Auswahl der wissenschaftlichen Literatur zum Thema „Polynomial growth of the norm of the solution u“

We consider a model of population dynamics whose mortality function is unbounded and the solution is not regular near the maximum age. A continuous-time discontinuous Galerkin method to approximate the solution is described and analyzed. Our results show that the scheme is convergent, in L∞(L2) norm, at the rate of r + 1/2 away from the maximum age and that it is convergent at the rate of l - 1/(2q) + α/2 in L2(L2) norm, near the maximum age, if u ∈ L2(Wl,2q), where q ≥ 1, 1/2 ≤ l ≤ r + 1, r is the degree of the polynomial of the approximation space, and α is the growth rate of the mortality function; this estimate is super-convergent near the maximum age. Strong stability of the scheme is shown.

For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.

"Let $P : \CI(M; E) \to \CI(M; F)$ be an order $\mu$ differential operator with coefficients $a$ and $P_k := P : H^{s_0 + k +\mu}(M; E) \to H^{s_0 + k}(M; F)$. We prove polynomial norm estimates for the solution $P_0^{-1}f$ of the form $$\|P_0^{-1}f\|_{H^{s_0 + k + \mu}(M; E)} \le C \sum_{q=0}^{k} \, \| P_0^{-1} \|^{q+1} \,\|a \|_{W^{|s_0|+k}}^{q} \, \| f \|_{H^{s_0 + k - q}},$$ (thus in higher order Sobolev spaces, which amounts also to a parametric regularity result). The assumptions are that $E, F \to M$ are Hermitian vector bundles and that $M$ is a complete manifold satisfying the Fr\'echet Finiteness Condition (FFC), which was introduced in (Kohr and Nistor, Annals of Global Analysis and Geometry, 2022). These estimates are useful for uncertainty quantification, since the coefficient $a$ can be regarded as a vector valued random variable. We use these results to prove integrability of the norm $\|P_k^{-1}f\|$ of the solution of $P_k u = f$ with respect to suitable Gaussian measures."

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓp summability of the (∥ψj∥L∞)j ≥ 1 implies the ℓp summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n− s of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L∞(U,V) or s = (1/p)−(1/2) in L2(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓp summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψj|)j ≥ 1. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, these results imply that for all 0 <p< 2, the ℓp summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψj∥L∞)j ≥ 1 is ℓq summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

This work proposes and investigates the existence and uniqueness of solutions of Riccati Fractional Differential Equation (RFDE) with constant coefficients using Banach’s fixed point theorem. This theorem is the uniqueness theorem of a fixed point on a contraction mapping of a norm space. Furthermore, the combined theorem of the Adomian Decomposition Method (ADM) and Kamal’s Integral Transform (KIT) is used to convert the solution of the Fractional Differential Equation (FDE) into an infinite polynomial series. In addition, the terms of an infinite polynomial series can be decomposed using ADM, which assumes that a function can be decomposed into an infinite polynomial series and nonlinear operators can be decomposed into an Adomian polynomial series. The final result of this study is to find a solution of the RFDE approach to the economic growth model with a quadratic cost function using the combined ADM and KIT. The results showed that the RFDE solution on the economic growth model using the combined ADM and KIT showed a very good performance. Furthermore, the numerical solution of RFDE on the economic growth model is presented at the end of this work.

Abstract For an entire function solution of generalized bi-axisymmetric potential equation, we obtain a relationship between the generalized growth characteristics and polynomial approximation errors in sup norm by using the general functions introduced by Seremeta [On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion, Amer. Math. Soc. Transl. 88 (1970), no. 2, 291–301].

We study the fourth order Kirchhoff equation Δ2u−(a+b∫Ω|∇u|2)γΔu=f(u) in Ω with −Δu>0 and u>0 in Ω, and Δu=u=0 on ∂Ω, where f(t)=α1tθ+λtq+μt+g(t) for t≥0, g has subcritical growth, α>0, λ>0, μ≥0, 0<θ<1, 0<q<1, γ≥0, a>0, b≥0. We use the Galerkin projection method to show the existence of solution under some boundedness restriction on α,λ,μ. In some cases we study the behavior of the norm of the solution u as λ→0 and as λ→∞. Similar issues are addressed for the equation (a+b∫Ω|∇u|2)γΔ2u−ϱΔu=f(u), ϱ≥0.

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

AbstractWe construct unbounded positive C2-solutions of the equation in (equipped with Euclidean metric go) such that K is bounded between two positive numbers in , the conformal metric is complete, and the volume growth of g can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on u, we obtain growth estimate on the -norm of the solution and show that it has slow decay.

A classical solution is considered for the Cauchy problem:(utt−Δu)t+utt−αΔu=f(x,t),x∈ℝ3,t>0;u(x,0)=f0(x),ut(x,0)=f1(x), andutt(x)=f2(x),x∈ℝ3, whereα=const,0<α<1. The above equation governs the propagation of time-dependent acoustic waves in a relaxing medium. A classical solution of this problem is obtained in the form of convolutions of the right-hand side and the initial data with the fundamental solution of the equation. Sharp time estimates are deduced for the solution in question which show polynomial growth for small times and exponential decay for large time whenf=0. They also show the time evolution of the solution whenf≠0.

Présentation du domaine: Il s'agit d'étudier des opérateurs différentiels sur des variétés riemanniennes singulières et leurs applications. Parmi les opérateurs les plus importants, on trouve les opérateurs de Laplace et de Dirac. Il y a beaucoup de connexions entre les deux types d'opérateurs, à cause de la formule de Lichnerowicz, un mathématicien français du dernier siècle. Pourtant, les opérateurs de Laplace ont été beaucoup plus étudiés que les opérateurs de Dirac. Les opérateurs de Dirac, aussi appelé opérateurs d'Atiyah--Singer, sont des opérateurs fondamentaux dans la géométrie riemannienne et dans la théorie de l'indice. Ce sont des opérateurs associés à une métrique et à un fibré de Clifford doté d'une connexion admissible. Leurs généralisations est l'objet principal dans la théorie de Kasparov, qui est un utile fondamental dans les algèbres d'opérateurs. Il y a beaucoup de gens qui pensent que les opérateurs de Dirac joueront un rôle central dans le programme de Grothendieck: généraliser le théorème de Riemann--Roch aux variétés algébriques singulières. Les opérateurs de Dirac ont donc été beaucoup étudiés dans les mathématiques fondamentales, ainsi que dans ces applications. Les opérateurs de Maxwell et de de Rham sont des cas particuliers des opérateurs de Dirac. Les opérateurs de Dirac apparaissent dans beaucoup d'applications dans d'autres domaines des mathématiques et physique théorique, comme la théorie des champs dans l'espace-temps courbe ou la théorie de la relativité générale. Ces opérateurs constituent donc un lien entre les mathématiques fondamentales et ces applications. Sujet de thèse: Il y a beaucoup de résultats sur l'analyse des opérateurs de Dirac, mais la plupart d'eux sont sur des variétés compactes lisses, avec ou sans bord. Cependant, il est important d'étudier ces opérateurs pour des variétés non compactes ou non lisses. Par exemple, les applications aux variétés algébriques et au programme de Grothendieck nécessitent le cas non lisse. Le sujet que nous proposons est d'utiliser les résultats et les techniques introduites par Monique Dauge et ses collaborateurs pour étudier les singularités des opérateurs de Dirac dans un domaine polyédrique et d'autres domaines singuliers. Un problème particulier est d'obtenir l'application au calcul de l'homologie de Rham avec des complexes finis, comme dans les travaux récents de Douglas Arnold. Pour la régularité des solutions de l'équation de Dirac, nous proposons d'utiliser les méthodes introduites récemment par Bernd Amman et Nadine Grosse ou par Victor Nistor et Nadine Grosse dans des articles récents. Nous allons étudier aussi les opérateurs de Dirac avec des potentiels et terms non linéaires. Un problème concret ici est d'étudier de modèles non linéaires couplés avec Maxwell, par exemple les modèles de magnéto-hydrodynamique et l'équation de Vlasov--Maxwell
General description of the domain. The general question that will be pursued as part of the thesis will be to study differential operators on Riemannian spaces and their applications. The Laplace and Dirac operators are among the most important differential operators arising in applications. There are many connections between these two types of operators, due to Lichnerowicz' formula, a French mathematician of the last century. However, Laplace operators have been much more studied than the Dirac operators. Dirac operators, also called Atiyah--Singer operators, are fundamental operators in Riemannian geometry and in index theory. These operators are associated to a metric and a Clifford bundle with an admissible connection. Their generalizations are the main object in Kasparov's theory, which is a fundamental theory in Operator Algebras. It is believed that the Dirac operators will play a central role in Grothendieck's program to generalize the Riemann--Roch theorem to singular algebraic varieties. Dirac operators have therefore been much studied in theoretical mathematics, as well as in its applications. The Maxwell and de Rham operators are special cases of Dirac operators. Dirac operators appear in many applications in other domains of mathematics and theoretical physics, such as field theory in curved space-time or the theory of general relativity. These operators are thus a link between the fundamental mathematics and its applications. Theses subject: There are many results on the analysis of the Dirac operators, but most of them are on smooth compact varieties, with or without boundary. However, it is important to study these operators for non-compact or non-smooth spaces (or varieties). For example, applications to algebraic varieties and to the Grothendieck program require the case of non smooth varieties. The subject we propose to use the results and the techniques introduced by Monique Dauge and her collaborators to study the singularities of the Dirac operators in a polyhedral domain and other singular domains. A particular problem is to obtain the application to the calculation of de Rham's homology with finite complexes, as in the recent works of Douglas Arnold. For the regularity of the solutions of the Dirac equation we propose to use the methods recently introduced by Bernd Amman and Nadine Grosse or by Victor Nistor and Nadine Grosse in recent articles. The thesis will also study the Dirac operators with nonlinear terms and potentials. A concrete problem here is to study nonlinear models coupled with Maxwell's equation, which arrise, for example in magneto-hydrodynamic models and in the Vlasov-Maxwell equation