A Regularity Theory for an Initial Value Problem with a Time-measurable Pseudo-differential Operator in a Weighted $$L_p$$-space

Lectures on pseudo differential operators.- Singular integral operators on the circle.- Pseudo differential operators.- Elliptic and hypoelliptic operators.- The initial value problem. Hyperbolic operators.- Elliptic boundary value problems petrowsky parabolic operators.- Propagation of singularities wave front sets.- The sharp Garding inequality.

We will now turn to a class of operators which — roughly speaking (details below) — are locally presentable in the form $$\left( {Pu} \right)\left( x \right)\,: = \int {ei < x,\varepsilon > p\left( {x,\varepsilon } \right)} \mathop u\limits^ \wedge \left( \varepsilon \right)d\varepsilon$$ where $$ \hat u\left( x \right): = \int {e^{ - i\left\langle {x,\xi } \right\rangle } u\left( \xi \right)d\xi } $$ is the Fourier transform of u (see the crash course above in I.8), p is the “amplitude” of the operator and $< x, \xi > = x_1 \xi _1 + ... + x_n \xi _n$ its “phase function”.

There is a close relationship among classical spaces of functions and distributions, pseudodifferential operators of classical type, and the regularity theory of elliptic operators. Scales of spaces such as the standard Sobolev or H/51der spaces indexed by the degree of smoothness are mapped among themselves by standard pseudodifferential operators. This is true in particular of the (approximate) inverses of elliptic operators, a fact which is equivalent to the classical regularity theory. A similar relationship exists among certain spaces defined by non-isotropic quasihomogeneous smoothness conditions, the corresponding types of parabolic or other semi-elliptic operators, and the corresponding regularity theory. It is the purpose of this paper to develop an analogous theory for the spaces of functions and distributions naturally associated to general classes of pseudodifferential operators, with applications to the regularity theory of hypoelliptic operators.

We construct parametrices for initial value problems of the form [Formula: see text] where $(z,x)\in \mathbb{R} \times \mathbb{R} ^{n}$ , A(z,x,Dx) is a family of order 1 pseudodifferential operators with homogeneous real principal symbol a(z,x,ξ), and B(z,x,Dx) is a family of order γ&gt;0 pseudodifferential operators with non‐negative homogeneous real principal symbol b(z,x,ξ). The parametrix is a family of pseudodifferential operators when A=0, and a Fourier integral operator with real phase function if A≠0. A priori this leads to symbols of type $(\rho,\delta)=(1-\frac{\gamma}{2},\frac{\gamma}{2})$ , which limits our construction to γ&lt;1, and leads to operators with a complicated symbol calculus in the case γ=1. With an additional assumption on B we obtain symbols of type $(\rho,\delta)=(1-\frac{\gamma}{L},\frac{\gamma}{L})$ , for some L≥2. The assumption implies in particular that the first L−1 derivatives of b vanish where b=0. Parametrices for (*) are constructed for the case when 2γ&lt;L.

We introduce basic concepts of generalized Differential Geometry of Frölicher and diffeological spaces; we consider formal and non-formal pseudodifferential operators in one independent variable, and we use them to build regular Frölicher Lie groups and Lie algebras on which we set up the Kadomtsev-Petviashvili hierarchy. The geometry of our groups allows us to prove smooth versions of the algebraic Mulase factorization of infinite dimensional groups based on formal pseudodifferential operators, and also an Ambrose-Singer theorem for infinite dimensional bundles. Using these tools we sketch proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also introduce a version of the KP hierarchy on infinite dimensional groups of series of non-formal pseudodifferential operators and we solve its initial value problem.

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical examples like (semi)classical limits of Schrödinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. © 1997 John Wiley & Sons, Inc.

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical examples like (semi)classical limits of Schrodinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. © 1997 John Wiley & Sons, Inc.

SynopsisIn 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.

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off. Our results place certain recent attempts to study inflation in the context of nonlocal field theories on a much firmer mathematical footing.

Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Pade-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.

We begin Chapter 2 with simple examples of initial and boundary value problems, solution operators of which have singularities of one or another type in the dual variable. The presence of a singularity often causes a failure of well posedness of the problem in the sense of Hadamard. Let A be a linear differential operator mapping a function space X into another function space F.

We study the- initial and boundary value problem for the equation utt = L[u] with Dirichlet boundary data where L is the sum of a strongly elliptic formally self-adjoint differential operator of order 2m and of an arbitrary differential operator of order q ≦m. We give a weak formulation of this problem in an appropriate Hilbert space. The main ideas of the existence, uniqueness and regularity theory are sketched. Detailed proofs will appear in two subsequent papers. Our existence argument makes essential use of L2-spaces of vector-valued functions with values in a Hilbert space. The theory of these "Bochner spaces" can be developed by employing methods of L. Schwartz's theory of distributions. In particular, our analysis avoids Lebesgue and Bochner integration

We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate primarily on boundaries that are geometrically determined by the outermost normal observer to spacelike slices of the foliation. We present high-order-derivative boundary conditions for the gauge, constraint violating and gravitational wave degrees of freedom of the formulation. Second order derivative boundary conditions are presented in terms of the conformal variables used in numerical relativity simulations. Using Kreiss–Agranovich–Métivier theory we demonstrate, in the frozen coefficient approximation, that with sufficiently high order derivative boundary conditions the initial boundary value problem can be rendered boundary stable. The precise number of derivatives required depends on the gauge. For a choice of the gauge condition that renders the system strongly hyperbolic of constant multiplicity, well-posedness of the initial boundary value problem follows in this approximation. Taking into account the theory of pseudo-differential operators, it is expected that the nonlinear problem is also well-posed locally in time.

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and different type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Preface.- Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds.- The Singular Functions of Branching Edge Asymptotics.- The Heat Kernel and Green Function of the Sub-Laplacian on the Heisenberg Group.- Metaplectic Equivalence of the Hierarchical Twisted Laplacian.- The Heat Kernel and Green Function of a Sub-Laplacian on the Hierarchical Heisenberg Group.- Lp-Bounds for Pseudo-Differential Operators on the Torus.- Multiplication Properties in Gelfand-Shilov Pseudo-Differential Calculus.- Operator Invariance.- Initial Value Problems in the Time-Frequency Domain.- Polycaloric Distributions and the Generalized Iterated Heat Operator.- Smoothing Effect and Fredholm Property for First-Order Hyperbolic PDEs.- A Note on Wave-Front Sets of Roumieu Type Ultradistributions.- Ordinary Differential Equations in Algebras of Generalized Functions.- Asymptotically Almost Periodic Generalized Functions.- Wave Equations and Symmetric First-Order Systems in Case of Low Regularity.- Concept of Delta-Shock Type Solutions to Systems of Conservation Laws and the Rankine-Hugoniot Conditions.- Classes of Generalized Functions with Finite Type Regularities.- The Wave Equation with a Discontinuous Coefficient Depending on Time Only: Generalized Solutions and Propagation of Singularities.- Gerenalized Solutions of Abstract Stochastic Problems.- Nonhomogeneous First-Order Linear Malliavin Type Differential Equation.