Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition (original) (raw)
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On the Stokes equation with Neumann boundary condition
Regularity and Other Aspects of the Navier-Stokes Equation, 2005
In this paper, we study the nonstationary Stokes equation with Neumann boundary condition in a bounded or an exterior domain in R n , which is the linearized model problem of the free boundary value problem. Mainly, we prove L p-L q estimates for the semigroup of the Stokes operator. Comparing with the non-slip boundary condition case, we have the better decay estimate for the gradient of the semigroup in the exterior domain case because of the null force at the boundary.
Mathematics, 2022
In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in Lp in time and Lq in space framework in a uniformly H∞2 domain Ω⊂RN for N≥4. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the Lq-Lr estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction N≥4 is required to deduce an estimate for the nonlinear term G(u) arising from divv=0. However, we establish the results in the half space R+N for N≥3 by reducing the linearized problem to the problem with G=0, where G is the right member corresponding to G(u).
Journal of Elliptic and Parabolic Equations, 2021
We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on \mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)Lσ2(Rd).Suchestimatesarewell−knownforellipticequationsintheformofpointwiseheatkernelboundsandforellipticsystemsintheformofintegratedoff−diagonalestimates.Onourwaytounveilthisoff−diagonalbehaviorweproveresolventestimatesinMorreyspacesL σ 2 ( R d ) . Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spacesLσ2(Rd).Suchestimatesarewell−knownforellipticequationsintheformofpointwiseheatkernelboundsandforellipticsystemsintheformofintegratedoff−diagonalestimates.Onourwaytounveilthisoff−diagonalbehaviorweproveresolventestimatesinMorreyspaces\mathrm {L}^{2 , \nu } ({\mathbb {R}}^d)L2,ν(Rd)withL 2 , ν ( R d ) withL2,ν(Rd)with0 \le \nu < 2$$ 0 ≤ ν < 2 .
Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms
2017
In this paper, we derive sharp bounds on the semigroup of the linearized Navier-Stokes equations near a stationary boundary layer on the half space. The bounds are obtained uniformly in the inviscid limit. Delicate boundary layer norms are introduced in order to capture the true boundary layer behavior of vorticity near the boundary. As an immediate application, we construct an approximate solution which exhibits an L^∞ instability of Prandtl's layers.
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids
In this chapter, we provide a review of results on the global well-posedness and optimal decay rate of strong solutions to the compressible Navier-Stokes equations in several type of domains: (1) whole space (Theorems 6, 7, 8, 9, 10, 11, and 12), (2) exterior domains (Theorems 13 and 14), (3) half-space (Theorem 15), (4) bounded domains (Theorem 16), and (5) infinite layers. Global well-posedness for the compressible viscous barotropic fluid motion with nonslip boundary condition was for the first time proved in the early 1980s by Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the assumption that the H 3 norm of the initial data is small. In Theorems 1, 2, 3, and 4, we revisit the same problem as in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the weaker assumptions, namely, that the H 2 norm of initial data is small. This is an improvement of the result in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) in view of the regularity assumption of the initial data. To show the methods, we perform the proof of Theorems 1, 2, 3, and 4 in all essential details. In this process, the L p-L q decay properties of solutions to the linearized equations are proved by using the cutoff technique and combining the local energy decay and the result in the whole space. This result was first proved by Kobayashi and Shibata (Commun Math Phys 200:621-659, 1999) under some additional assumption, and in this chapter, this assumption is eliminated by using a bootstrap argument. In the final section of this chapter, the optimal decay rate of the H 2 norm of solution of the nonlinear problem is proved by combining the L p-L q decay properties of the linearized equations with some energy inequality of exponential decay type under the assumption that the initial data belong to the intersection space of H 2 and L 1. The main idea of this part of the proof is to combine the L p-L q decay properties of the Stokes semigroup and some Lyapunov-type energy inequality.
On the decay of solutions to the 2D Neumann exterior problem for the wave equation
Journal of Differential Equations, 2003
We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution. r
In this paper, we report the recent development of the study of the Stokes equation with Neumann boundary condition which is obtained as a hnearized equation of the free boundary value problem for the Navier-Stokes equation. Especially, we are concerned with the resolvent problem of the reduced Stokes equation with Neumann boundary condition, the generation of the Stokes semigroup which is analytic on the solenoidal space and the Lp−LqL_{p^{-}}L_{q}Lp−Lq estimate of the Stokes semigroup both in a bounded domain and in an exterior domain. Especially, comparing with the nonshp boundary condition case, we have the better decay