arXiv subjects

Kazuyuki Tsuda

Publications and source records attributed to Kazuyuki Tsuda.

8 recordsLinked to original sources

arXiv2026

Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary

In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partialΩ(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).

math.AP↗

arXiv2026

The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to $\mathscr H^\infty$-calculus to Fractional Domains

We consider the Stokes operator $A$ on smooth exterior domains $Ω$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{κ,q}_w(Ω)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^θ$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^θu\|_{L^q_w}$) as well as homogeneous norm ($\|A^θu\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-Δ_{q,w})^θ) = [L^q_{w},\widehat{\mathcal D}(-Δ_{q,w})]_θ$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{σ,w}(Ω)$.

math.AP↗

arXiv2026

On dynamic stability of energetically stable equilibria of the Navier-Stokes-Korteweg flows

We consider the Navier-Stokes-Korteweg equations in a bounded domain or a periodic cell. The pressure considered in this paper may not be monotone with respect to the density so that there exist non-constant equilibria allowing two-phases. Using a simple Hilbert space framework, we prove that if an isolated equilibrium is energetically stable and non- degenerate, it is exponentially stable under the isothermal Navier-Stokes-Korteweg flows when the space dimension is less than or equal to three. For non-isolated case, we prove that a global-in-time solution near an energetically stable equilibrium converges to possibly another equilibrium exponentially fast. No smallness assumptions on equilibria are imposed. For the proof we apply a (generalized) stability principle due to J. Prüss, M. Wilke and G. Simonett (2013).

math.AP↗

arXiv2026

The time periodic problem for the Navier-Stokes equations in exterior domains in weighted spaces

The paper considers the time periodic problem of the Navier-Stokes system in an exterior domain under time periodic external forces. Existence of periodic mild solutions is obtained in the critical scale invariant space $C(\mathbb{R};L^n)$ $(n \geq 4)$ if the external force is small without exploiting any divergence form as in the study of Okabe and Tsutsui (2017) for the whole space case in Lorentz spaces. Previous studies mainly rely on either potential theoretical estimates or time-space integral estimates in Lorentz spaces introduced by Yamazaki (Math. Ann.(2000)). To the best of our knowledge, there are no results using Muckenhoupt weights in $L^q$ class for $1< q <\infty$ to construct time periodic solutions of the Navier-Stokes equations in the exterior domain case. In this article, a new method based on radially symmetric Muckenhoupt weights in space is used. To apply these weights, we reconsider weighted $L^p$-$L^q$ decay estimates for the Stokes semigroup. This important result was announced by Kobayashi and Kubo (2012-2015) about ten years ago with a sketch of the proof by Kubo. In this paper, we give a rigorous proof of the result and, as an important application, solve the time periodic problem for the Navier-Stokes equations on an exterior domain.

math.AP↗

arXiv2026

Perturbation theory of the compressible Navier-Stokes equations and its application

In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is considered for a perturbation system from a stationary solution $u_ω$ belonging to the weak $L^n$ space. Decay rates of the perturbation including $L^\infty$ norm are obtained which coincide with those of the heat kernel except a bit loss. The proof is based on deriving suitable resolvent estimates with perturbation terms in the low frequency part having a parabolic spectral curve. Our method can be applicable to dispersive hyperbolic systems like wave equations with strong damping. Indeed, a parabolic type decay rate of a solution is obtained for a damped wave equation including variable coefficients which satisfy spatial decay conditions.

math.AP↗

arXiv2025

Time periodic problem of compressible Euler equations with damping on the whole space

In this article, time periodic problem of the compressible Euler equations with damping on the whole space is studied. It is well known that in the Euler system, long-time behavior of solutions is a more delicate problem due to lack of the viscosity. By virtue of a damping effect, time global solutions barely exist. Under such circumstances, existence of a time periodic solution is obtained for sufficiently small time periodic external force when the space dimension is greater than or equal to $3$. In addition, its stability is also obtained. The solution is asymptotically stable under sufficiently small initial perturbations and the $L^\infty$ norm of the perturbation decays as time goes to infinity. The potential theoretical estimates work well on a low frequency part of solutions, while a new energy estimate with weights is established to avoid derivative loss.

math.AP↗

arXiv2019

Asymptotic profile for diffusion wave terms of the compressible Navier-Stokes-Korteweg system

Asymptotic profile for diffusion wave terms of solutions to the compressible Navier-Stokes-Korteweg system is studied on $R^2$. The diffusion wave with time decay estimate is studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002) and Kobayashi and Tsuda (2018) for the compressible Navier-Stokes system and the compressible Navier-Stokes-Korteweg system. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space-time $L^2$ of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by $L^2$ on space, a decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz's energy estimate, and the Fefferman-Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

math.AP↗

arXiv2019

Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition

Global existence of solutions to the compressible Navier-Stokes-Korteweg system around a constant state is studied. This system describes liquid-vapor two phase flow with phase transition as diffuse interface model. In previous works they assume that the pressure is a monotone function for change of density similarly to the usual compressible Navier-Stokes system. On the other hand, due to phase transition the pressure is accurately non-monotone function and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. It is shown that in the critical case for small data whose momentum has derivative form there exist global $L^2$ solutions and the parabolic type decay rate of the solutions is obtained. The proof is based on decomposition method for solutions to a low frequency part and a high frequency part.

math.AP↗