Globally Solvable Research Articles

Globally solvable time-periodic evolution equations in Gelfand–Shilov classes

AbstractIn this paper we consider a class of evolution operators with coefficients depending on time and space variables $$(t,x) \in {\mathbb {T}}\times {\mathbb {R}}^n$$ ( t , x ) ∈ T × R n , where $${\mathbb {T}}$$ T is the one-dimensional torus, and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand–Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on $${\mathbb {R}}^n$$ R n .

Global classical solvability and stabilization in a two-dimensional chemotaxis–fluid system with sub-logarithmic sensitivity

In this paper, we consider the following system: [Formula: see text] in a smoothly bounded domain [Formula: see text] with [Formula: see text] and a given function [Formula: see text] with [Formula: see text] It is proved that if [Formula: see text] then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if [Formula: see text] then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria [Formula: see text] in [Formula: see text] as [Formula: see text] with [Formula: see text]

On the global solvability of porous media type equations with space dependent advection flux

We show that the Cauchy problem for advection-diffusion equations with nonlinear diffusion term div(|u|α∇u) and advection flux f(x,t,u) of order O(|u|1+k) for large u is globally solvable for arbitrary initial data u0∈L1(Rn)∩L∞(Rn) when k∈[0,α+1/n). Some important related results are also given.

Global solvability and global hypoellipticity of complex vector fields on surfaces

We consider smooth, nonvanishing complex (not essentially real) vector fields L=X+iY that satisfy the Nirenberg-Treves condition (P) and are allowed to possess some closed one-dimensional orbits in the sense of Sussmann that we call clean orbits. Concerning global solvability, we deal with two different cases —depending on whether the surface is compact or not— and prove two positive results which extend and unify several known results on the subject.In our positive results, it is assumed that X∧Y vanishes of order greater than 1 on all closed one-orbits, on the other hand, we prove that global solvability cannot hold if there exists at least one closed one-orbit on which X∧Y vanishes of order 1.We also give a characterization of globally hypoelliptic complex vector fields L in terms of the topological type of the equivalence classes determined by a standard equivalence relation defined on the set of non elliptic points of L. Some of the consequences are that globally hypoelliptic vector fields are globally solvable and that if a surface M carries a globally hypoelliptic vector field, it must be parallelizable.

Global solvability of real analytic involutive systems on compact manifolds

The focus of this work is the smooth global solvability of a linear partial differential operator $${\mathbb {L}}$$ associated to a real analytic closed non-exact 1-form b—defined on a real analytic closed n-manifold—that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co-rank one. We define an appropriate covering projection $$\widetilde{M}\rightarrow M$$ such that the pullback of b has a primitive $$\widetilde{B}$$ and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of $$\widetilde{B}$$ are connected. As a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator. When M is orientable we prove a connection between the global solvability of $${\mathbb {L}}$$ and that of $${\mathbb {L}}^{n-1}$$ which is the last non-trivial operator of the complex, in particular, we prove that $${\mathbb {L}}$$ is globally solvable if and only if $${\mathbb {L}}^{n-1}$$ is globally solvable. In the smooth category, we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a Morse 1-form, i.e., when the structure is of Mizohata type.

Global Conjugation of Solutions of a Hyperbolic Problem along an Unknown Contact Boundary

We consider the problem of local and global solvability of a nonlinear problem with a free (unknown) line of discontinuity of initial data for a hyperbolic system of first-order quasilinear equations with two independent variables. For the problem to be globally solvable, one should additionally require that the coefficients and free terms be monotone and of constant sign.

Global solvability for first order real linear partial differential operators II

Let L be a real C∞ vector field on a smooth manifold X, vanishing at exactly one point x0. From the pioneering work of B. Malgrange (1955–1956) [6], we know that solvability of P=L+c on C∞(X), for c∈C∞(X,C), implies that: (a) X is L-convex. Also, it follows: (b) a non-resonance condition for the jet-solvability at x0.In a previous paper, in addition to (a) and (b), the authors showed that P is globally solvable on C∞ if we assume: (c) a non-resonance condition in order to linearize L near x0; that (d) the only relatively compact orbit of L is {x0}; and that (e) c is real.Here we obtain the same conclusion without (c) and (e).

Globally solvable systems of complex vector fields

We consider a class of involutive systems of n smooth vector fields on the n+1 dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

A majorant criterion for the total preservation of global solvability of controlled functional operator equation

We prove the unique solvability of a nonlinear controlled functional operator equation in a Banach ideal space. We also establish sufficient conditions for the global solvability of all controls from a pointwise bounded set, provided that some majorant equation for the given family of these controls is globally solvable. We give examples of controlled boundary value problems reducible to the considered equation.

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C ∞ ( X ) , for an open subset X of R n . Let P = L + c be a linear partial differential operator with real coefficients on a C ∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C ∞ ( X ) . Based on Harvey–Treves's result we prove sufficient conditions for the global solvability of P on C ∞ ( X ) , in the spirit of geometrical Duistermaat–Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.

Global Solvability of Nonlinear Diffusion Equations with Forcing at the Boundary

Under the influence of a sufficiently “weak” nonlinear source term, it is by now well known that a degenerate diffusion equation is globally solvable. A similar result is known when the nonlinear source is present as a forcing term at the boundary. Such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle. However, these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present. In this work, we obtain the global existence of solutions for such problems by deriving a differential inequality involving theLr+1norms of solutions. The result is similar to that described above for equations with a nonlinear source term and, furthermore, establishes the global existence of “small” solutions even for “strong” forcings. The technique also applies to a more general class of problems involving nonlinear reaction, diffusion, and convection.

Global solvability of the laplacians on pseudo-Riemannian symmetric spaces

Let G be a noncompact semisimple Lie group with finite center and H an open subgroup of the fixed point group of an involution of G. G H becomes a pseudo-Riemannian manifold. We prove that the Laplacian P on G H is globally solvable in the sense that PC ∞( G H ) = C ∞( G H ) . This generalizes the global solvability of the Casimir operators on non-compact semisimple Lie groups with finite center due to J. Rauch and D. Wigner.

Globally hypoelliptic and globally solvable first-order evolution equations

We consider global hypoellipticity and global solvability of abstract first order evolution equations defined either on an interval or in the unit circle, and prove that it is equivalent to certain conditions bearing on the total symbol. We relate this to known results about hypoelliptic vector fields on the 2-torus.

Biinvariant operators on nilpotent Lie groups

The purpose of this article is to prove P-convexity for biinvariant differential operators on connected simply connected nilpotent Lie groups. More precisely, we show that for any compact subset K of a connected simply connected nilpotent Lie group N, and for any non-zero biinvariant differential operator P on N, there is a compact subset L ~ K with the property that whenever the support of Pu is contained in L for a C OO function of compact support u on N, then the support of u is contained in L. I am grateful to M. Duflo, to A. Cerezo, and to F. Rouvi6re for several helpful discussions. Solubility properties of biinvariant operators have been considered by several authors. S. Helgason 1,6] proves local solvability of biinvariant operators on semisimple Lie groups. Rais 1,8] proves the existence of a fundamental solution for a biinvariant operator on a connected simply connected nilpotent Lie group. Duflo and Rais 1,4] prove the local solvability of biinvariant operators on a solvable Lie group and Rouvi6re [9] proves semi-global solvability for biinvariant operators on simply connected solvable groups. Finally, Duflo 1,3] proves local solvability of biinvariant operators on any Lie group whatsoever. Semi-global solvability is in general false even for noncompact simple groups as was demonstrated by A. Cerezo and F. Rouvi6re 1,2]. Finally, even local solvability of left invariant operators is frequently false as was shown by L. Hormander, c.f. 1,6-1 and independently by A. Cerezo and F. Rouvi6re I-1]. From our result and that of Rais 1,8-1 or Rouvi6re 1,9-1, we conclude the global solvability of biinvariant operators on simply connected nilpotent Lie groups, i.e. that for any C oO function f and nonzero biinvariant operator P on a simply connected nilpotent Lie group N, there exists a Coo function u on N such that Pu =f. For simply connected abelian Lie groups, this reduces to the theorem of Malgrange and Ehrenpreis that constant coefficient differential operators on R n are globally solvable, c.f. 1,11-1. Thus our Theorem2 can be regarded as a generalization of the Malgrange-Ehrenpreis theorem. Henceforward N will denote a connected simply connected nilpotent Lie group, and 9l its Lie algebra. We write exp: 91--*N for the exponential map of 91

Global Solvability of the Casimir Operator

Let G be a semisimple Lie group, g its Lie algebra. The Killing form on g is a nondegenerate symmetric bilinear form which induces an isomorphism a: g* -. g. The Casimir operator is the unique biinvariant differential operator on G which annihilates constants and whose principal symbol is the pullback of the Killing form under the isomorphism a. Let C denote the Casimir operator and C-(G) the space of infinitely differentiable functions on G. THEOREM. If G is a non-compact semisimple Lie group with finite center, f e C-(G), then there is a u e C-(G) such that Cu = f. Since G is not compact, C is not elliptic but ultrahyperbolic. Analogous questions have been considered by S. Helgason [3] and A. Cerezo and F. Rouviere [1]. Helgason proves local solvability of all biinvariant differential operators and Cerezo and Rouviere prove global solvability of the Casimir operator on complex semisimple Lie groups. The latter authors prove that the imaginary part of the complex Casimir operator on SL(2, C) is not globally solvable. This shows that not every biinvariant operator is globally solvable. Henceforward G will denote a group satisfying the hypotheses of the theorem and g will denote the Lie algebra of left invariant vector fields on G. Proof. The remainder of the paper is devoted to a proof of this theorem. In spite of its length the proof is simple in outline. A result from functional analysis and Hormander's propagation of singularities theorem can be combined to show that it suffices to demonstrate three things.