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Yoshihiro Shibata - One of the best experts on this subject based on the ideXlab platform.
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on the local wellposedness of free boundary problem for the navier stokes equations in an Exterior Domain
Communications on Pure and Applied Analysis, 2018Co-Authors: Yoshihiro ShibataAbstract:This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the fluid initially occupies an Exterior Domain \begin{document} $Ω$ \end{document} in \begin{document} $N$ \end{document} -dimensional Euclidian space \begin{document} $\mathbb{R}^N$ \end{document} .
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About Compressible Viscous Fluid Flow in a 2-dimensional Exterior Domain
Spectral Theory Mathematical System Theory Evolution Equations Differential and Difference Equations, 2012Co-Authors: Yuko Enomoto, Yoshihiro ShibataAbstract:We report our results [5, 6, 7] concerning a global in time unique existence theorem of strong solutions to the equation describing the motion of compressible viscous fluid flow in a 2-dimensional Exterior Domain for small initial data and some decay properties of the analytic semigroup associated with Stokes operator of compressible viscous fluid flow in a 2-dimensional Exterior Domain.Our results are an extension of the works due to Matsumura and Nishida [13] and Kobayashi and Shibata [10] in a 3-dimensional Exterior Domain to the 2-dimensional case.W e also discuss some analytic semigroup approach to the compressible viscous fluid flow in a bounded Domain, which was first investigated by G.S tromer [20, 21, 22].
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Local energy decay of solutions to the oseen equation in the Exterior Domains
Indiana University Mathematics Journal, 2004Co-Authors: Yuko Enomoto, Yoshihiro ShibataAbstract:In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional Exterior Domain (n ≥ 3). In the three dimensional Exterior Domain case, Kobayashi and Shibata [16] already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shibata to the n-dimensional case but also the complete study of the local energy decay theorem for the Oseen equation with optimal time decay rates. The local energy decay gives us a crucial step to obtain the Lp-Lq estimates of the Oseen semigroup, which enable us to prove the unique existence of globally in time solutions to the Navier-Stokes equation in an Exterior Domain with small initial data in the L n framework, and their properties of time decay.
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decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an Exterior Domain in ℝ3
Communications in Mathematical Physics, 1999Co-Authors: T Kobayashi, Yoshihiro ShibataAbstract:We consider the equations of motion of compressible viscous and heat-conductive gases in an Exterior Domain in ℝ3. We give the L_q−L_p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.
Yong Zhou - One of the best experts on this subject based on the ideXlab platform.
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asymptotic behavior of d solutions to the steady navier stokes flow in an Exterior Domain of a half space
Zeitschrift für Angewandte Mathematik und Physik, 2019Co-Authors: Peter Wittwer, Yong ZhouAbstract:We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded Domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an Exterior Domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.
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on fujita critical exponent for a nonlinear ultraparabolic equation in an Exterior Domain
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Mohamed Jleli, Bessem Samet, Yong ZhouAbstract:Abstract In this paper, we consider the nonlinear ultraparabolic equation { ∂ t u + ∂ s u − Δ u = | u | p , t > 0 , s > 0 , x ∈ D c , u ( t , s , x ) = f ( x ) , t > 0 , s > 0 , x ∈ ∂ D , u ( t , 0 , x ) = u 1 ( t , x ) , t > 0 , x ∈ D c , u ( 0 , s , x ) = u 2 ( s , x ) , s > 0 , x ∈ D c , where D = B ( 0 , 1 ) ‾ is the closed unit ball in R N , N ≥ 2 , D c is its complement, p > 1 , u i ≥ 0 , i = 1 , 2 , and ∫ ∂ D f ( x ) d S x > 0 . We derive the critical exponent for the considered problem in the sense of Fujita. We discuss separately the cases N = 2 and N ≥ 3 . To the best of our knowledge, this is the first work dealing with the blow-up of solutions to multi-time equations in an Exterior Domain.
Motohiro Sobajima - One of the best experts on this subject based on the ideXlab platform.
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finite time blowup of solutions to semilinear wave equation in an Exterior Domain
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Motohiro Sobajima, Kyouhei WakasaAbstract:Abstract We consider the initial-boundary value problem of semilinear wave equation with nonlinearity | u | p in Exterior Domain in R N ( N ≥ 3 ) . Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in R N . At least in the case N = 4 , their estimates are sharp in view of the work by Zha–Zhou [21] . The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa–Yordanov [15] .
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remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional Exterior Domain
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:Abstract In this paper we consider the following initial-boundary value problem with the power type nonlinearity | u | p with 1 p ≤ 2 in a two-dimensional Exterior Domain (0.1) { τ ∂ t 2 u ( x , t ) − Δ u ( x , t ) + e i ζ ∂ t u ( x , t ) = λ | u ( x , t ) | p , ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , u ( x , 0 ) = e f ( x ) , x ∈ Ω , ∂ t u ( x , 0 ) = e g ( x ) , x ∈ Ω , where Ω is given by Ω = { x ∈ R 2 ; | x | > 1 } , ζ ∈ [ − π 2 , π 2 ] , λ ∈ C and τ ∈ { 0 , 1 } switches the parabolicity, dispersivity and hyperbolicity. Remark that 2 = 1 + 2 / N is well-known as the Fujita exponent. If p > 2 , then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata [11] ), and if p 2 , then global-in-time solutions cannot exist for any positive initial data (see Ogawa–Takeda [22] and Lai–Yin [14] ). The result is that for given initial data ( f , τ g ) ∈ H 0 1 ( Ω ) × L 2 ( Ω ) satisfying ( f + τ g ) log | x | ∈ L 1 ( Ω ) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p = 2 : LifeSpan ( u ) ≤ exp [ exp ( C e − 1 ) ] . The crucial idea is to use test functions which approximates the harmonic function log | x | satisfying Dirichlet boundary condition and the technique modified from [9] .
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finite time blowup of solutions to semilinear wave equation in an Exterior Domain
arXiv: Analysis of PDEs, 2018Co-Authors: Motohiro Sobajima, Kyouhei WakasaAbstract:We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in Exterior Domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in $\mathbf{R}^N$. At least in the case $N=4$, their estimates are sharp in view of the work by Zha--Zhou (2015). The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa--Yordanov.
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remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional Exterior Domain
arXiv: Analysis of PDEs, 2017Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional Exterior Domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial data $(f,tau g)in H^1_0(Omega)times L^2(Omega)$ satisfying $(f+tau g)log |x|in L^1(Omega)$ with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to the problem is given as the following {it double exponential type} when $p=2$: [ lifespan(u) leq exp[exp(Cep^{-1})] . ] The crucial idea is to use test functions which approximates the harmonic function $log |x|$ satisfying Dirichlet boundary condition and the technique for derivation of lifespan estimate in Ikeda-Sobajima(arXiv:1710.06780).
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diffusion phenomena for the wave equation with space dependent damping in an Exterior Domain
Journal of Differential Equations, 2016Co-Authors: Motohiro Sobajima, Yuta WakasugiAbstract:Abstract In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an Exterior Domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.
Ryo Ikehata - One of the best experts on this subject based on the ideXlab platform.
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global existence of weak solutions for two dimensional semilinear wave equations with strong damping in an Exterior Domain
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Ryo Ikehata, Yuki InoueAbstract:Abstract We consider two-dimensional mixed problems in an Exterior Domain for a semilinear strongly damped wave equation with a power-type nonlinearity | u | p . If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p of the nonlinear term satisfies p > 6 .
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decay of solutions for a semilinear system of elastic waves in an Exterior Domain with damping near infinity
Nonlinear Analysis-theory Methods & Applications, 2007Co-Authors: Ruy Coimbra Charao, Ryo IkehataAbstract:Abstract We show that the solutions of a semilinear system of elastic waves in an Exterior Domain with a localized damping near infinity decay in an algebraic rate to zero. We impose an additional condition on the Lame coefficients. It seems that this restriction cannot be overcome by using the two-finite-speed propagation of the elastic model, since we do not assume compact support on the initial data and because the dissipation does not have compact support. The decay rates obtained for the total energy of the linear problem and the L 2 -norm of the solution improve previous results. For the semilinear problem the decay rates in this paper seem to be the first contribution, mainly in the context of initial data without compact support and localized dissipation.
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global existence of solutions for 2 d semilinear wave equations with dissipation localized near infinity in an Exterior Domain
Mathematical Methods in The Applied Sciences, 2006Co-Authors: Ryo IkehataAbstract:We shall derive some global existence results to semilinear wave equations with a damping coefficient localized near infinity for very special initial data in H×L2. This problem is dealt with in the two-dimensional Exterior Domain with a star-shaped complement. In our result, a power p on the non-linear term |u|p is strictly larger than the two-dimensional Fujita-exponent. Copyright © 2005 John Wiley & Sons, Ltd.
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global existence of solutions for semilinear damped wave equation in 2 d Exterior Domain
Journal of Differential Equations, 2004Co-Authors: Ryo IkehataAbstract:Abstract We consider a mixed problem of a damped wave equation utt−Δu+ut=|u|p in the two dimensional Exterior Domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|p satisfies p ∗ =2 . For this purpose we shall deal with a radially symmetric solution in the Exterior Domain. A new device developed in Ikehata–Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.
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Global existence of solutions for semilinear damped wave equation in 2-D Exterior Domain
Journal of Differential Equations, 2004Co-Authors: Ryo IkehataAbstract:We consider a mixed problem of a damped wave equation utt−Δu+ut=|u|p in the two dimensional Exterior Domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|p satisfies p∗=2
T Kobayashi - One of the best experts on this subject based on the ideXlab platform.
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some estimates of solutions for the equations of motion of compressible viscous fluid in the three dimensional Exterior Domain
Journal of Differential Equations, 2002Co-Authors: T KobayashiAbstract:Abstract We consider the equations of motion of compressible viscous fluid in an Exterior Domain in R 3 . We give the L q − L p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem. In particular, we shall give L 1 estimates, which implies the diffusion wave of compressible Navier–Stokes flow.
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decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an Exterior Domain in ℝ3
Communications in Mathematical Physics, 1999Co-Authors: T Kobayashi, Yoshihiro ShibataAbstract:We consider the equations of motion of compressible viscous and heat-conductive gases in an Exterior Domain in ℝ3. We give the L_q−L_p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.
