Local Existence of Strong Solutions to the 3D Zakharov-Kuznestov Equation in a Bounded Domain

We consider here the local existence of strong solutions for the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-pi /2, pi /2)^d, d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution on a short time interval, whose length only depends on the given data. We use the parabolic regularization of the ZK equation to derive the global and local bounds independent of epsilon for various norms of the solution. In particular, we derive the local bound of the nonlinear term by a singular perturbation argument. Then we can pass to the limit and hence deduce the local existence of strong solutions

The Existence of Strong Solutions to the 3D Zakharov-Kuznestov Equation in a Bounded Domain

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-\pi /2, \pi /2)^d, d=1,2 supplemented with suitable boundary conditions. We prove that there exists a solution u \in \mathcal C ([0, T]; H^1(\dom)) to the initial and boundary value problem for the ZK equation in both dimensions 2 and 3 for every T>0. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in 3D. More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures. At the same time, the uniqueness of solutions is still open in 2D and 3D due to the partially hyperbolic feature of the model.Comment: arXiv admin note: substantial text overlap with arXiv:1307.682

Initial and Boundary value problems for the Deterministic and Stochastic Zakharov-Kuznetsov Equation in a Bounded Domain

Thesis (Ph.D.) - Indiana University, Mathematics, 2015We study in this thesis the well-posedness and regularity of the Zakharov-Kuznetsov (ZK) equation in the deterministic and stochastic cases, subjected to a rectangular domain in space dimensions 2 and 3. Mainly we have established the existence, in 3D, and uniqueness, in 2D, of the weak solutions, and the local and global existence of strong solutions in 3D. Then we extend the results to the stochastic case and obtain in 3D the existence of martingale solutions, and in 2D the pathwise uniqueness and existence of pathwise solutions. The main focus is on the mixed features of the partial hyperbolicity, nonlinearity, nonconventional boundary conditions, anisotropicity and stochasticity, which requires methods quite different than those of the classical models of fluid dynamics, such as the Navier-Stokes equation, Primitive Equation and related equations

Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and oceans, we study their time discretization by an implicit Euler scheme. From deterministic viewpoint the 3D Primitive Equations are studied with physically realistic boundary conditions. From probabilistic viewpoint we consider a wide class of nonlinear, state dependent, white noise forcings. The proof of convergence of the Euler scheme covers the equations for the oceans, atmosphere, coupled oceanic-atmospheric system and other geophysical equations. We obtain the existence of solutions weak in PDE and probabilistic sense, a result which is new by itself to the best of our knowledge

Body image construction and mental health levels among college students: a data survey of Chinese university students

BackgroundWith the rapid changes in body image construction brought about by the upgrading of consumption in China, trend-seeking college students are faced with mental health problems brought about by the pursuit of the “ideal body type,” which cannot be ignored. This study aims to explore the relationship between body image construction and mental health among college students. This study utilized data from the Survey on Physical Activity and Mental Health of College Students. A total of 1,192 students were randomly selected as the survey sample, and 1,044 valid samples were obtained. The mean age of the respondents was 19.34 years.MethodsFirst, we categorized body image constructs into three categories based on the differences between subjective and objective body image: high acceptance, low acceptance, and consistency. Second, to ensure analytical rigor and minimize potential confounders, we used a generalized propensity score weighting model. Finally, we used a causal mediation framework to investigate the potential causal mechanisms between the independent variable (perceived body image bias) and the dependent variable (mental health) in order to better understand the “net effect.”Results(1) There is a significant correlation between college students’ body image perceptual bias and mental health, i.e., the higher the individual’s acceptance of his/her own body image, the higher the level of mental health, and vice versa. (2) Students in humanities and social sciences are more likely to have increased psychological burden due to poor negative body image. (3) In the mediation analysis, although the causal mediating effect of physical exercise was not significant, family and peer support in physical exercise played an important mediating role, especially the influence of peers was more significant.ConclusionThe construction of body image is a double-edged sword that can either promote positive individual development or lead to self-depreciation. Creating a positive climate for physical activity has a positive impact on college students’ mental health compared to participation in physical activity behaviors. While improving students’ media literacy on college campuses, it is important to enhance adaptive guidance to promote their physical and mental health and personal development

A multilayer network model of the coevolution of the spread of a disease and competing opinions

During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we construct a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. On one layer, we consider the concurrent evolution of two opinions — pro-physical-distancing and anti-physical-distancing — that compete with each other and have mutual immunity to each other. The disease evolves on the other layer, and individuals are less likely (respectively, more likely) to become infected when they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. Through numerical simulations, we illustrate the influence of opinion dynamics on the spread of the disease from complex interactions both between the two conflicting opinions and between the opinions and the disease. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease

The impact of Bacillus subtilis DSM32315 and L-Threonine supplementation on the amino acid composition of eggs and early post-hatch performance of ducklings

Poultry requires Threonine, an essential amino acid, and its metabolites for proper metabolic function. Threonine is crucial in the biosynthesis of mucin, which is essential for intestinal health and nutrient absorption. Bacillus subtilis (B. subtilis) is a potential substitute for antibiotic growth promoters in the poultry industry. The current study was designed to evaluate the simultaneous effect of L-Threonine (Thr) and B. subtilis DSM32315 supplementation on laying duck breeders in order to maximize performance. A total number of 648 female 23-week-old Longyan duck breeders were assigned to a 3 × 2 factorial design with six replicates of 18 birds per replicate. L-Thr was added to the control diet at concentrations of 0, 0.7, and 1.4 g/kg, equating to 3.9, 4.6, and 5.3 g Thr/kg, with or without B. subtilis strain DSM 32315 (0.0 and 0.5 g/kg). Increasing Thr concentrations improved egg production and ducklings’ hatchling weight (p < 0.05). In addition, L-Thr supplementation resulted in a tendency for decreased feed conversion ratio without affecting egg quality. There was no significant effect (p > 0.05) of the dietary Thr levels on egg yolk and albumen amino acid concentrations. In contrast, the addition of B. subtilis decreased the concentrations of amino acids, excluding proline, in the egg white (albumen) and the egg yolk (p < 0.05). Furthermore, the supplementation of B. subtilis decreased (p < 0 0.001) the hatching weight of ducklings. The addition of B. subtilis without L-Thr decreased (p < 0.05) the hatchability of fertile eggs and the hatching weight of ducklings compared to those of ducks fed dietary L-Thr along with B. subtilis (p < 0.001). The combining L-Thr at 0.7 g/kg with B. subtilis DSM 32315 at 0.5 g/kg could increase eggshell quality, hatchability, and hatching weight. The current study revealed that the combination supplemented of L-Thr and B. subtilis DSM 32315 is recommended due to its positive effects on the eggshell percentage, hatchability and the body weights of newly hatched ducklings when dietary Thr was added at a rate of 0.7 g/kg and B. subtilis DSM 32315 at 0.5 g/kg. In addition, adding L-Thr separately at 0.7 g/kg could improve the egg production of duck breeders. Further studies are required to find the proper dosages of B. subtilis DSM 32315 with co-dietary inclusion of limiting amino acids in the diets of duck breeders. The findings of these trials will support feed additive interventions to transition into antibiotic-free diets