Operator-valued Fourier Multiplier 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.

Periodic solutions for the Blackstock–Crighton–Westervelt equation

We investigate the Blackstock–Crighton–Westervelt equation which models nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We prove existence and regularity, in a Lp−Lq setting, of time-periodic solutions for a given sufficiently small time-periodic forcing data, and homogeneous Dirichlet boundary conditions over a cylindrical domain. We show maximal Lp-regularity for the abstract linearized model. We use techniques of operator-valued Fourier multiplier theorems combined with a generalized version of the implicit function theorem.

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.

On periodic solutions for one-phase and two-phase problems of the Navier\u2013Stokes equations

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal L_p–L_q regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of {mathcal R}-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the {{mathcal {R}}}-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; {{mathcal {R}}} boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. ({{mathcal {R}}}-solvers and their application to periodic L_p estimates, Preprint in 2019) for the L_p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

L^{p}-L^{q}-Maximal regularity of the Van Wijngaarden\u2013Eringen equation in a cylindrical domain

We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

Lp-Lq-Well Posedness for the Moore–Gibson–Thompson Equation with Two Temperatures on Cylindrical Domains

We examine the Cauchy problem for a model of linear acoustics, called the Moore–Gibson–Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove Lp-Lq-well-posedness, and we provide maximal regularity estimates which are optimal thanks to the theory of operator-valued Fourier multipliers.

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.

The [formula omitted]-bounded operator families arising from the study of the barotropic compressible flows with free surface

In this paper, we study some model problem associated to the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account. To obtain the maximal Lp−Lq regularity property of the model problem, we prove the existence of R-bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.

$$L^p$$-Maximal Regularity for a Class of Degenerate Integro-differential Equations with Infinite Delay in Banach Spaces

Using the theory of operator-valued Fourier multipliers, we establish characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We are concerned with the spaces $$L^p({\mathbb {R}},X), \, 1\leqslant p<\infty $$ where X is a given Banach space. When X is a UMD space and $$1<p<\infty $$, we obtain concrete conditions for well-posedness based on the concept of R-boundedness (or Rademacher boundedness) for operator families. We rely on a transfer to weighted vector-valued $$L^p$$-spaces in order to prove the results. The results are applied to some concrete equations. The equation that we consider appear in several models in the applied sciences, particularly physics, rheology, material science and more generally in phenomena where memory effects play an important role.

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.

Operator-valued Fourier multipliers in vector-valued function spaces and application

Operator-valued Fourier multipliers in vector-valued function spaces and application

НЕКОТОРЫЕ КРАЕВЫЕ ЗАДАЧИ ДЛЯ ОПЕРАТОРНОДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ТИПА СОБОЛЕВА

Рассматривается вопрос о разрешимости краевых задач для операторнодифференциального уравнения вида But Lu f, где B, L : X X (X банахово пространство) замкнутые операторы такие, что D(L) D(B) (D(L), D(B) области определения соответствующих операторов), с краевыми условиями Bu(0) Bu0 или TR0 Bu()d() Bu0, где функция ограниченной вариации. Уточняются некоторые известные результаты о разрешимости начальнокраевых задач для операторнодифференциальных уравнений типа Соболева в случае произвольного убывания (роста) резольвенты соответствующего линейного пучка. Получены теоремы о существовании и единственности решений задачи типа Коши и нелокальной краевой задачи общего вида, в том числе при определенных условиях показана максимальная регулярность решений. Последние результаты основаны на теореме Михлина для операторнозначных мультипликаторах Фурье. В отличие от предыдущих результатов в качестве функциональных пространств используются пространства Соболева Бесова. We consider the solvability of boundary value problems for operatordifferential equation of the form But Lu f, where X is a Banach space, B, L : X X are closed operators such that D(L) D(B) (D(L), D(B) are domains of the corresponding operators), with boundary conditions Bu(0) Bu0 or R0T Bu()d() Bu0, where is a function of bounded variation. Some wellknown results on solvability of initial boundary value problems for operatordifferential equations of Sobolev type are refined in the case of arbitrary decrease (growth) of the resolvent of the corresponding linear pencil. Existence and uniqueness theorems of solutions to the Cauchytype problems and general nonlocal boundary value problems are obtained and the maximal regularity of solutions is proven under certain conditions. The results rely on Mikhlin theorems for operatorvalued Fourier multipliers. In contrast to the previous results, the function spaces are the Sobolev Besov spaces.

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.

Well-posedness for degenerate third order equations with delay and applications to inverse problems

In this paper, we study well-posedness for the following third-order in time equation with delay $$\left( {0.1} \right)\;\alpha \left( {Mu'} \right)''\left( t \right) + \left( {Nu'} \right)'\left( t \right) = \beta Au\left( t \right) + \gamma Bu{'(t)} + Gu{'_t}+F{u_t} + f\left( t \right),\;t \in \left[ {0,2\pi } \right]$$ where α, β, γ are real numbers, A and B are linear operators defined on a Banach space X with domains D(A) and D(B) such that $$D(A)\cap{D(B)}\subset{D(M)}\cap{D(N)};$$ u(t)is the state function taking values in X and ut: (−∞, 0] → X defined as ut(θ) = u(t+θ) for θ < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue–Bochner spaces $$L^p(\mathbb{T}, X)$$ , periodic Besov spaces $$B^s_{p,q}(\mathbb{T}, X)$$ and periodic Triebel–Lizorkin spaces $$F^s_{p,q}(\mathbb{T}, X)$$ . A novel application to an inverse problem is given.

Weighted norm inequalities for Weyl multipliers and fourier multipliers on the Heisenberg group

In this paper, we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show, in the context of a theorem of Weis on operator-valued Fourier multipliers, that R-boundedness of the derivative of the multiplier is not necessary for boundedness of the multiplier transform.

Stability theory for semigroups using (Lp,Lq) Fourier multipliers

We study polynomial and exponential stability for C0-semigroups using the recently developed theory of operator-valued (Lp,Lq) Fourier multipliers. We characterize polynomial decay of orbits of a C0-semigroup in terms of the (Lp,Lq) Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator.As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.