Taiwan-Waseda joint workshop on partial differential equations
and mathematical modelings
Location: Building 55, Room 02-01, Nishi-Waseda Campus, Waseda University
Date: August 30, 2024
Organizers：Hideo Kozono, Kousuke Kuto, Ken-ichi Maruno (Waseda Univ.)
10am-10:50am
Ying-Chieh Lin (National University of Kaohsiung)
Global entropy solution and relaxation limit for Greenberg-Klar-Rascle multi-lane
traffic flow model
11am-11:30am
Jou-Chun Kuo (Waseda University)!! approach to the compressible viscous fluid flows in the half-space
11:30pm-12pm
Tomoharu Kinoshita (Waseda University)
Existence and multiplicity of radially symmetric solutions for nonlinear Schrödingerequations
Lunch time
1:30pm-2:20pm
Shih-Wei Chou (Soochow University)
Global Transonic Solutions of the Hot Jupiter Model for Atmospheric Escape Behavior
2:30pm-3pm
Tatsuki Yamamoto (Waseda University)
Nonhomogeneous boundary value problem for the steady Navier-Stokes equations in
multiply-connected domains
3:20pm-4:10pm
Hsin-Yi Lee (National Chiayi University)!
Global Classical (" ) Solutions for Non-isentropic Gas in the Nozzle Flows
4:20pm-4:50pm
Yoshiki Iida (Waseda University)
On the global well-posedness of the primitive equations on non-flat layers
4:50pm-5:20pm
Taichi Eguchi (Waseda University)
The energy equality of the fractional Navier—Stokes equations in the framework ofthe Lorentz—Besov spaces

Title: Global entropy solution and relaxation limit for Greenberg-Klar-Rascle multilane traffic flow model
Speaker: Ying-Chieh Lin (National University of Kaohsiung)
Abstract: In this talk, we consider the Greenberg-Klar-Rascle multi-lane traffic flow
model. This model is a relaxation system with the equilibrium state that is a
discontinuous function of the car density. We study the existence of global entropy
solutions and the relaxation limit for the GKR model. To construct the approximate
solutions, we find two sequences of invariant regions under some suitable condition ofinitial data. As the relaxation time approaches 0, we prove that the limit of the entropy
solutions for the GKR model is a weak solution of its equilibrium equation. It isinteresting that the equilibrium equation is a scalar conservation law with discontinuousflux.

L1 approach to the compressible viscous fluid flowsin the half-space
Jou-Chun Kuo
Waseda Universitykuojouchun@asagi.waseda.jp

This talk is based on a joint work with Professor Emeritus Yoshihiro Shibata (WasedaUniversity).
This talk is devoted to proving the local well-posedness for the Navier-Stokes equations describing the isotropic motion of the compressible viscous fluid flows in thehalf-space with non-slip boundary conditions. We construct a unique strong solution
s+1s
in the L1 in time and Bq,1
⇥ Bq,1in space maximal regularity class. Here, 1 < q < 1and 1 + N/q  s < 1/q, where N
2 is the space dimension. The approach is bymeans of the analytic semigroup theory and we use Lagrange transformation in order
to eliminate the convection term v · r⇢.

1Existence and multiplicity of radially symmetricsolutions for nonlinear Schrödinger equations
Tomoharu Kinoshita
Waseda University
In this talk, we study the nonlinear Schrödinger equations with Berestycki-Lionstype nonlinearities and show the existence and multiplicity of radially symmetric solutions by variational method. Azzollini-Pomponio(2010) and Liu-Liu-Liao(2019) considered the case that the potential has constant sign. In our case, we deal with thepotential which is allowed to be sign changing.

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Title: Global Transonic Solutions of the Hot Jupiter Model for Atmospheric Escape
Behavior
Speaker: Shih-Wei Chou (Soochow University)
Abstract: In this talk, we present modifications to the original hot-Jupiter model, which
addresses the problem of hydrodynamic escape in the planetary atmosphere. The model
incorporates the Euler equation with gravity, tidal effects, and heat. We employ thegeneralized Glimm technique to prove the existence of transonic solutions to thisproblem.By adjusting the dilation of the characteristic fields, we enhance the accuracyof the Glimm–Goodman wave interaction estimates. This improvement allows us toestablish a more general admissible condition for stabilizing the generalized Glimmscheme. Additionally, we derive the exact relationship between the lower bound of thegas velocity in the subsonic state and the adiabatic constant of the gas. Under certain
constraints on the transonic initial and boundary data, the limit of the approximatedsolutions represents an entropy transonic solution with bounded variations.

Nonhomogeneous boundary value problem for thesteady Navier-Stokes equations in multiply-connecteddomains
Tatsuki YamamotoWaseda University, Japan
We consider the nonhomogeneous boundary value problem for the steady NavierStokes equations under the slip boundary conditions in a two-dimensional boundeddomain with multiple boundary components. By the incompressibility condition of thefluid, the total flux of the given boundary datum through the boundary must be zero.
We prove that this problem has a solution if the friction coeﬃcient is suﬃciently large
compared with the kinematic viscosity constant and the curvature of the boundary. No
additional assumption (other than the necessary requirement of zero total flux throughthe boundary) is imposed on the boundary data. We also show that such an assumption
on the friction coeﬃcient is redundant for the existence of a solution in the case when
the fluxes across each connected component of the boundary are suﬃciently small, or
the domain and the given data satisfy certain symmetry conditions. This talk is basedon the joint work with Prof. Giovanni P. Galdi (University of Pittsburgh).

1Title: Global Classical (C! ) Solutions for Non-isentropic Gas in the Nozzle Flows
Speaker: Hsin-Yi Lee (National Chiayi University)
Abstract: In this talk, we investigate the global existence of classical solutions for nonisentropic gas flows through a duct. This problem can be described as an initial-boundary
value problem for the full compressible Euler equations with the geometric source inLagrangian coordinates, which can be viewed as a hyperbolic system of balance laws when
the Riemann invariants are applied. We prove the global existence theorem for classicalsolutions under appropriate conditions on entropies, ducts, and initial and boundary data.
Our analysis mainly depends on the local existence theorem and uniform a priori estimates,
which can be obtained by using the method of characteristics and introducing generalized
Lax transformations. Furthermore, the long-time behavior of global classical solutions

along all characteristic curves and vertical lines is also determined.
This is joint work with Shih-Wei Chou, John M. Hong, and Shih-Ming Wang.

On the global well-posedness of the primitive
equations on non-flat layers— Taiwan-Waseda joint workshop on partial diﬀerential equations andmathematical modelings —Yoshiki Iida (Waseda university)
July 5, 2024

Abstract
In this talk, we consider the primitive equations on non-flat layers. Thissystem is deduced from the anisotropic Navier–Stokes equations via hydrostatic approximation. Firstly, we discuss how to define the hydrostatic
Helmholtz projection and the hydrostatic Stokes operator without the flatness of the layers, and we then prove the maximal Lq -regularity of the corresponding linearized equation via H ∞ -calculus. By combining the maximal
regularity with suitable nonlinear estimates, it is shown that the system admits a unique strong solution locally in time. Secondly, in order to prolong
the existence time of the solution, we establish H 1 L2 ∩ L2 H 2 -a priori estimates. The key idea is to decompose the horizontal velocity field into thehorizontally divergence-free part and the remaining part. We also mention
long time behavior of the solution.

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The energy equality of the fractional Navier–Stokes
equations in the framework of the Lorentz-Besov
spacesTaichi Eguchi
Waseda University, Japan
hasegawa-t@akane.waseda.jp
We find a new suﬃcient condition to establish the energy equality of the 3D fractional Navier–Stokes equations with the viscous term ν(−∆)α u, 0 ! α, 0 < ν in thewhole space. In the case of the Navier–Stokes equations (α = 1), Cheskidov–Luo (2020)
obtained a suﬃcient condition for the validity of its energy equality in the framework of
the Lorentz–Besov spaces. In this talk, considering a class of the Lorentz–Besov spaces
corresponding to that of Cheskidov–Luo, we obtain a new suﬃcient condition for theenergy equality of the fractional Navier–Stokes equations for 0 ! α. Note that we may
regard our result as a generalization of Cheskidov–Luo.

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