principle method. It shows that in the equilibrium, (i) as time goes by, the optimal insider trading intensity is exponentially increasing even up to infinity while both the market liquidity and the residual information are exponentially decreasing even down to zero; (ii) the more accurate information observed by market makers, the stronger optimal insider trading intensity is such that the total expect profit of the insider is decreasing even go to zero while both the market liquidity and the residual information are decreasing; (iii) the longer the mean of random time, the weaker the optimal insider trading intensity is while the more both the residual information and the expected profit are, but there is a threshold of trading time, half of the mean of the random time, such that if and only if after it the market liquidity is increasing with the mean of random time increasing.

In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in $\mathbb{T}^2$ or $\mathbb{R}^2$ and $\mathbb{R}^3$.

A bio-safe dengue control strategy is to use Wolbachia, which can induce incomplete cytoplasmic incompatibility (CI) and reduce the mating competitiveness of infected males. In this work, we formulate a delay differential equation model, including both the larval and adult stages of wild mosquitoes, to assess the impacts of CI intensity ξ and mating competitiveness θ of infected males on the suppression efficiency. Our analysis identifies a CI intensity threshold ξ^* below which a successful suppression is impossible. When ξ≥ξ^*, the wild population will be eliminated ultimately if the releasing level exceeds the release amount threshold R^* uniformly. The dependence of R^* on ξ and θ, and the impact of temperature on suppression are further exhibited through numerical examples. Our analyses indicate that a slight reduction of ξ is more devastating than significantly decrease of θ in the suppression efficiency. To suppress more than 95% wild mosquitoes during the peak season of dengue in Guangzhou, the optimal starting date for releasing is sensitive to ξ but almost independent of θ. One percent reduction of ξ from 1 requires at least one week earlier in the optimal releasing starting date from 7 weeks ahead of the peak season of dengue.

We study the L²-supercritical nonlinear Schrödinger equation (NLS) with a partial confinement, which is the limit case of the cigar-shaped model in Bose-Einstein condensate (BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution flow, we show a sharp condition for global existence.

The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.

The traditional simple random sampling (SRS) design method is inefficient in many cases. Statisticians proposed some new designs to increase efficiency. In this paper, as a variation of moving extremes ranked set sampling (MERSS), double MERSS (DMERSS) is proposed and its properties for estimating the population mean are considered. It turns out that, when the underlying distribution is symmetric, DMERSS gives unbiased estimators of the population mean. Also, it is found that DMERSS is more efficient than the SRS and MERSS methods for usual symmetric distributions (normal and uniform). For asymmetric distributions considered in this study, the DMERSS has a small bias and it is more efficient than SRS for usual asymmetric distribution (exponential) for small sample sizes.

In this paper, the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description, numerical simulation and theoretical analysis. Two important factors, heat and magnetic influences are involved. The mathematical model is formulated by four nonlinear partial differential equations (PDEs), determining four major physical variables. The influences of magnetic fields are supposed to be weak, and the strength is parallel to the $z$-axis. The elliptic equation is treated by a block-centered method, and the law of conservation is preserved. The computational accuracy is improved one order. Other equations are convection-dominated, thus are approximated by upwind block-centered differences. Upwind difference can eliminate numerical dispersion and nonphysical oscillation. The diffusion is approximated by the block-centered difference, while the convection term is treated by upwind approximation. Furthermore, the unknowns and adjoint functions are computed at the same time. These characters play important roles in numerical computations of conductor device problems. Using the theories of priori analysis such as energy estimates, the principle of duality and mathematical inductions, an optimal estimates result is obtained. Then a composite numerical method is shown for solving this problem.

A path-factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor of $G$ means a path factor in which each component is a path with at least $k$ vertices. A graph $G$ is a $P_{\geq k}$-factor covered graph if for any $e\in E(G)$, $G$ has a $P_{\geq k}$-factor including $e$. Let $\beta$ be a real number with $\frac{1}{3}\leq\beta\leq1$ and $k$ be a positive integer. We verify that (\romannumeral1) a $k$-connected graph $G$ of order $n$ with $n\geq5k+2$ has a $P_{\geq3}$-factor if $|N_G(I)|>\beta(n-3k-1)+k$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$; (\romannumeral2) a $(k+1)$-connected graph $G$ of order $n$ with $n\geq5k+2$ is a $P_{\geq3}$-factor covered graph if $|N_G(I)|>\beta(n-3k-1)+k+1$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$.

In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term, where the initial data lies in $W^{1,\infty}(\mathbb{R}^n) \cap C^1(\mathbb{R}^n)$. We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

In this paper, it is proved that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity, different far field states and periodic perturbations, not only exists globally in time, but also converges towards the viscous shock wave of the corresponding Riemann problem as time goes to infinity. Furthermore, the decay rate is shown. The proof is given by a technical energy method.

In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail.

In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled Nonlinear-Einstein-Klein-Gordon (NLEKG) system in the presence of a negative cosmological constant. We prove the well posedness of the NLEKG system in the Schwarzschild-AdS spacetimes and that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which ϕ = 0 identically) are asymptotically stable. Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a SchwarzschildAdS spacetime. Bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.

In this paper, we design the differentially private variants of the classical Frank-Wolfe algorithm with shuffle model in the optimization of machine learning. Under weak assumptions and the generalized linear loss (GLL) structure, we propose a noisy Frank-Wolfe with shuffle model algorithm (NoisyFWS) and a noisy variance-reduced Frank-Wolfe with the shuffle model algorithm (NoisyVRFWS) by adding calibrated laplace noise under shuffling scheme in the $\ell_{p} (p\in [1,2])$-case, and study their privacy as well as utility guarantees for the Hölder smoothness GLL. In particular, the privacy guarantees are mainly achieved by using advanced composition and privacy amplification by shuffling. The utility bounds of the NoisyFWS and NoisyVRFWS are analyzed and obtained the optimal excess population risks $\mathcal{O}(n^{-\frac{1+\alpha}{4\alpha}}+\frac{\log(d)\sqrt{\log (1/\delta)}}{n\epsilon})$ and $\mathcal{O}(n^{-\frac{1+\alpha}{4\alpha}}+\frac{\log(d)\sqrt{\log (1/\delta)}}{n^{2}\epsilon})$ with gradient complexity $\mathcal{O}(n^{\frac{(1+\alpha)^{2}}{4\alpha^{2}}})$ for $\alpha \in [1/\sqrt{3},1]$. It turns out that the risk rates under shuffling scheme are a nearly-dimension independent rate, which is consistent with the previous work in some cases. In addition, there is a vital tradeoff between $(\alpha,L)$-Hölder smoothness GLL and the gradient complexity. The linear gradient complexity $\mathcal{O}(n)$ is showed by the parameter $\alpha=1$.

In this paper, solutions of the Camassa-Holm equation near the soliton $Q$ is decomposed by pseudo-conformal transformation as follows: $\lambda^{1/2}(t)u(t,\lambda(t)y+x(t))=Q(y)+\varepsilon(t,y)$, and the estimation formula with respect to $\varepsilon(t,y)$ is obtained: $|\varepsilon(t,y)|\leq Ca_3Te^{-\theta|y|}+|\lambda^{1/2}(t)\varepsilon_0|$. For the CH equation, we prove that the solution of the Cauchy problem and the soliton $Q$ is sufficiently close as $y\rightarrow\infty$, and the approximation degree of the solution and $Q$ is the same as that of initial data and $Q$, besides the energy distribution of $\varepsilon$ is consistent with the distribution of the soliton $Q$ in $H^2$.

In this paper, we are concerned with a predator-prey model with Holling type II functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined. We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.

This paper introduces two local conditional dependence matrices based on Spearman’s ρ and Kendall’s τ given the condition that the underlying random variables belong to the intervals determined by their quantiles. The robustness is studied by means of the influence functions of conditional Spearman’s ρ and Kendall’s τ. Using the two matrices, we construct the corresponding test statistics of local conditional dependence and derive their limit behavior including consistency, null and alternative asymptotic distributions. Simulation studies illustrate a superior power performance of the proposed Kendall-based test. Real data analysis with proposed methods provides a precise description and explanation of some financial phenomena in terms of mathematical statistics.

Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we reconstruct the transition probability function that has a central role in the renormalization group using a stochastic differential equation. From a probabilistic perspective, the renormalization procedure can be characterized by a discrete-time Markov chain. Therefore, we focus on this stochastic dynamic, and establish the local Poincaré inequality by calculating the Bakry-Émery curvature for two point functions. Finally, we choose an appropriate coupling relationship between parameters $K$ and $T$ to obtain the Poincaré inequality of two point functions for the limiting system. Our method extends the classic Bakry-Émery criterion, and the results provide a new perspective to characterize the renormalization procedure.

This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/Allen-Cahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time $t\rightarrow+\infty$.

In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in $L^2$ norm. Finally, numerical examples show the effectiveness and practicability for some actual problems.

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $\sum\limits_{A} x^{|A|}$, where the sum is over all independent sets $A$ of $G$. In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal. Although this unimodality conjecture has attracted many researchers' attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal. In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.

The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.

We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.

We consider the following quasilinear Schrödinger equation involving $p$-Laplacian \begin{align*} -\Delta_p u +V(x)|u|^{p-2}u-\Delta_p(|u|^{2\eta})|u|^{2\eta-2}u=\lambda\frac{|u|^{q-2}u}{|x|^{\mu}}+\frac{|u|^{2\eta p^*(\nu)-2}u}{|x|^\nu}\quad\text{in}\ \mathbb{R}^N, \end{align*} where $ N> p>1,\ \eta\ge \frac{p}{2(p-1)}$, $p< q<2\eta p^*(\mu)$, $p^*(s)=\frac{p(N-s)}{N-p}$, and $\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,p)$. Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.

We investigate the global structures of the non-selfsimilar solutions for $n$-dimensional ($n$-D) non-homogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a $({n-1})$-dimensional sphere. We first obtain the expressions of $n$-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves, we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the $n$-D shock waves. The asymptotic behaviors with geometric structures are also proved.