In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of one-sided coupled operator matrices provides an excellent framework to study the well-posedness of such problems. We show that with this machinery even operator splitting methods can be treated conveniently and rather efficiently. We consider three specific examples: the Lie (sequential), the Strang and the weighted splitting, and prove the convergence of these methods along with error bounds under fairly general assumptions.
We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.
In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.
In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $n\geqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.
In this article we first derive some sufficient conditions to establish the monotonicity and comparison principles of the semi-flow generated by non-densely defined Cauchy problems. We apply our results to a class of age structured population models. As a consequence we obtain a monotone semi-flow theory and some comparison principles for age structured models.
In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. By applying this result, the Cauchy problem for systems of Boussinesq equations of finite or infinite orders are studied
Due to the seminal works of Hochbruck and Ostermann exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than $π/2$ (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing on the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the sem…
Let $M$ be a compact abstract $CR$ manifold of arbitrary $CR$ codimension. Under certain conditions on the Levi form we prove the infinite dimensionality of some global cohomology groups of $M$.
In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we…
A general class of singular abstract Cauchy problems is considered which naturally arises in applications to certain Free Boundary Problems. Existence of an associated evolution operator characterizing its solutions is established and is subsequently used to derive optimal regularity results. The latter are well known to be important basic tools needed to deal with corresponding nonlinear Cauchy Problems such as those associated to Free Boundary Problems.
We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.
We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we can apply the Laplace transform to any $n+1$ dimensional linear higher-order partial differential equations $P(\partial_x,\partial_t)u=f(x,t)$ directly, without using the Fourier transform. By making introduction of abstract operators $G(\partial_x,t):=\mathcal{L}^{-1}[1/P(\partial_x,s)]$, the analytic solutions of initial value problems are expressed in th…
We study maximal regularity in interpolation spaces for the sum of three closed linear operators on a Banach space, and we apply the abstract results to obtain Besov and Hölder maximal regularity for complete second order Cauchy problems under natural parabolicity assumptions. We discuss applications to partial differential equations.
For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.
This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.
The abstract first order Cauchy problem is solved in terms of Taylor's series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order lower than the Cauchy problem are built-in to the problem, it suffices to solve the higher order derivatives in terms of the lower order ones, in a simple way.
The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
In this paper, the existence, the uniqueness and estimates of solution to the integral Cauchy problem for linear and nonlinear abstract wave equations are proved. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.
In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order $α\in(0,1]$, and discuss the continuity of the respective solutions regarding the parameter $α$. I also present a study about the continuity of the Mittag-Leffler families of operators (for $α\in(0,1]$), induced by sectorial operators.
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
In this paper, we prove global-in-time $\dot{\mathrm{H}}^{α,q}$-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X , provided $q\in(1,+\infty) , $α\in(-1+1/q,1/q)$. We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space.In order to put our result in perspective, we also provide a short review on L q-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation th…
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are obtained. These problems have a numeros applications which occur in fluid mechanics, environmental engineering and atmospheric dispersion of pollutants.
This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces $\mathcal{EV}(Ω,E)$ of $\mathcal{C}^{\infty}$-smooth vector-valued functions whose growth on strips along the real axis with holes $K$ is induced by a family of continuous weights $\mathcal{V}$. Vector-valued means that these functions have values in a locally convex Hausdorff space $E$ over $\mathbb{C}$. We characterise the weights $\mathcal{V}$ which give a counterpart of the Grothendieck-Köthe-Silva duality $\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong\mathscr{A}(K)$ with non-…
Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=Δu -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data.
We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by Hörmander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by Métivier). As an example of application, we apply our …
The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says that by assuming the regularity of initial data, asymptotic profiles (of arbitrary order) are explicitly written by using the semigroup $e^{-tA}$ generated by $-A$. To prove this, a kind of maximal regularity for $e^{-tA}$ is used.
In this paper, we prove local existence and uniqueness of analytic sharp-front solutions to a generalised SQG equation by the use of an abstract Cauchy--Kowalevskaya theorem. Here, the velocity is determined by $u = |\nabla|^{-2β}\nabla^\perpθ$ which (for $1<β\leq 2$) is more singular than in SQG. This is achieved despite the appearance of pseudodifferential operators of order higher than one in our equation, by recasting our equation in a suitable integral form. We also provide a full proof of the abstract version of the Cauchy--Kowalevskaya theorem we use.
In this paper, regularity properties, Strichartz type estimates to solutions of multipoınt Cauchy problem for linear and nonlinear abstract wave equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
In this paper, we discuss the Cauchy problem for a degenerate parabolic hyperbolic equation with a multiplicative noise. We focus on the existence of a solution. Using nondegenerate smooth approximations, Debussche, Hofmanová and Vovelle [8] proved the existence of a kinetic solution. On the other hand, we propose to construct a sequence of approximations by applying a time splitting method and prove that this converges strongly in $L^1$ to a kinetic solution. This method will somewhat give us not only a simpler and more direct argument but an improvement over the existence result.
In this paper, regularity properties of Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained.
In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representat…
General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in Gene…
This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.
We construct with full rigorous mathematical proof a family of approximate solutions to the Cauchy problem for the standard system of two fluid flows with energy equations and we pass to the limit by weak compactness to obtain Radon measures that satisfy the equations in a natural weak sense. Our method provides a convergent numerical method for the numerical calculation of these Radon measures by reducing the system of partial differential equations in the case of these approximate solutions to a system of ordinary differential equations. We observe numerically on the standard Toumi shock tub…
In this paper, the regularity properties of Cauchy problem for linear and nonlinear nonlocal wave equations are studied.The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform are operator functions defined in Hilbert space H together with some growth conditions. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data and the operator functions. By selecting the space H and the operators, the wide class of wave equations in the field of physics are obtained.
For an elliptic complex of first order differential operators on a smooth manifold, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. T…
The main purpose of this paper is to give a general regularity result for Cauchy-Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic $L^p$-regularity results hold true under one crucial assumption: The totally real submanifold has to be modelled on an $L^p$-space or a closed subspace thereof. Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-ma…
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even tho…
We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the 3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available…
A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by [T. M. Elgindi, F. Rousset, Commun. Pure Appl. Math. 68 (2015), 2005--2021] where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by [A. V. Bhave, R. C. Armstrong, R. A. Brown, J. Chem. Phys. 95(1991), 2988--3000], the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the n…
We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof of maximal regularity for closed and maximal accretive operators following from Kato's inequality for fractional powers and almost orthogonality arguments.
The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified integral equation method to regularize the nonlinear problem with globally and locally Lipschitz source terms. Convergence estimates are established under priori assumptions on exact solution. A numerical test is provided to illustrate that the proposed method is feasible and effective. These results extend some earlier works on a Cauchy problem for elliptic …
We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order operator to show that it generates an analytic semigroup.
We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem for the abstract wacve eqation of the form $Pu_{tt}+Au=F(u) \ (*)$ in a Hilbert space, where $P,A$ are positive linear operators and $F(\cdot)$ is a continuously differentiable gradient operator can be obtained from the result of H.A. Levine on the growth of solutions of the Cauchy problem for (*). This result is applied to the study of inital boundary value problems for nonlinear Klein-Gordon equations, generalized Boussinesq equations and nonlinear plate equations. A result on blow up of solutions …
In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age and the space variables. We handle this linear problem by re-writing it as a non-densely defined abstract Cauchy problem. To that aim we develop a new result on the closedness of a commutative sum of two non-densely defined operators by using the theory of integrated semigroups. As an application of this abstract result, we are able to associate a suitable integrated semigroup to some age-structured problem with spatial diffusion and equipped with non-homogeneous bo…
