A second-order Cauchy problem in a scale of Banach spaces and application to Kirchhoff equations

In a scale of Banach spaces we study the Cauchy problem for the equation u^′′=A(Bu(t),u), where A is a bilinear operator and B is a completely continuous operator. Obtained results are applied to prove existence of solutions in the Gevrey class for Kirchhoff equations.     

Uniform stability of solutions of Boltzmann equation for soft potential with external force

The study of Cauchy problem of the Boltzmann equation is important in both theory and applications. Existence of global solutions to the equation and uniform stability of solutions in the absence of external force were introduced in the previous work on the Boltzmann equation. In this paper, we will investigate the uniform stability of solutions in L¹ for the Cauchy problem of the Boltzmann equation when there is an external force for the case of soft potentials.     

The Boltzmann equation without angular cutoff in the whole space: I,Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.     

Sul problema di Cauchy per le equazioni a derivate parziali

Sunto Si considerano equazioni alle derivate parziali in più variabili indipendenti, lineari e non lineari, di ordine qualsiasi; si dimostrano teoremi di esistenza e unicità per il problema di Cauchy non caratteristico. Il metodo di questo lavoro consiste nel ricondurre il problema di Cauchy allo studio di una trasformazione in sé di uno spazio funzionale, strettamente connesso alle classi di Gevrey.

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Summary We consider linear and non linear partial differential equations of any order, in several independént variables. We prove some existence and uniqueness theorems for the non-characteristic Cauchy problem. Our method consist in the reduction of the Cauchy problem to a study of a mapping of a function space into itself; this function space is closely related to Gevrey classes.

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Questo lavoro fa parte dell'attività del gruppo di ricerca n. 23 del C.N.R.     

Über Anfangs-und Eigenwertprobleme aus der Neutronentransporttheorie

Based on the theory of semi-groups in Hilbert space, a proof is given for the existence of a unique solution of an abstract Cauchy problem arising in the transport theory of mono-energetic neutrons, corresponding to the time-dependent linear Boltzmann equation in the general three-dimensional geometry. The spectral properties of the Boltzmann operator are investigated, an explicit representation of the solution is obtained by the perturbation theory for semi-groups of linear operators and alternatively an expansion in a series of eigenfunctions is given.     

Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 838–852, June, 2007.     

Existence of local solutions for the Boltzmann equation without angular cutoff

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Almost global well-posedness of Kirchhoff equation with Gevrey data

The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains.     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

Spectrum structure and decay rate estimates on the Landau equation with Coulomb potential

In this paper we first deduce the estimates on the linearized Landau operator with Coulomb potential and then analyze its spectrum structure by using semigroup theory and linear operator perturbation theory. Based on these estimates, we give the precise time decay rate estimates on the semigroup generated by the linearized Landau operator so that the optimal time decay rates of the nonlinear Landau equation follow. In addition, we present a similar result for the non-angular cutoff Boltzmann equation with soft potentials.

     

Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand–Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.     

Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.     

Intorno alle classi funzionali di Gevrey

Sunto Si espongono teoremi, esempi ed osservazioni a proposito di certe-classi funzionali, analoghe a quelle di Gevrey, che si presentano nella teoria delle equazioni differenziali ipoellittiche, di alcune equazioni integrodifferenziali, e dei problemi di Cauchy.

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Summary This paper collect theorems, examples and observations concerning some classes of functions, at first considered by Gevrey, of interest in the theory of hypoelliptic differential equations, of some integrodifferential equations, and of Cauchy problems.

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Questo lavoro ha avuto origine da una tesi di laurea diretta dal prof.Carlo Pucci nell'anno accademico 1961–1962, e fa parte delle attività del gruppo di ricerca matematica n. 23 del C. N. R.     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Equivalent forms of Levi condition

In this paper we consider hyperbolic differential operators with characteristic roots of constant multiplicity and we prove the equivalence of some conditions, called Levi conditions, for the correctness of the Cauchy problem inC ^∞ and in Gevrey classes.

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Sommario In questo articolo prendiamo in considerazione operatori differenziali iperbolici con caratteristiche di molteplicità costante e dimostriamo l’equivalenza di alcune condizioni, note come condizioni di Levi, necessarie e sufficienti per la buona positura del problema di Cauchy nelle classiC ^∞ e Gevrey.

     

Maximum entropy principle closure for 14-moment system for a non-polytropic gas

In this paper, we consider a rarefied polyatomic gas with a non-polytropic equation of state. We use the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes. The closed partial differential system is symmetric hyperbolic and the Cauchy problem is well-posed. In the limiting case of polytropic gas in which the internal energy is a linear function of the temperature, we find, as a special case, the previous results of Pavić et al. (Physica A 392:1302–1317, 2013). This paper, therefore, completes the equivalence between the closure obtained in the phenomenological rational extended thermodynamics theory and the one obtained by the MEP for general non-polytropic gas.

     

On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order

We shall consider the Cauchy problem for weakly hyperbolic equations of higher order with coefficients depending only on time. The regularities of the distinct characteristic roots and the multiple characteristic roots independently influence Gevrey well posedness of the Cauchy problem.     

Gevrey Class Regularity of a Semigroup Associated with a Nonlinear Korteweg-de Vries Equation

In this paper, the authors consider the Gevrey class regularity of a semigroup associated with a nonlinear Korteweg-de Vries(Kd V for short) equation. By estimating the resolvent of the corresponding linear operator, the authors conclude that the semigroup generated by the linear operator is not analytic but of Gevrey class δ∈( 3/2, ∞) for t 0.     

On the Cauchy problem for non-effectively hyperbolic operators,the Gevrey 5 well-posedness

In this paper, we prove that for non-effectively hyperbolic operators with smooth double characteristics exhibiting a Jordan block of size 4 on the double manifold, the Cauchy problem is well-posed in the Gevrey 5 class, beyond the generic Gevrey class 2 (see, e.g., [5]). Moreover, we show that this value is optimal, due to certain geometric constraints on the Hamiltonian flow of the principal symbol. These results, together with results already proved, give a complete picture of the well-posedness of the Cauchy problem around hyperbolic double characteristics.     

Gevrey空间中抽象柯西-柯瓦列夫斯卡娅定理:能量方法

本文研究了非线性柯西问题的适定性问题.利用经典的能量法和抽象柯西-柯瓦列夫斯卡娅定理,得到非线性柯西问题在Gevrey空间中是适定的.推广了已有文献在非线性柯西问题适定性方面的研究.