Asymptotic behavior of spherically symmetric solutions to the compressible Navier Stokes equation with external forces

We study the large time behavior of an isentropic and spherically symmetric motion of compressible viscous gas in a eld of external force over an unbounded exterior domain in R^n (n \ge 2). The typical example of this problem appears in analysis of the behavior of atmosphere around the earth. First, we show that there exists a stationary solution satisfying an adhesion boundary condition and a positive spatial asymptotic condition. Then, it is shown that this stationary solution is a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data can be chosen arbitrarily large if it belongs to the suitable Sobolev space. Moreover, if the external force is attractive, it also can be arbitrarily large. This condition includes the most typical external force, i.e., the gravitational force. In the proof of the stability theorem, it is the essential step to obtain the uniform positive lower bound for the density. It is derived through the energy method with the aid of a representation formula for the density

Large-time behaviour of the spherically symmetric solution to an outflow problem for isentropic model of compressible viscous fluid

We study the large time behaviour of a spherically symmetric motion of out-flowing isentropic and compressible viscous gas. The fluid occupies an unbounded exterior domain in $\mathbb{R}^n \; (n \ge 2)$, and it flows out from an inner sphere centred at the origin of radius $r=1$. The unique existence of a stationary solution satisfying the outflow boundary condition has been obtained by I. Hashimoto and A. Matsumura in 2021. The main aim of present paper is to show that this stationary solution becomes a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data is chosen arbitrarily large if it belongs to the suitable weighted Sobolev space. The main strategy is to approximate the unbounded exterior problem by solving a sequence of outflow-inflow initial boundary value problems posed in finite annular domain. Then the solution is obtained as a limit of these approximate solutions. The key argument for the stability theorem is based on the derivation of a-priori estimates in the weighted Sobolev space, executed under the Lagrangian coordinate. The essential step of the proof is to obtain the point-wise upper and lower bound for the density. It is derived through employing a representation formula of the density with the aid of the weighted energy method.Comment: 42 page

Convergence rate of solutions toward stationary solutions to the compressible Navier–Stokes equation in a half line

AbstractWe study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method

Issues in teaching to the writing test : preparing students for the TOEFL(R); iBT independent writing task

Students in the Department of International Studies at Doshisha Women’s College are required to study overseas for one year at a college or university in an English-speaking country. For this reason, their first year is devoted to test preparation. Teaching such test preparation courses forces the instructors to face several issues related to "teaching to the test" such as effectiveness and the relationship to usual language teaching. These issues have been discussed in the literature to some extent but only a few studies address them in a specific context. This paper examines a course designed to prepare students for the independent writing task of the TOEFL iBT; it presents a detailed description of how the course is taught based on the reflections of four instructors, focusing on three issues related to teaching to the test: teaching writing as opposed to teaching to the prompt; the effect of the test rubric and the holistic scoring policy on teaching; and the use of textbooks. We conclude that teaching to the test is compatible with sound instructional practice.論文 (Article

Shock Profiles for Non Equilibrium Radiating Gases

We study a model of radiating gases that describes the interaction of an inviscid gas with photons. We show the existence of smooth traveling waves called 'shock profiles', when the strength of the shock is small. Moreover, we prove that the regularity of the traveling wave increases when the strength of the shock tends to zero

Discovery of gene expression-based pharmacodynamic biomarker for a p53 context-specific anti-tumor drug Wee1 inhibitor

<p>Abstract</p> <p>Background</p> <p>Wee1 is a tyrosine kinase regulating S-G2 cell cycle transition through the inactivating phosphorylation of CDC2. The inhibition of Wee1 kinase by a selective small molecule inhibitor significantly enhances the anti-tumor efficacy of DNA damaging agents, specifically in p53 negative tumors by abrogating S-G2 checkpoints, while normal cells with wild-type p53 are not severely damaged due to the intact function of the G1 checkpoint mediated by p53. Since the measurement of mRNA expression requires a very small amount of biopsy tissue and is highly quantitative, the development of a pharmacodynamic (PD) biomarker leveraging mRNA expression is eagerly anticipated in order to estimate target engagement of anti-cancer agents.</p> <p>Results</p> <p>In order to find the Wee1 inhibition signature, mRNA expression profiling was first performed in both p53 positive and negative cancer cell lines treated with gemcitabine and a Wee1 inhibitor, MK-1775. We next carried out mRNA expression profiling of skin samples derived from xenograft models treated with the Wee1 inhibitor to identify a Wee1 inhibitor-regulatory gene set. Then, the genes that were commonly modulated in both cancer cell lines and rat skin samples were extracted as a Wee1 inhibition signature that could potentially be used as a PD biomarker independent of p53 status. The expression of the Wee1 inhibition signature was found to be regulated in a dose-dependent manner by the Wee1 inhibitor, and was significantly correlated with the inhibition level of a direct substrate, phosphorylated-CDC2. Individual genes in this Wee1 inhibition signature are known to regulate S-G2 cell cycle progression or checkpoints, which is consistent with the mode-of-action of the Wee1 inhibitor.</p> <p>Conclusion</p> <p>We report here the identification of an mRNA gene signature that was specifically changed by gemcitabine and Wee1 inhibitor combination treatment by molecular profiling. Given the common regulation of expression in both xenograft tumors and animal skin samples, the data suggest that the Wee1 inhibition gene signature might be utilized as a quantitative PD biomarker in both tumors and surrogate tissues, such as skin and hair follicles, in human clinical trials.</p