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Daisuke Kawagoe

Publications and source records attributed to Daisuke Kawagoe.

14 recordsLinked to original sources
arXiv2026

$H^s_x$ regularity of solutions to the stationary Boltzmann equation with the incoming boundary condition

We consider the stationary Boltzmann equation with the angular cutoff cross section in a bounded convex domain under the incoming boundary condition. In this article, we discuss the fractional Sobolev regularity of the solution without assuming the positivity of the Gaussian curvature on the boundary. For a boundary data sufficiently smooth and close to the standard Maxwellian, the solution has $H^{1-}_x$ regularity for hard potentials and soft potentials ($-2 \leq γ\leq 1$), while $H^{((4 + γ)/2)-}_x$ regularity is obtained for very soft potentials ($-3 < γ< -2$). We first show the well-posedness of the linearized problem on a weighted $L^2$ space and develop the $L^2-L^\infty$ estimate without the stochastic cycle. We next investigate $H^s_x$ regularity of the solution to the linearized problem. The idea of the celebrated velocity averaging lemma plays a key role in our analysis. We finally derive a bilinear estimate to extend the result on the linearized problem to the weakly nonlinear problem.

math.AP↗
arXiv2025

Remarks on propagation of discontinuities in stationary radiative transfer

We consider the stationary transport equation with the incoming boundary condition. We are interested in discontinuities of the solution. Under the generalized convexity condition, it is known that it has only boundary-induced discontinuities, which are discontinuities arising from discontinuous boundary data, they propagate along positive characteristic lines, and we can reconstruct the attenuation coefficient from boundary measurements by the inverse X-ray transform. In this article, we observe that coefficient-induced discontinuities, discontinuities of the solution arising from discontinuous coefficients, would also appear without the generalized convexity condition. If the set of discontinuous points of the coefficients contains at most finite number of flat parts, coefficient-induced discontinuities do not affect the inverse X-ray transform. We also remark that, under the generalized convexity condition, a three dimensional inverse problem can be reduced to the two dimensional one. A numerical experiment is exhibited.

math.AP↗
arXiv2025

On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation

We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $εΔ$, where $ε$ is a positive parameter and $Δ$ is the Laplacian. In this article, we show the $L^2$ strong convergence of solutions as the parameter $ε$ tends to $0$, and discuss its convergence rates assuming $H^1$ or $H^2$ regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates.

math.AP↗
arXiv2025

On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations \cite{Evans}. In this article, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski \cite{CP} to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished.

math.AP↗
arXiv2024

A revisit on well-posedness of a boundary value problem of a stationary advection equation without the separation condition

We consider a boundary value problem of a stationary advection equation in a bounded domain with Lipschitz boundary. It is known to be well-posed in $L^p$-based function spaces for $1 < p < \infty$ under the separation condition of the inflow and the outflow boundaries. In this article, we provide another sufficient condition for the well-posedness with $1 \leq p \leq \infty$.

math.AP↗
arXiv2023

Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation

We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$.

math.AP↗
arXiv2023

Nonexistence of multi-dimensional solitary waves for the Euler-Poisson system

We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons.

math.AP↗
arXiv2018

Spectral structure of the Neumann--Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann--Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann--Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.

math.SP↗
arXiv2018

Surface Riesz transforms and spectral property of elastic Neumann--Poincaé operators on less smooth domains in three dimensions

It is known that the Neumann--Poincaré operator for the Lamé system of linear elasticity is polynomially compact and, as a consequence, that its spectrum consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters, if the boundary of the domain where the operator is defined is $C^\infty$-smooth. We extend this result to less smooth boundaries, namely, $C^{1, α}$-smooth boundaries for some $α> 0$. The results are obtained by proving certain identities for surface Riesz transforms, which are singular integral operators of nonconvolution type, defined by the matrix tensor on a given surface.

math.FA↗
arXiv2018

Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography

We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct attenuation coefficient from jumps in boundary measurements.

math.AP↗
arXiv2018

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.

math.AP↗
arXiv2017

Propagation of boundary-induced discontinuity in stationary radiative transfer

We consider the boundary value problem of the stationary transport equation in the slab domain of general dimensions. In this paper, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the stationary transport equation. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along positive characteristic lines as that of the solution to the stationary transport equation. Our analysis does not depend on the celebrated velocity averaging lemma, which is different from previous works. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.

math-ph↗