Nash Moser Theorem Research Articles

Local Solutions of RG Flow Equations from the Nash–Moser Theorem

We prove local existence of solutions of a functional renormalisation group equation for the effective action of an interacting quantum field theory, when a suitable local potential approximation is considered. To obtain this equation in a Lorentzian setting a quantum state for the theory is selected and a regulator consisting in a mass is added to the action. The flow equation for mass rescalings is then studied using the renown Nash–Moser theorem.

Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems

We investigate a general question about the size and regularity of the data and the solutions in implicit function problems with loss of regularity. First, we give a heuristic explanation of the fact that the optimal data size found by Ekeland and Sere with their recent non-quadratic version of the Nash-Moser theorem can also be recovered, for a large class of nonlinear problems, with quadratic schemes. Then we prove that this heuristic observation applies to the singular perturbation Cauchy problem for the nonlinear Schrodinger system studied by Metivier, Rauch, Texier, Zumbrun, Ekeland, Sere. Using a free flow component decomposition and applying an abstract Nash-Moser-Hormander theorem, we improve the existing results regarding both the size of the data and the regularity of the solutions.

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.

Long time dynamics of Schrödinger and wave equations on flat tori

We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a tϵ(∀ϵ>0) upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions.Both results are based on “clusterization properties” of the eigenvalues of the Laplacian on a flat torus and on suitable “separation properties” of the singular sites of Schrödinger and wave operators, which are integers, in space–time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [20] to control the speed of growth of the Sobolev norms, and Berti–Corsi–Procesi abstract Nash–Moser theorem [8] to construct quasi-periodic solutions.

The Nash–Moser theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash–Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space Hk-nil2(g,g), and certain subspaces of it, that provide fine information about the deformations of g in the variety of k-step nilpotent Lie algebras.Then we focus on degenerations and rigidity in the variety of k-step nilpotent Lie algebras of dimension n with n≤7 and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

An application of the Nash–Moser theorem to the vacuum boundary problem of gaseous stars

We have been studying spherically symmetric motions of gaseous stars with physical vacuum boundary governed either by the Euler–Poisson equations in the non-relativistic theory or by the Einstein–Euler equations in the relativistic theory. The problems are to construct solutions whose first approximations are small time-periodic solutions to the linearized problem at an equilibrium and to construct solutions to the Cauchy problem near an equilibrium. These problems can be solved when 1/(γ−1) is an integer, where γ is the adiabatic exponent of the gas near the vacuum, by the formulation by R. Hamilton of the Nash–Moser theorem. We discuss on an application of the formulation by J.T. Schwartz of the Nash–Moser theorem to the case in which 1/(γ−1) is not an integer but sufficiently large.

On spherically symmetric solutions of the Einstein–Euler equations

We construct spherically symmetric solutions to the Einstein-Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying so called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash-Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the non-relativistic Euler-Poisson equations.

Smooth solutions of the one-dimensional compressible Euler equation with gravity

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is difficult to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Due to this reason, we try to find a family of solutions expanded by a small parameter and apply the Nash–Moser Theorem to justify this expansion. Note that the application of Nash–Moser Theorem is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

On Spherically Symmetric Motions of the Atmosphere Surrounding a Planet Governed by the Compressible Euler Equations

We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds

We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

Asymmetrical Three-Dimensional Travelling Gravity Waves

We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non-symmetric with respect to the propagation direction of the waves and we consider a general non-resonant situation. Defining a couple of amplitudes \({\varepsilon_{1},\varepsilon_{2}}\) along the basis of wave vectors which satisfy the dispersion relation, we first give the formal asymptotic expansion of bifurcating solutions in powers of \({\varepsilon_{1},\varepsilon_{2}.}\) Then, introducing an additional equation for the unknown diffeomorphism of the torus associated with an irrational rotation number, which allows us to transform the differential at the successive points of the Newton iteration method into a differential equation with two constant main coefficients, we are able to use a descent method leading to an invertible differential. Then, by using an adapted Nash–Moser theorem, we prove the existence of solutions with the above asymptotic expansion for values of the couple \({(\varepsilon_{1}^{2},\varepsilon_{2}^{2})}\) in a subset of the first quadrant of the plane with asymptotic full measure at the origin.

Linearization of vector fields and embedding of diffeomorphisms in flows via Nash–Moser theorem

In the first part of this paper we give suitable spectral properties of the adjoint operators induced by appropriate perturbations of some hyperbolic linear vector fields. These properties are useful to prove general facts based on the Nash–Moser inverse function theorem. In the second part of this work we study circumstances where a global linearization of a vector field X in a real numerical space is feasible and where some diffeomorphisms which are close to exp ( X ) can be embedded in a flow.

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.

Local Controllability of a One-Dimensional Beam Equation

We prove that the beam equation with clamped ends is locally controllable in a $H^{5+\epsilon} \times H^{3+\epsilon} ((0,1),\mathbb{R})$-neighborhood of a particular trajectory of the free system, with $\epsilon>0$ and with control functions in $H^{1}_{0}((0,T),\mathbb{R})$. Ball,. Marsden, and Slemrod already proved that this equation is not controllable in $H^{2}_{0} \times L^{2}((0,1),\mathbb{R})$ with control functions in $L^{r}_{\text{loc}}(\mathbb{R},\mathbb{R})$, $r>1$. This article justifies that their negative result is due to a choice of functional spaces which does not allow controllability. Our proof uses moment theory and the Nash-Moser theorem.

A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations

We study the well-posedness of the initial value problem for a wide class of singular evolution equations. We prove a general well-posedness theorem under three assumptions easy to check: the first controls the singular part of the equation, the second the behavior of the nonlinearities, and the third one assumes that an energy estimate can be found for the linearized system. We allow losses of derivatives in this energy estimate and therefore construct a solution by a Nash-Moser iterative scheme. As an application to this general theorem, we prove the well-posedness of the Serre and Green-Naghdi equation and discuss the problem of their validity as asymptotic models for the water-waves equations.

Remarks on the Hölder continuity of solutions to elliptic equations in divergence form

The classical proofs of the De Giorgi–Nash–Moser Theorem are based on the iteration of some inequality through countably many concentric balls. In this note, we present a new approach to the Holder continuity of solutions to elliptic equations in divergence form, which avoids any form of discrete iteration. In particular, we prove that a suitable energy function satisfies a differential inequality, whose integration yields a new proof of the crucial step in the regularity result.

ON THE LOCAL SOLVABILITY FOR A NONLINEAR WEAKLY HYPERBOLIC EQUATION WITH ANALYTIC COEFFICIENTS

We prove the local solvability in C ∞ of the following degenerate hyperbolic quasilinear equation in one space dimension: where the coefficient a(t, x) is nonnegative and real analytic in both variables, while f satisfies suitable Levi conditions. The main tools used are weighted energy estimates and the Nash-Moser 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

Negative results on the Nash-Moser theorem for Köthe sequence spaces and for spaces of ultradifferentiable functions

Counterexamples to the Nash-Moser inverse function theorem are given for Kothe sequence spaces extending a result of Lojasiewicz and Zehnder. As an application also negative results are obtained for spaces of ultradifferentiable functions of Beurling type, in particular for the Gevrey classes.

On weakly harmonic maps and Noether harmonic maps from a Riemann surface into a Riemannian manifold

I want to present some results on the regularity for harmonic maps between a surface of dimension two and a Riemannian manifold. First of all, we recall what harmonic maps between Riemannian manifolds are. Let (M, g) and (N , g) be two Riemannian manifolds of dimension m and n respectively, and assume that N is compact and is isometrically embedded in some Euclidean space R (which is always possible thanks to the Nash–Moser theorem). We introduce the Dirichlet functional on the set of maps between M and N . To do this we define the energy density of a map u from M into N at any point x of M by e(u)(x) = 1 2hij [u(x)]g (x)uαu j β