Nash-Moser Implicit Function Theorem Research Articles

Deformations Of Multivalued Harmonic Functions

Abstract We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.

Asymptotic stability of a free boundary problem for the growth of multi-layer tumours in the necrotic phase

In this paper we study a free boundary problem for the growth of multi-layer tumours in the necrotic phase. The tumour region is strip-like and divided into a necrotic region and a proliferating region with two free boundaries. The upper free boundary is the tumour surface and is governed by a Stefan condition. The lower free boundary is the interface separating the necrotic region from the proliferating region, and its evolution is implicit and intrinsically governed by an obstacle problem. By the Nash–Moser implicit function theorem we show that the lower free boundary is smoothly dependent on the upper free boundary, and by using analytic semigroups theory we obtain asymptotic stability of the unique flat stationary solution.

A Nash–Moser–Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs

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 theorem to a control and a Cauchy problem for quasi-linear perturbations of KdV equations, improving the regularity of a previous result. With respect to other approaches to control and Cauchy problems, the application of our theorem requires lighter assumptions to be verified.

Geometric flows and Strominger systems

A geometric flow on (2, 2)-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton’s version of the Nash–Moser implicit function theorem.

Stationary free surface viscous flows without surface tension in three dimensions

We consider an incompressible, viscous, finite depth fluid flowing down a three-dimensional channel. In the absence of surface tension, we prove the existence of a unique stationary solution in weighted Sobolev spaces. The result is based on a thorough study of the linearized problem, particularly the pseudodifferential operator relating the normal velocity of the fluid and the normal component of the associated stress tensor along the free surface, and requires the use of the Nash-Moser implicit function theorem.

Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the exact controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models (Beauchard, 2005, 2008, 2006) [14,15,17], in non-optimal spaces, in long time and the proof relied on the Nash–Moser implicit function theorem in order to deal with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized, showing there is actually no loss of regularity. Then, the same strategy is applied to nonlinear Schrödinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems.

An abstract Nash–Moser theorem with parameters and applications to PDEs

We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.

Controllablity of a quantum particle in a 1D variable domain

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function φ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given φ0 close enough to an eigenstate corresponding to the length l =1a ndφf close enough to another eigenstate corresponding to the length l = 1, there exists a continuous function l :( 0 ,T ) → R ∗ with T> 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from φ0 to φf in time T.I n particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way. Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.

Cantor families of periodic solutions for completely resonant nonlinear wave equations

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ 0 close enough to an eigenstate and ψ f close enough to another eigenstate, the wave function can be moved exactly from ψ 0 to ψ f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash–Moser implicit function theorem, the return method and expansion to the second order.

Local controllability of a 1-D Schrödinger equation

We consider a nonrelativistic charged particle in a 1-D box of potential. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the return method, a Nash–Moser implicit function theorem and moment theory.

Constrained Wave Equations and Wave Maps

In this paper we establish that wave maps can be obtained by a penalization method if the initial data is well prepared. When the data is not well prepared, we prove that the solution of the penalized equation converges weakly to the solution of the system of coupled equations obtained in [11] by a multi-scale formal analysis. In particular, the interaction between the rapid normal oscillations and the tangential motions creates a new term in the limit system whose well-posedness is proved by using the Nash-Moser Implicit Function Theorem.

An application of the Nash-Moser theorem to ordinary differential equations in Fréchet spaces

A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main

Second Order Hamiltonian Equations on [formula omitted]∞ and Almost-Periodic Solutions

Motivated by problems arising in nonlinear PDE′s with a Hamiltonian structure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type ẍ = ∂xV, [x ∈ TN, ∂x ≡ (∂x1, ..., ∂xN)]. Extending methods from quantitative perturbation theory (Kolmogorov-Arnold-Moser theory, Nash-Moser implicit function theorem, etc.) we construct uncountably many almost-periodic solutions for the infinite dimensional system ẍi = ƒi(x), i ∈ Zd, x ∈ TZd (endowed with the compact topology); the Hamiltonian structure is reflected by ƒ being a "generalized gradient." Such a result is derived under (suitable) analyticity assumptions on ƒi but without requiring any "smallness conditions."

Realization of Arbitrary Vector Fields on Center Manifolds of Parabolic Dirichlet BVPs

Consider the following parabolic equation: u t − Lu = σ(x, u, ∇u), t ≥ 0, x ∈ Ω u(x, t) = 0, t ≥ 0, x ∈ ∂ΩHere Ω ⊂ R N is a smooth bounded domain, L is a second-order self-adjoint uniformly elliptic differential operator and σ: Ω × R × R N → R is some nonlinearity which explicitly depends on the gradient of the solution. Using the Nash-Moser implicit function theorem we prove in this paper that if L and Ω satisfy the so-called Poláčik condition and X 1 denotes the kernel of L with Dirichlet boundary conditions (in this case the dimension of X 1 is necessarily N or N + 1) then for every sufficiently smooth and "small" vector field υ defined in a neighborhood of zero in X 1 there exists a nonlinearity σ and a local center manifold M σ of ( P σ) such that υ is exactly the reduced vector field of ( Pσ) on M σ. This result implies, in particular, that arbitrary chaotic behavior of solutions of ODEs is also observable in suitable scalar parabolic equations with Dirichlet boundary conditions and gradient dependent nonlinearities.

On the existence of small amplitude solitary waves with strong surface tension

An existence theory for small amplitude solitary waves with surface tension effects included is developed for large values of the surface tension parameter ( β > 1 3 ). Using ideas of Beale, the Nash-Moser implicit function theorem is applied to justify the well-known approximation of Korteweg and deVries. Some of the recent results of Amick and Kirchgässner are thereby achieved more directly and additional insight obtained for the open case β < 1 3 .

Unsteady flow in porous media with a free surface

The short time existence of the unsteady free boundary appearing in the porous media is discussed by use of Nash-Moser's implicit function theorem. The uniqueness, existence and the semibounded estimates for the solution of the linearized equation play an important role in the proof of the existence theorem. Also Nash-Moser's implicit function theorem is modified in an applicable form.

A new method for the existence of solutions of nonlinear differential equations

Many problems is analysis reduce to solving an equation Au=f, where A is an operator on a space into another space, u is the unknown, and f is a given element. The existence of a local solution near a trivial solution u0 of Au, = 0, where 0 is near f, is usually solved by the classical implicit function theorem. As a matter of fact, the Picard iteration procedure is used. Unfortunately, this procedure cannot be applied in some problems of partial differential equations. These cases occur if the loss of derivatives appears [6, 7, 31. To get around this difficulty a more sophisticated iteration procedure is needed. Moser [ 1 ] has found such a method which enables one to handle a loss of derivatives by using a rapidly converging iteration scheme. The Nash-Moser technique is nicely described in the papers [S, lo]. We prove in this paper several theorems on the existence of solutions, with the application to several types of specific problems. The main interest is that in many cases these results can be used in situations which up to now have required the Nash-Moser implicit function theorem, yet they have classical proofs. We also apply abstract results to show the existence of local invariant manifolds for singularly perturbed ordinary differential equations and we give a simple proof of a theorem of J. Moser for invariant tori [l]. The plan of our paper is as follows. In Section 2 we present proofs of several existence theorems in an abstract setting in a scale of Hilbert spaces. The hypotheses are those commonly found in problems which have up to now required the application of the Nash-Moser technique, however, the results have classical proofs based on the degree of mappings [S]. We shall give in the proof of Theorem 2.8 a new iteration procedure which retains supergeometric convergence properties. We require obligations only for the linearization of our equation. This is the main advance of this theorem. Section 3 of the paper presents applications of the theorems of Section 2. There are demonstrated proofs of the above mentioned theorems of

The existence of internal solitary waves in a two-fluid system near the KdV limit

Abstract Two fluid layers of constant density lying one over the other on top of a rigid horizontal lower boundary with either a free upper surface or a rigid upper boundary can support solitary waves. The existence of a unique branch of such waves emanating from the horizontal flow at a critical speed U ∗ is demonstrated in both cases by use of the Nash—Moser implicit function theorem. These results complement the global results of Amick and Turner (1986) and are analogous to the work of Friedrichs and Hyers (1954) and Beale (1977) for surface waves. It is also noted that the most obvious variational principle which characterizes these waves as constrained extremals (Benjamin, 1984) is of indefinite type, having a Hessian with infinitely many positive and infinitely many negative eigenvalues.

A perturbation study of the obstacle problem by means of a generalized implicit function theorem

The Nash-Moser Implicit Function Theorem is applied to the study of the perturbation of the geometrical domain of an Obstacle Problem, considered as a Free Boundary Problem in an annular region of the plane. Applications to Optimal Control are derived and numerical results are given.