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In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ(ρ) = ρ θ, λ(ρ) = (θ − 1)ρ θ in RN ${\mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ., vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ=23 $\theta =\frac{2}{3}$ , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ=53 $\gamma =\frac{5}{3}$ , respectively.

Most college students have grown up using technology and consequently, they are proficient with its many uses and applications. The use of this technology provides many benefits to college students' learning, both in and out of the classroom. However, despite the numerous benefits of technology, these digital activities can also lead to much digital distraction. Digital distractions may include Internet surfing, watching movies, checking texts, reading, and sending emails, and perusing social media. These digital distractions often result in limited attention and engagement during class, as well as less learning and academic achievement overall. Digital distraction can result from such factors as anxiety and depression, motivational variables, the need to keep up and the fear of missing out, emotional numbing and procrastination as well as an overreliance on multitasking. It is suggested here that the use of self-regulated learning (SRL) strategies may be effective in assisting college students in avoiding digital distractions, both in and out of the classroom. SRL strategies include forethought, planning, activation; monitoring, control and reflection on the learner's cognition, motivation/affect, and behavior. Through the use of these SRL strategies, college students may be taught to decrease digital distractions and, thus, experience higher levels of learning and academic performance.

International students who pursue their academic goals in United States are prone to difficulties when attempting to build social resources and adjust to the new culture. Social media is a practical means of connection due to its ease of use and accessibility. Previous research has indicated contradictory effects of social media use on academic engagement. In addition to the direct effect, this research examined social media use influences on international students' learning engagement by mediating social capital and cultural adjustment. A total of 209 international students completed a web-based survey distributed via e-mail and social media between November 2021 and May 2022. Data were analyzed using Structural Equation Model. Results showed that only purposely using social media to collaborate with learning counterparts or materials directly improves international students' learning engagement. Other uses of social media (e.g., expanding "new" resources, solidifying "close" relationships) have no significant direct effects. Nonetheless, they are essential to improving levels of learning engagement via the mediation of "bridging" capital (social resources attributed to expanding relationships) and students' cultural adjustment in the U.S. International students' "bonding" capital (social resources available through trustworthy relationships) and "home" cultural retention showed little direct or indirect effects on learning engagement. This study recognizes the importance of social resources and cultural adjustment for international students. Also, this study provides valuable information to educators and administrators, as there is a need to identify the underlying mechanisms to contribute feasible learning intervention approaches and alleviate negative effects for international students.

Purpose: The purpose of this study was to examine the relationship between factors in the extended technology acceptance model (TAM) model and teachers' self-efficacy in remote teaching during the COVID-19 pandemic. In addition, the authors sought to listen to classroom teachers as they expressed their unbiased views of the advantages, disadvantages and challenges of teaching remotely during the COVID-19 pandemic. Design/methodology/approach: A survey was employed to examine the relationship between factors in the extended TAM model and teachers' self-efficacy in remote teaching during the COVID-19 pandemic using the 49-item questionnaire. A multiple regression analysis using a stepwise procedure was used to examine the relationship between factors in the extended TAM model and teachers' self-efficacy. Three open-ended questions closely examined remote teaching during the pandemic, related to challenges, advantages and disadvantages. Findings: Qualitative findings challenges included Internet connection, lack of interaction and communication and challenges with motivation and student engagement. Disadvantages included teachers' level of self-efficacy in using technology to teach, lack of support and resources to teach online and the struggle to motivate and engage students. Perceived benefits included flexibility for the teacher and differentiation, rich resources and a way to support learners when in-person instruction is not possible. Research limitations/implications: The data suggest that instead, during COVID-19, many teachers were learning about the platforms simultaneously as they were instructing students. Practical implications: To ensure quality remote instruction and that students receive the support to make instruction equitable, teachers need to perceive that their instructional technology needs are met to focus on teaching, learning and needs of their students. Social implications: Teachers need opportunities to explore the platforms and to experience success in this environment before they are exposed to the high stakes of preparing students to meet K-12 standards. Originality/value: Instructional delivery has not explored teacher motivational and instructional teaching self-efficacy related to satisfaction with the learning management system (LMS).

In this paper, we present some analytical solutions to the one-dimensional compressible isothermal Navier–Stokes equations with Maxwell's law in the real line. First, we construct two analytical solutions by using a self-similar ansatz, one blows up in finite time and the other exists globally-in-time. Second, we construct two global analytical solutions with different large initial data by using a non-self-similar ansatz. [ABSTRACT FROM AUTHOR]