arXiv subjects
Publications and source records attributed to Naoki Tsuge.
In this paper, we study quasi-linear hyperbolic systems. Our goal in this paper is to provide a new proof of local existence of a classical solution for the system. Most difficult point is to prove the convergence of the derivative of approximate solutions by the Arzela-Ascoli theorem.
Our goal in this paper is to prove the global existence of a classical solution for the isentropic nozzle flow. Regarding this problem, there exist some global existence theorems of weak solutions. However, that of classical solutions does not have much attention until now. When we consider the present problem, the main difficulty is to obtain the uniform bound of solutions and their derivatives. To solve this, we introduce an invariant region depending on the space variable and a functional satisfying the Riccati equation along the characteristic lines.
In this paper, we are concerned with the one-dimensional initial boundary value problem for isentropic gas dynamics. Through the contribution of great researchers such as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay theory of solutions was established. They treated with the Cauchy problem and the corresponding initial data have the small total variation. On the other hand, the decay for initial data with large oscillation has been open for half a century. In addition, due to the reflection of shock waves at the boundaries, little is known for the decay of the boundary value problem on a bounded interval. Our goal is to prove the existence of a global attractor, which yields a decay of solutions for large data. To construct approximate solutions, we introduce a modified Godunov scheme.
In the field of differential equations, particularly fluid dynamics, many researchers have shown an interest in the behavior of time periodic solutions. In this paper, we study isentropic gas flow in a bounded interval and apply a time periodic outer force. This motion is described by the compressible Euler equation with the outer force. Our purpose in this paper is to prove the existence of a time periodic solutions. Unfortunately, little is known for the system of conservation laws until now. The problem seems to lie in fact that the equation does not possesses appropriate decay estimates.
In this paper, we are concerned with the Cauchy problem for isentropic gas dynamics. Through the contribution of many researchers such as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay of solutions was established. They treated with initial data with the small total variation. On the other hand, the decay for large initial data has been open for half a century. Our goal is to provide a new method to analyze this problem. We prove the existence of a global attractor, which yields a decay of solutions for large data. To construct approximate solutions, we introduce a modified Godunov scheme.
We are concerned with isentropic gas flow in the Laval nozzle with a friction due to viscosity. It is well known that the flow attains the sonic state at the throat, where the cross section is minimum in the Laval nozzle. However, the present friction changes the position of the sonic state into downstream, which is called chooking. Our goal in this paper is to investigate this phenomena mathematically. From the mathematical point of view, the friction is different from normal frictions and difficult to treat with. In spite of its physical importance, the friction has not received much attention until now. For the case without friction, the global existence of a solution was obtained by author. However, for the case with the friction, there are only restrictive results. The most difficult point is to obtain the bounded estimate of solutions. To solve this problem, we introduce an invariant region depending on the mass. Adjusting the invariant region, we invent a new difference scheme, which yields approximate solutions including the mass.
We study a model of vendors competing to sell a homogeneous product to customers spread evenly along a circular city. This model is based on Hotelling's celebrated paper in 1929. Our aim in this paper is to present a necessary and sufficient condition for the equilibrium. This yields a representation for the equilibrium. To achieve this, we first formulate the model mathematically. Next, we prove that the condition holds if and only if vendors are equilibrium.
We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.
We study the compressible Euler equations in the isentropic nozzle flow. The global existence of an $L^{\infty}$ solution has been proved in (Tsuge in Nonlinear Anal. Real World Appl. 209: 217-238 (2017)) for large data and general nozzle. However, unfortunately, this solution does not possess finiteness of energy. Although the modified Godunov scheme is introduced in this paper, we cannot deduce the energy inequality for the approximate solutions. Therefore, our aim in the present paper is to derive the energy inequality for an $L^{\infty}$ solution. To do this, we introduce the modified Lax Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. We shall first deduce from the formula the energy inequality. Next, applying the compensated compactness method, the approximate solution converges to a weak solution. The energy inequality also holds for the solution as the limit. As a result, since our solutions are $L^{\infty}$, they possess finite energy and propagation, which are essential to physics.
We study a model of vendors competing to sell a homogeneous product to customers spread evenly along a linear city. This model is based on Hotelling's celebrated paper in 1929. Our aim in this paper is to present a necessary and sufficient condition for the equilibrium. This yields a representation for the equilibrium. To achieve this, we first formulate the model mathematically. Next, we prove that the condition holds if and only if vendors are equilibrium.
We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These phenomena are governed by the compressible Euler equation, which is one of crucial equations in inhomogeneous conservation laws. In this paper, we consider its unsteady flow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. The theorem has been proved in (Tsuge in Arch. Ration. Mech. Anal. 209:365-400 (2013)). However, this result is limited to small data. Our aim in the present paper is to remove this restriction, that is, we consider large data. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to lie in a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties.