Let Ω ⊂ ℝN be open, bounded with boundary ∂Ω of Class C2, let a(t) be of bounded variation, and https://www.w3.org/1998/Math/MathML"> g ∈ C 2 Ω ¯ × ℝ N …
Let Ω ⊂ ℝN be open, bounded with boundary ∂Ω of Class C2, let a(t) be of bounded variation, and https://www.w3.org/1998/Math/MathML"> g ∈ C 2 Ω ¯ × ℝ N ; ℝ N . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419914/8d73d2ed-d0a7-4013-a8ba-50e0dc128626/content/ieq1071.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We consider the quasilinear Volterra equation 1.1 https://www.w3.org/1998/Math/MathML"> u t ( t , x ) = ∫ 0 t d a ( τ ) { div g ( x , ∇ u ( t − τ , x ) ) + f ( t − τ , x ) } , t ∈ J , x ∈ Ω , u ( t , x ) = 0 , t ∈ J , x ∈ ∂ Ω , u ( 0 , x ) = u 0 ( x ) , x ∈ Ω , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419914/8d73d2ed-d0a7-4013-a8ba-50e0dc128626/content/eqn1_1m.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where J = [0, T], and u0, f are given functions. The initial-boundary value problem (1.1) arises as a model problem in several fields, like the theory of viscoelasticity or heat conduction in materials with memory, and has been studied by many authors; cp. Mac Camy [14,15], Dafermos and Nohel [5], Engler [8], Londen and Nohel [12], Lunardi and Sinestrari [13], StafFans [23], the recent monograph Renardy, Hrusa and Nohel [21], and the references given there.
Abstract We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and flows along $$\Sigma $$ …
Abstract We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> without boundary and flows along $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> . Local-in-time well-posedness is established in the framework of $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> - $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math> -maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> , and we show that each equilibrium on $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.
A recent vector-valued multiplier theorem for operator-valued Fourier multipliers in LP (ℝ; X), 1 < p < ∞, X a UMD-space, is combined with the vector-valued transference principle to prove …
A recent vector-valued multiplier theorem for operator-valued Fourier multipliers in LP (ℝ; X), 1 < p < ∞, X a UMD-space, is combined with the vector-valued transference principle to prove existence of operator-valued R-bounded H ∞ -functional calculi. These results are applied to maximal Lp -regularity of abstract operator equations as well as to evolutionary integral equations.
Classes of second order, one- or two phase- elliptic systems with time-fractional boundary conditions are studied. It is shown that such problems are well posed in an $L_q$-setting, and stability …
Classes of second order, one- or two phase- elliptic systems with time-fractional boundary conditions are studied. It is shown that such problems are well posed in an $L_q$-setting, and stability is considered. The tools employed are sharp results for elliptic boundary and transmission problems and for the resulting Dirichlet-Neumann operators, as well as maximal $L_p$-regularity of evolutionary integral equations, based on modern functional analytic tools like $\mathcal{R} $-boundedness and the operator-valued $\mathcal{H} ^\infty $-functional calculus.
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on …
We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray--Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.
The vector-valued Fourier multiplier theorem due to L. Weis (2001) for $L_p({\mathbb R};X)$, $1 \lt p \lt \infty $, $X$ of class UMD, is extended in an elementary way to the …
The vector-valued Fourier multiplier theorem due to L. Weis (2001) for $L_p({\mathbb R};X)$, $1 \lt p \lt \infty $, $X$ of class UMD, is extended in an elementary way to the weighted spaces $L_{p,\mu }({\mathbb R};X)$ with $1/p \lt \mu \lt 1+1/p$. Th
Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent …
Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.
Isothermal incompressible multi-component two-phase flows with mass transfer, chemical reactions, and phase transition are modeled based on first principles. It is shown that the resulting system is thermodynamically consistent in …
Isothermal incompressible multi-component two-phase flows with mass transfer, chemical reactions, and phase transition are modeled based on first principles. It is shown that the resulting system is thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional, and the equilibria are identified. It is proved that the problem is well-posed in an $L_p$-setting, and generates a local semiflow in the proper state manifold. It is further shown that each non-degenerate equilibrium is dynamically stable in the natural state manifold. Finally, it is proved that a solution, which does not develop singularities, exists globally and converges to an equilibrium in the state manifold.
Of concern is the motion of two fluids separated by a free interface in a porous medium,where the velocities are given by Darcy's law.We consider the case with and without …
Of concern is the motion of two fluids separated by a free interface in a porous medium,where the velocities are given by Darcy's law.We consider the case with and without phase transition.It is shown that the resulting models can be understood as purely geometric evolution laws,where the motion of the separating interface depends in a non-local way on the mean curvature.It turns out that the models are volume preserving and surface area reducing,the latter property giving rise to a Lyapunov function.We show well-posedness of the models, characterize all equilibria,and study the dynamic stability of the equilibria.Lastly, we show that solutions which do not develop singularities exist globallyand converge exponentially fast to an equilibrium.
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This …
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven {\em without} any structural assumptions on the Leslie coefficients and in particular {\em without} assuming Parodi's relation.
Let $X$ be a complex Banach space and$A:\,D(A) \to X$ a quasi-$m$-sectorial operatorin $X$. This paper is concerned with theidentification of diffusion coefficients$\nu > 0$ in the initial-value problem:\[ (d/dt)u(t) …
Let $X$ be a complex Banach space and$A:\,D(A) \to X$ a quasi-$m$-sectorial operatorin $X$. This paper is concerned with theidentification of diffusion coefficients$\nu > 0$ in the initial-value problem:\[ (d/dt)u(t) + {\nu}Au(t) = 0,\quad t \in (0,T), \quad u(0) = x \in X,\]with additional condition $\|u(T)\| = \rho$,where $\rho >0$ is known. Except forthe additional condition, the solution to theinitial-value problem is given by$u(t) := e^{-t\,{\nu}A} x\in C([0,T];X) \cap C^{1}((0,T];X)$.Therefore, the identification of $\nu$ is reducedto solving the equation$\|e^{-{\nu}TA}x\| = \rho$.It will be shown that the unique root$\nu = \nu(x,\rho)$depends on $(x,\rho)$ locally Lipschitzcontinuously if the datum $(x,\rho)$ fulfillsthe restriction $\|x\|> \rho$. This extendsthose results inMola [6](2011).
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, …
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, and the implicit function theorem.
Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics (Prüss et al., Evol Equ Control Theory 1:171–194, 2012; Prüss and Shimizu, J Evol …
Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics (Prüss et al., Evol Equ Control Theory 1:171–194, 2012; Prüss and Shimizu, J Evol Equ 12:917–941, 2012; Prüss et al., Commun Part Differ Equ 39:1236–1283, 2014; see also Prüss et al., Interfaces Free Bound 15:405–428, 2013) is extended to the case of temperature-dependent surface tension. We prove well-posedness in an L p -setting, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally, and if its limit set contains a stable equilibrium it converges to this equilibrium in the natural state manifold for the problem as time goes to infinity.
In this chapter we investigate the spectral properties of the linearizations Lj of the six problems at a given equilibrium. We show that the dimension of the kernel N(Lj) equals …
In this chapter we investigate the spectral properties of the linearizations Lj of the six problems at a given equilibrium. We show that the dimension of the kernel N(Lj) equals the dimension of the tangent space of the manifold of equilibria ε, the eigenvalue 0 is semi-simple for Lj, and the intersection of the spectrum of Lj with the imaginary axis is {0}.
This chapter is devoted to maximal L p -regularity of one-phase linear generalized Stokes problems on domains $$ {\Omega} \subset {\mathbb{R}}^{n} $$ which are either $$ {\mathbb{R}}^{n} $$ , $$ …
This chapter is devoted to maximal L p -regularity of one-phase linear generalized Stokes problems on domains $$ {\Omega} \subset {\mathbb{R}}^{n} $$ which are either $$ {\mathbb{R}}^{n} $$ , $$ {\mathbb{R}}_{ + }^{n} $$ , or domains with compact boundary $$ \partial {\Omega} $$ of class C 3, i.e., interior or exterior domains. Here we only consider the physically natural boundary conditions no-slip, pure slip, outflow, and free. As in Chap. 6, our approach is based on vector-valued Fourier multiplier theory, perturbation, and localization. It turns out that due to the divergence condition (and the pressure), the analysis for the half-space as well as the localization procedure are much more involved than in the previous chapter. Nevertheless, besides some extra compatibility condition which comes from the divergence condition, the main results will parallel those in Chap. 6.
In this chapter we study local well-posedness and regularity of the solutions of Problems (P1)~(P6). Here we employ without further comments the notations introduced in Chapters 1 and 2, in …
In this chapter we study local well-posedness and regularity of the solutions of Problems (P1)~(P6). Here we employ without further comments the notations introduced in Chapters 1 and 2, in particular those in connection with Conditions (H1)~(H6) from Chapter 1, the Hanzawa transform, and the transformed problems on the fixed domain Ω\Σ in Section 1.3. In the first section of this chapter we reformulate Problems (P1)~(P6) in a way which is amenable to a joint analysis, which will be based on maximal Lp-regularity as well as on the contraction mapping principle in Section 9.2, and on the implicit function theorem for dependence on the data in Section 9.3. For regularity we employ in Section 9.4 the so-called parameter trick, which is also based on maximal Lp-regularity and the implicit function theorem. This way we can show that the solutions obtained in Section 9.2 are in fact classical solutions. The proofs for the technical results on the nonlinearities are postponed to the last section of this chapter.
In this chapter, operator-valued Fourier multiplier results for vector-valued L p -spaces are derived and discussed. These form the basic tools for the proof of various results on maximal L …
In this chapter, operator-valued Fourier multiplier results for vector-valued L p -spaces are derived and discussed. These form the basic tools for the proof of various results on maximal L p -regularity which are needed for the nonlinear problems.
In this chapter we introduce some basic tools from operator and semigroup theory. The class of sectorial operators is studied in detail, its functional calculus is introduced, leading to analytic …
In this chapter we introduce some basic tools from operator and semigroup theory. The class of sectorial operators is studied in detail, its functional calculus is introduced, leading to analytic semigroups and complex powers.
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili
The purpose of this introductory chapter is to explain the problems to be considered in the main part of this book in some detail. We derive their physical origin from …
The purpose of this introductory chapter is to explain the problems to be considered in the main part of this book in some detail. We derive their physical origin from first principles, discuss some of the main structural properties of the models, and describe the strategies of our analytical approach. All the notions and properties relating to differential geometry of hypersurfaces will be introduced and explained in Chapter 2.
In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in …
In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in Chapter 1.
In this chapter we consider abstract quasilinear parabolic problems of the form.
In this chapter we consider abstract quasilinear parabolic problems of the form.
In this chapter we introduce the necessary background in differential geometry of closed compact hypersurfaces in ℝ n . We investigate the differential geometric properties of embedded hypersurfaces in n-dimensional …
In this chapter we introduce the necessary background in differential geometry of closed compact hypersurfaces in ℝ n . We investigate the differential geometric properties of embedded hypersurfaces in n-dimensional Euclidean space, introducing the notions of Weingarten tensor, principal curvatures, mean curvature, tubular neighbourhood, surface gradient, surface divergence, and Laplace-Beltrami operator.
Now we are in position to study maximal L p -regularity for linear two-phase Stokes problems. There are two problems of relevance, the standard one, and another, nonstandard problem, which …
Now we are in position to study maximal L p -regularity for linear two-phase Stokes problems. There are two problems of relevance, the standard one, and another, nonstandard problem, which we call the asymmetric two-phase Stokes problem.
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This …
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven {\em without} any structural assumptions on the Leslie coefficients and in particular {\em without} assuming Parodi's relation.
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili
In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasili
We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the …
We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lpregularity for the linearized problem, which is fully developed in this paper as well.
In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted …
In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted to concrete realizations of linear parabolic evolution equations by general parabolic systems.There we discuss the various function spaces that are needed and useful, and the generation of analytic semigroups by general elliptic boundary value problems.The last volume contains the abstract nonlinear theory as well as various applications to concrete systems, illustrating the scope and the flexibility of the general results.Of course, each one of the three volumes contains much material of independent interest related to our main subject.In writing this book I had help from many friends, collegues, and students.It is a pleasure to thank all of them, named or unnamed.I am particularly indebted to P. Quittner and G. Simonett, who critically and very carefully read, not only the whole manuscript of this first volume but also many earlier versions that were produced over the years and will never be published, and pointed out numerous mistakes and improvements.Large parts of the first volume, and of earlier versions as well, were also read and commented on by D. Daners, J. Escher, and P
During the last two decades the theory of abstract Volterra equations has under gone rapid development. To a large extent this was due to the applications of this theory to …
During the last two decades the theory of abstract Volterra equations has under gone rapid development. To a large extent this was due to the applications of this theory to problems in mathematical ph
Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier–Stokes problems on domains with compact $C^{\mbox{3-}}$-boundary is proven in an $L_p$-setting for any space dimension $n\geq2$. The …
Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier–Stokes problems on domains with compact $C^{\mbox{3-}}$-boundary is proven in an $L_p$-setting for any space dimension $n\geq2$. The stress tensor is assumed to be of the generalized Newtonian type, i.e., $\cS=2\mu(|{\mathcal E}|_2^2){\mathcal E} -\pi I$, ${\mathcal E}=\frac{1}{2}(\nabla u+\nabla u^{\sf T}),$ where $|{\mathcal E}|_2^2=\sum_{i,j=1}^n \varepsilon_{ij}^2$ denotes the Hilbert–Schmidt norm of the rate of strain tensor ${\mathcal E}$. The viscosity function $\mu\in C^{2-}({\mathbb R}_+)$ is subject only to the condition $\mu(s)>0$, $\mu(s)+2s\mu^\prime(s)>0$, $s\geq 0,$ which for the standard power-law–like function $\mu(s)=\mu_0(1+s)^{\frac{d-2}{2}}$ merely means $\mu_0>0$ and $d\geq 1$. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.
Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class …
Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.
We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing …
We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously.
Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), …
Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t )) x ( t ) + f ( x ( t ))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.
Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, …
Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal $L_p$-regularity is shown. By means of this purely operator theoretic approach, classical results on $L_p$-regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface diffusion for the diffusion equation is included.
Abstract This is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals. It also includes some new results concerning disclinations (line defects) which …
Abstract This is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals. It also includes some new results concerning disclinations (line defects) which have not been published elsewhere.
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of …
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved …
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique ofmaximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.
The presence of surfactants, ubiquitous at most fluid/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the …
The presence of surfactants, ubiquitous at most fluid/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface — so-called Marangoni effects. The mathematical model governing the dynamics of such systems is studied for the case in which the surfactant is soluble in one of the adjacent bulk phases. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk fluids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. Employing L p -maximal regularity we obtain well-posedness of this model for a certain initial configuration. The proof is based on recent L p -theory for two-phase flows without surfactant.
Some classical examples of transference The general transference result Multipliers defined by the action of locally compact Abelian groups Transference from the integers and the Maximal Ergodic Theorem Ergodic flows …
Some classical examples of transference The general transference result Multipliers defined by the action of locally compact Abelian groups Transference from the integers and the Maximal Ergodic Theorem Ergodic flows and the theory of $H^p$ spaces Integral transforms with zonal kernels Integral transforms with zonal kernels (continued) Kernels having certain invariance properties with respect to representations of $G$ The group SL$(2, \mathbf C)$ Some aspects of harmonic analysis on SL$(2, \mathbf C)$ Bibliography.
A recent result of Kalton and Weis is extended to the case of non-commuting operators, employing the commutator condition of Labbas and Terreni, or of Da Prato and Grisvard. Under …
A recent result of Kalton and Weis is extended to the case of non-commuting operators, employing the commutator condition of Labbas and Terreni, or of Da Prato and Grisvard. Under appropriate assumptions it is shown that the sum of two non-commuting operators admits an $\mathcal H^\infty$-calculus. The main results are then applied to a parabolic problem on a wedge domain.