View source for Gevrey class

An intermediate space between the spaces of smooth (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200401.png" />-) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see [[#References|[a13]]], in which regularity estimates of the heat kernel are deduced).

Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200403.png" />, the Gevrey class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200404.png" /> (of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200405.png" />) is defined as the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200406.png" /> such that for every compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200407.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200408.png" /> satisfying

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g1200409.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004010.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004011.png" /> one recovers the space of all real-analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004012.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004014.png" /> contains non-zero functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004015.png" /> being the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004016.png" />-functions with compact support. There are various equivalent ways to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004017.png" /> (cf. [[#References|[a27]]]). Introducing the natural inductive topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004018.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004019.png" />, one can define the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004020.png" /> of Gevrey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004022.png" />-ultra-distributions as the dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004023.png" />. The space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004024.png" />-ultra-distributions contains the Schwartz distributions (cf. also [[Generalized functions, space of|Generalized functions, space of]]). The Gevrey classes are the most simple case of classes of ultra-differentiable functions (or Denjoy–Carleman classes; see, e.g., [[#References|[a17]]]). Since the scale of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004025.png" /> starts from the analytic functions (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004026.png" />) and ends in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004027.png" />-category (setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004028.png" />), the Gevrey classes play an important role in various branches of partial and ordinary differential equations; namely, whenever the properties of certain differential operators (mappings) differ in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004029.png" /> and in the analytic category, it is natural to investigate the behaviour of such operators (mappings) in the scale of Gevrey classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004030.png" /> and, if possible, to find the critical value(s) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004031.png" />, i.e. those for which a change of behaviour occurs. In particular, all weak solutions to the [[Heat equation|heat equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004032.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004033.png" />, while they are not real-analytic. In the scale of Gevrey spaces, the result is sharp; namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004034.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004035.png" /> (and hence for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004036.png" />), but, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004038.png" />.

==Applications.==
The Gevrey classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004040.png" />, have numerous applications, a few of the main applications being listed below.

===Gevrey micro-local analysis.===
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004041.png" /> one says that a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004042.png" />-ultra-distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004043.png" /> is (micro-locally) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004046.png" />-regular at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004047.png" /> if there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004049.png" />, an open cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004051.png" />, and a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004052.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004053.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004055.png" /> denotes the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004057.png" />. This definition is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004058.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004059.png" /> wave front set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004061.png" /> is the smallest closed conic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004063.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004064.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004065.png" />-regular at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004066.png" />. Here, being a conic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004067.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004068.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004070.png" />. For equivalent definitions, see [[#References|[a14]]], [[#References|[a27]]].

Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004071.png" /></td> </tr></table>

be a linear partial differential operator (cf. also [[Linear partial differential equation|Linear partial differential equation]]), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004074.png" />. The presence of the imaginary unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004075.png" /> allows one to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004076.png" /> via the Fourier transform, namely

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004077.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004078.png" />. This definition is valid for pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]) as well, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004079.png" /> is a suitable symbol from the Hörmander classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004080.png" /> or from other classes (see [[#References|[a27]]] for more details and references). The characteristic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004081.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004082.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004083.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004084.png" /> standing for the principal symbol (cf. also [[Symbol of an operator|Symbol of an operator]]; [[Principal part of a differential operator|Principal part of a differential operator]]). The operator is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004086.png" />-hypo-elliptic (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004088.png" />-micro-locally hypo-elliptic) in an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004089.png" /> (respectively, in an open conic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004090.png" />) if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004091.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004092.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004093.png" />) necessarily <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004094.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004095.png" />).

Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004096.png" /> is called of principal type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004097.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004098.png" /> is not linearly dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g12004099.png" />. The operator is called of multiple characteristics type if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040100.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040101.png" />. The properties of operators of principal type are basically the same in the analytic-Gevrey category and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040102.png" />-category. An essential difference occurs in the case of multiple characteristics. For operators with constant multiple real or complex characteristics, modelled by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040104.png" /> being an operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040105.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040106.png" /> is a first-order operator modelled by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040107.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040109.png" />, the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040110.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040112.png" />, is governed by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040113.png" />, independently of the lower-order terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040114.png" />. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040115.png" />, then the lower-order terms affect both the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040116.png" />-hypo-ellipticity and the propagation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040117.png" /> singularities, cf. [[#References|[a5]]], [[#References|[a21]]], [[#References|[a27]]]. In fact, often one is interested in finding a critical index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040118.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040120.png" /> certain properties are complementary. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040122.png" /> is even, there are examples of operators analytic and Gevrey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040123.png" />-hypo-elliptic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040124.png" /> but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040125.png" />- and Gevrey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040126.png" />-hypo-elliptic for large values of the Gevrey index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040127.png" /> (see [[#References|[a21]]], [[#References|[a27]]] for more details and references).

As to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040128.png" />-hypo-ellipticity for operators of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040130.png" /> being analytic vector fields satisfying the Hörmander bracket hypothesis, a typical pattern of behaviour is the following one: There is a critical index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040131.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040132.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040133.png" />-hypo-ellipticity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040134.png" /> holds, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040135.png" /> it does not (cf. [[#References|[a2]]], [[#References|[a7]]], [[#References|[a23]]]).

Gevrey singularities appear in the study of initial-boundary value problems for hyperbolic equations in domains with analytic diffractive boundaries (cf. [[#References|[a18]]] and the references therein). In particular, for the [[Wave equation|wave equation]], Gevrey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040136.png" />-singularities along the diffractive analytic boundary appear, and this fact is used in scattering theory (cf. [[#References|[a1]]]).

===Gevrey solvability.===
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040137.png" /> is called (locally) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040138.png" />-solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040139.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040140.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040141.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040142.png" />.

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040143.png" />-solvability implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040144.png" />-solvability for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040145.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040146.png" /> is not solvable in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040147.png" />-category one looks for an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040148.png" /> such that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040149.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040150.png" />-solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040151.png" /> and not for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040152.png" />. The model operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040153.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040154.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040156.png" /> being even, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040157.png" />-solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040158.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040159.png" /> they need not be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040160.png" />-solvable (cf. [[#References|[a6]]], [[#References|[a21]]], [[#References|[a27]]]).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040161.png" />-solvability for semi-linear partial differential operators, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040162.png" />, is proved in [[#References|[a12]]].

===Hyperbolic equations.===
The Gevrey classes serve as a framework for the well-posedness of the [[Cauchy problem|Cauchy problem]] for weakly hyperbolic linear partial differential operators (cf. also [[Linear hyperbolic partial differential equation and system|Linear hyperbolic partial differential equation and system]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040163.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040164.png" /></td> </tr></table>

Weak hyperbolicity means that the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040165.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040166.png" /> are real. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040167.png" /> is the maximal multiplicity of the real roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040168.png" />, then the Cauchy problem is always well-posed in the framework of the Gevrey classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040169.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040170.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040171.png" />, one can point out specific lower-order terms such that the existence fails. More subtle estimates for the critical Gevrey index are obtained by using the distance between the roots or via additional restrictions on the lower-order terms (so-called Levi-type conditions). See [[#References|[a3]]], [[#References|[a14]]], [[#References|[a4]]], [[#References|[a15]]], [[#References|[a19]]], [[#References|[a24]]], [[#References|[a21]]] for more details and references. Local Gevrey well-posedness for weakly hyperbolic non-linear systems is shown in [[#References|[a16]]] (see also [[#References|[a12]]]). Goursat problems for Kirchoff-type equations in Banach spaces of Gevrey functions (cf. also [[Kirchhoff formula|Kirchhoff formula]]; [[Goursat problem|Goursat problem]]) have been studied in [[#References|[a10]]].

===Divergent series and singular differential equations.===
One may also define formal Gevrey spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040172.png" />, e.g. the set of all formal power series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040173.png" /></td> </tr></table>

where for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040174.png" /> the following estimates hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040175.png" /></td> </tr></table>

Such formal Gevrey spaces are used in the study of divergent series and singular ordinary linear differential equations with Gevrey coefficients (see [[#References|[a26]]] and the references therein). The Fredholm property in such type of Gevrey spaces of certain singular analytic partial differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040176.png" /> has been studied by means of Toeplitz operators (cf. [[#References|[a22]]]).

===Dynamical systems.===
The framework of Gevery classes is used in the study of normal forms of analytic perturbations of (non-) integrable (non-) Hamiltonian systems. Roughly speaking, one obtains normal forms modulo exponentially small error terms of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040177.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040178.png" /> is small parameter, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040179.png" /> is related to Gevrey-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040180.png" />-type estimates, or so-called Nekhoroshev-type estimates (see e.g. [[#References|[a11]]] for Gevrey normal forms of billiard ball mappings and [[#References|[a25]]] on normal forms of perturbations of Hamiltonian systems).

===Evolution partial differential equations.===
In the study of the analytic regularity of solutions of semi-linear evolution equations (Navier–Stokes, Kuramoto–Sivashinksi, Euler, the [[Ginzburg–Landau equation|Ginzburg–Landau equation]]) with periodic boundary data for positive time, the term "Gevrey class" is used usually to denote the [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040182.png" />, (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040183.png" /> being the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040184.png" />-dimensional torus) of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040185.png" /> with the norm defined by means of the discrete Fourier transform

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040186.png" /></td> </tr></table>

for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040187.png" />. In these applications, the Gevrey index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040188.png" /> (the analytic category). See [[#References|[a8]]] for the [[Navier–Stokes equations|Navier–Stokes equations]]; [[#References|[a9]]] for recent results on semi-linear parabolic partial differential equations; and [[#References|[a20]]] for a generalized [[Euler equation|Euler equation]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Bardos, G. Lebeau, J. Rauch, "Scattering frequencies and Gevrey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120040/g120040189.png" /> singularities" ''Invent. Math.'' , '''90''' (1987) pp. 77–114 {{MR|906580}} {{ZBL|0723.35058}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Bove, D. Tartakoff, "Optimal non-isotropic Gevrey exponents for sums of squares of vector fields" ''Commun. Partial Diff. Eq.'' , '''22''' (1997) pp. 1263–1282 {{MR|1466316}} {{ZBL|0921.35043}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M.D. Bronstein, "The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity" ''Trans. Moscow Math. Soc.'' , '''1''' (1982) pp. 87–103 (In Russian) {{MR|0611140}} {{MR|0427842}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Colombini, E. Janelli, S. Spagnolo, "Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coeficients depending on time" ''Ann. Scuola Norm. Sup. Pisa'' , '''10''' (1983) pp. 291–312</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Cattabriga, L. Rodino, L. Zanghirati, "Analytic-Gevrey hypoellipticity for a class of pseudodifferential operators with multiple characteristics" ''Commun. Partial Diff. Eq.'' , '''15''' (1990) pp. 81–96 {{MR|1032624}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Cicognani, L. Zanghirati, "On a class of unsolvable operators" ''Ann. Scuola Norm. Sup. Pisa'' , '''20''' (1993) pp. 357–369 {{MR|1256073}} {{ZBL|0816.47051}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Christ, "Intermediate Gevrey exponents occur" ''Commun. Partial Diff. Eq.'' , '''22''' (1997) pp. 359–379 {{MR|1443042}} {{ZBL|0893.35021}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> C. Foias, R. Temam, "Gevrey class regularity for the solutions of the Navier-Stokes equations" ''J. Funct. Anal.'' , '''87''' (1989) pp. 359–369 {{MR|1026858}} {{ZBL|0702.35203}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> A. Ferrari, E. Titi, "Gevrey regularity for nonlinear analytic parabolic equations" ''Commun. Partial Diff. Eq.'' , '''23''' (1998) pp. 1–16 {{MR|1608488}} {{ZBL|0907.35061}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Gourdain, M. Mechab, "Problème de Goursat non-linéaire dans les espaces de Gevrey pour les équations de Kirchoff généralisées" ''J. Math. Pures Appl.'' , '''75''' (1996) pp. 569–593</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> T. Gramchev, G. Popov, "Nekhoroshev type estimates for billiard ball maps" ''Ann. Inst. Fourier (Grenoble)'' , '''45''' : 3 (1995) pp. 859–895 {{MR|1340956}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> T. Gramchev, L. Rodino, "Gevrey solvability for semilinear partial differential equations with multiple characteritics" ''Boll. Un. Mat. Ital. Sez. B (8)'' , '''2''' : 1 (1999) pp. 65–120 {{MR|1666731}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> M. Gevrey, "Sur la nature analytique des solutions des équations aux dérivées partielles" ''Ann. Ecole Norm. Sup. Paris'' , '''35''' (1918) pp. 129–190 {{MR|1509208}} {{ZBL|46.0721.01}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> H. Chen, L. Rodino, "General theory of PDE and Gevrey classes" , ''General theory of partial differential equations and microlocal analysis (Trieste, 1995)'' , ''Pitman Res. Notes Math.'' , '''349''' , Longman (1996) pp. 6–81 {{MR|1429633}} {{ZBL|0864.35130}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> V.Ya. Ivrii, "Correctness in Gevrey classes of the Cauchy problem for certain nonstrictly hyperbolic operators" ''Izv. Vyš. Učebn. Zaved. Mat.'' , '''189''' (1978) pp. 26–35 (In Russian) {{MR|0499786}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> K. Kajitani, "Local solutions of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes" ''Hokkaido Math. J.'' , '''12''' (1983) pp. 434–460 {{MR|0725589}} {{ZBL|1138.35053}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> H. Komatsu, "Ultradistributions I–III" ''J. Fac. Sci. Univ. Tokyo Sec. IA Math.'' , '''19; 24; 29''' (1973/77/82) pp. 25–105; 607–628; 653–717</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> B. Lascar, R. Lascar, "Propagation des singularités Gevrey pour la diffraction" ''Commun. Partial Diff. Eq.'' , '''16''' (1991) pp. 547–584 {{MR|1189413}} {{MR|1113098}} {{ZBL|0734.35166}} {{ZBL|0728.35155}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> J. Leray, Y. Ohya, "Equations et systèmes non-linèaires, hyperboliques non-strictes" ''Math. Ann.'' , '''170''' (1967) pp. 167–205</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> C.D. Levermore, M. Oliver, "Analyticity of solutions for a generalized Euler equation" ''J. Diff. Eq.'' , '''133''' (1997) pp. 321–339 {{MR|1427856}} {{ZBL|0876.35090}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> M. Mascarello, L. Rodino, "Partial differential equations with multiple characteristics" , ''Math. Topics'' , '''13''' , Akad. (1997) {{MR|1608649}} {{ZBL|0888.35001}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> M. Miyake, M. Yoshino, "Fredholm property of partial differential opertors of irregular singular type" ''Ark. Mat.'' , '''33''' (1995) pp. 323–341</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> T. Matsuzawa, "Gevrey hypoellipticity for Grushin operators" ''Publ. Res. Inst. Math. Sci.'' , '''33''' (1997) pp. 775–799 {{MR|1607020}} {{ZBL|0912.35166}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> S. Mizohata, "On the Cauchy problem" , Acad. Press &amp;Sci. Press Beijing (1985) {{MR|0860041}} {{ZBL|0616.35002}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> J.-P. Ramis, R. Schäfke, "Gevrey separation of fast and slow variables" ''Nonlinearity'' , '''9''' (1996) pp. 353–384 {{MR|1384480}} {{ZBL|0925.70161}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> J.-P. Ramis, "Séries divergentes et théorie asymptotiques" ''Bull. Sci. Math. France'' , '''121''' (1993) (Panoramas et Syntheses, suppl.) {{MR|1272100}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> L. Rodino, "Linear partial differential operators in Gevrey spaces" , World Sci. (1993) {{MR|1249275}} {{ZBL|0869.35005}} </TD></TR></table>