Operator-valued Fourier Multiplier Theorems Research Articles

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ( $P_4$ ): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$ , where $A$ , $B$ , $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$ , $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$ ) into $X$ , $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$ . We completely characterize the well-posedness of ( $P_4$ ) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

Well-posedness of fractional differential equations with finite delay in complex Banach spaces

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems.

Half-Space Model Problem for Navier-Lamé Equations with Surface Tension

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the <img src=image/13426244_01.gif>-boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of <img src=image/13426244_02.gif>-dimensional Euclidean space (<img src=image/13426244_02.gif>≥ 2) as one way to study the well-posedess. We investigate the <img src=image/13426244_01.gif>-boundedness in half-space domain case. The <img src=image/13426244_01.gif>-boundedness implies not only the generation of analytic semigroup but also the maximal <img src=image/13426244_03.gif> regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal <img src=image/13426244_03.gif> regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

Maximal Lp-Lq regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation

This paper is concerned with the generalized resolvent estimate and the maximal Lp-Lq regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of R-bounded solution operators for the model problems and the maximal Lp-Lq regularity for the system. The key step is to prove the maximal Lp-Lq regularity theorem for the linearized heat equation with the help of the R-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2&lt;p&lt;∞ and N&lt;q&lt;∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

Solutions of second-order degenerate equations with infinite delay in Banach spaces

We consider the well-posedness of the second-order degenerate differential equations with infinite delay (P2): (Mu′)′(t)+(Lu)′(t)=Au(t)+ ∫ −∞ta(t−s)Bu(s)ds+f(t), (0≤t≤2π) with periodic boundary conditions u(0)=u(2π), (Mu′)(0)=(Mu′)(2π), in Lebesgue–Bochner spaces Lp(𝕋;X) and periodic Besov spaces Bp,qs(𝕋;X), where A, B, L and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(L), D(A)∩D(B)≠{0} and a∈L1(ℝ+). We completely characterize the well-posedness of (P2) in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.

Solutions of third order degenerate equations with infinite delay in Banach spaces

We study the well-posedness of the third order degenerate differential equations with infinite delay$$(P_3): (Mu)'''(t) + (Lu)''(t) + (Bu)'(t)= Au(t) + \int _{-\infty }^t a(t-s)Au(s)ds + f(t){\text{ on }}[0, 2\pi ]$$in Lebesgue–Bochner spaces $$L^p(\mathbb{T};\; X)$$ and periodic Besov spaces $$B_{p,\,q}^s(\mathbb{T};\; X)$$, where A, B, L and M are closed linear operators on a Banach space X satisfying $$D(A)\subset D(B)\cap D(L)\cap D(M)$$ and $$ a\in L^1(\mathbb{R}_+)$$. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for $$(P_3)$$ to be $$L^p$$-well-posed(or $$B_{p,q}^s$$-well-posed). Concrete examples are also given to support our main abstract results.

Well-Posedness of Fractional Degenerate Differential Equations in Banach Spaces

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 &lt; β &lt; α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Discrete maximal regularity of time-stepping schemes for fractional evolution equations

In this work, we establish the maximal ell ^p-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order alpha in (0,2), alpha ne 1, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank–Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735–758, 2001. doi:10.1007/PL00004457) and its discrete analogue due to Blunck (Stud Math 146:157–176, 2001. doi:10.4064/sm146-2-3). These results generalize the corresponding results for parabolic problems.

Well-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaces

We study the well-posedness of degenerate fractional differential equations with infinite delay $(P_\alpha ): D^\alpha (Mu)(t) =Au(t)+\int _{-\infty }^t a(t-s)Au(s)\,ds+f(t)$, $0\leq t\leq 2\pi $, in Lebesgue-Bochner spaces $L^p(\mathbb {T}; X)$ and Besov spaces $B_{p,q}^s(\mathbb {T}; X)$, where $A$ and $M$ are closed linear operators on a Banach space~$X$ satisfying $D(A)\subset D(M)$, $\alpha >0$ and $a\in L^1(\mathbb {R}_+)$ are fixed. Using well known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of $(P_\alpha )$ in the above vector-valued function spaces on $\mathbb {T}$.

$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$

By using Blunck’s operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue sequence spaces for a discrete version of the Cauchy problem with fractional order 1<α≤2. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the UMD-class.

Maximal regularity for the thermoelastic plate equations with free boundary conditions

We consider the linear thermoelastic plate equations with free boundary conditions in the $$L_p$$ in time and $$L_q$$ in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform $$C^4$$ -domain, which includes the cases of a bounded domain and of an exterior domain with $$C^4$$ -boundary. Moreover, we prove uniform a priori estimates for the solution. The proof is based on the existence of $${\mathcal R}$$ -bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

On the free boundary problem for the Oldroyd-B Model in the maximal [formula omitted]–[formula omitted] regularity class

In the present work, we prove the local well-posedness of non-Newtonian compressible viscous barotropic fluid flow of Oldroyd-B type with free surface in a bounded domain of N-dimensional Euclidean space (N≥2). The key step is to prove the maximal Lp–Lq regularity theorem for the linearized equation with the help of the R-bounded solution operators for the corresponding resolvent problem and Weis’s operator valued Fourier multiplier theorem.

A-Stable Time Discretizations Preserve Maximal Parabolic Regularity

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank--Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge--Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal $L^p$-regularity in terms of $R$-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A($\alpha$)-stable higher-order BDF methods have maximal $\ell^p$-regularity under an $R$-boundedness condition in a larger sector. As an illustration of the use of maximal regularity ...

Maximal regularity in exponentially weighted Lebesgue spaces of the Stokes operator in unbounded cylinders

Abstract We study resolvent estimates and maximal regularity of the Stokes operator in Lq -spaces with exponential weights in the axial directions of unbounded cylinders of ℝ n , n ≥ 3. For a straight cylinder we use exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for cylinders with several exits to infinity we prove that the Stokes operator in Lq -spaces with exponential weights generates an exponentially decaying analytic semigroup and has maximal regularity. The proof for straight cylinders uses an operator-valued Fourier multiplier theorem and unconditional Schauder decompositions based on the ℛ-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the one-dimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the case without exponential weight.

Well-posedness of second order degenerate integro-differential equations in vector-valued function spaces

We consider the well-posedness of the second order degenerate integrodifferential equations (P2): , (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu′)(0) = (Mu′)(2π), in periodic Lebesgue-Bochner spaces Lp(𝕋, X), periodic Besov spaces (𝕋, X) and and periodic Triebel-Lizorkin spaces (𝕋, X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∊ L1 (ℝ+) and α is a scalar number. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for the well-posedness of (P2) in the above three function spaces.

On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal L p - L q regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).MSC: 35Q35, 76T10.

On a maximal [formula omitted] approach to the compressible viscous fluid flow with slip boundary condition

In this paper, we prove a local in time unique existence theorem for the compressible viscous fluids in the general domain with slip boundary condition. For the purpose, we use the contraction mapping principle based on the maximal Lp–Lq regularity by means of the Weis operator valued Fourier multiplier theorem for the corresponding time dependent problem. To obtain the maximal Lp–Lq regularity, we prove the sectorial R-boundedness of the solution operator to the generalized Stokes equations.

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\)). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\)–\(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\)-bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\)–\(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin caratheodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Holder spaces, but our approach is different from them.