Статті в журналах з теми "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.

<abstract><p>We consider the dynamic behavior of solutions for a nonclassical diffusion equation with memory</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{t}-\varepsilon(t) \triangle u_{t}- \triangle u-\int_{0}^{\infty}\kappa(s)\triangle u(t-s)ds+f(u) = g(x) $\end{document} </tex-math></disp-formula></p> <p>on time-dependent space for which the norm of the space depends on the time $ t $ explicitly, and the nonlinear term satisfies the critical growth condition. First, based on the classical Faedo-Galerkin method, we obtain the well-posedness of the solution for the equation. Then, by using the contractive function method and establishing some delicate estimates along the trajectory of the solutions on the time-dependent space, we prove the existence of the time-dependent global attractor for the problem. Due to very general assumptions on memory kernel $ \kappa $ and the effect of time-dependent coefficient $ \varepsilon(t) $, our result will include and generalize the existing results of such equations with constant coefficients. It is worth noting that the nonlinear term cannot be treated by the common decomposition techniques, and this paper overcomes the difficulty by dealing with it as a whole.</p></abstract>

Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analysing effective numerical methods which fully exploit these properties and, in turn, are immune to the growth in dimensionality.Part I of this article studies the smoothness and approximability of the solution map, that is, the map $a\mapsto u(a)$, where $a$ is the parameter value and $u(a)$ is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic, in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of $n$-term approximations to the solution map, for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best $n$-term approximation, sparsity, and $n$-widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms.Part II of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low-dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

We study non-negative solutions to the chemotaxis system [Formula: see text] under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞)2 with values in [0, ∞) and ℝ2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (⋆), in the special case [Formula: see text] reducing to a version of the standard Keller–Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by [Formula: see text], reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u0, v0) fulfilling a smallness condition on the norm of v0 in L∞(Ω), the corresponding initial-boundary value problem associated with (⋆) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo–Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) ≡ (μ, κ) in the large time limit, with μ := fΩu0 and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

Abstract We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

AbstractWe study the nonlinear elliptic problem-Δu = Fʹ (u) in ℝwhere F(u) is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist α ∈ ℝ(*) |u - α · x| ≤ C.He also showed the existence of solutions with any prescribed α ∈ ℝ

AbstractUnder mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

Abstract. In this paper, we propose a C 0 interior penalty method for m th-Laplace equation on bounded Lipschitz polyhedral domain in R d , where m and d can be any positive integers. The standard H 1 -conforming piecewise r -th order polynomial space is used to approximate the exact solution u , where r can be any integer greater than or equal to m . Unlike the interior penalty method in [T. Gudi and M. Neilan, An interior penalty method for a sixth-order elliptic equation , IMA J. Numer. Anal., 31(4) (2011), pp. 1734–1753], we avoid computing D m of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete H m -norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete H m -norm. Numerical experiments validate our theoretical estimate.

AbstractWe study the homogeneous Dirichlet problem for the evolution p(x, t)-Laplacian with the nonlinear source $$\begin{aligned} u_t-{\text {div}}\left( |\nabla u|^{p(x,t)-2}\nabla u\right) =f(x,t,u),\quad (x,t)\in Q=\Omega \times (0,T). \end{aligned}$$ u t - div | ∇ u | p ( x , t ) - 2 ∇ u = f ( x , t , u ) , ( x , t ) ∈ Q = Ω × ( 0 , T ) . Here, $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n is a bounded domain, $$n\ge 2$$ n ≥ 2 , and $$p(x,\!t)$$ p ( x , t ) is a given function $$p(\cdot ):Q\mapsto (\frac{2n}{n+2},p^+]$$ p ( · ) : Q ↦ ( 2 n n + 2 , p + ] , $$p^+<\infty $$ p + < ∞ . It is shown that the solution is stable with respect to perturbations of the exponent p(x, t), the nonlinear source f(x, t, u), and the initial datum. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the exponent p(x, t) and the data u(x, 0), f. Estimates on the rate of convergence of solutions of perturbed problems to the solution of the limit problem are derived.

Abstract We consider the growth of the convex viscosity solution of the Monge–Ampère equation $\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary $\{x_n=0\}$ and there exists some $\varepsilon>0$ such that $u=O(|x|^{3-\varepsilon })$ at infinity, then $u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations269(1) (2020), 326–348].

AbstractOn a complete Riemannian manifold (M, g), we consider $$L^{p}_{loc}$$ L loc p distributional solutions of the differential inequality $$-\Delta u + \lambda u \ge 0$$ - Δ u + λ u ≥ 0 with $$\lambda >0$$ λ > 0 a locally bounded function that may decay to 0 at infinity. Under suitable growth conditions on the $$L^{p}$$ L p norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized $$L^{p}$$ L p -preservation property that can be read as a Liouville-type property for nonnegative subsolutiuons of the equation $$\Delta u \ge \lambda u$$ Δ u ≥ λ u . An application of the analytic results to $$L^{p}$$ L p growth estimates of the extrinsic distance of complete minimal submanifolds is also given.

AbstractIn this paper, we consider the nonclassical diffusion equation with memory and the nonlinearity of the polynomial growth condition of arbitrary order in the time-dependent space. First, the well-posedness of the solution for the equation is obtained in the time-dependent space $\mathscr{U}_{t}$ U t . Then, we establish the existence and regularity of the time-dependent global attractor. Finally, we also conclude that the fractal dimension of the time-dependent attractor is finite.

Abstract This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the m m -order gradients of weak solution to a higher-order elliptic equation with p p -growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the L γ , q {L}^{\gamma ,q} -estimate of D m u {D}^{m}u , while the coefficient satisfies a small BMO semi-norm and the boundary of underlying domain is flat in the sense of Reifenberg. In particular, a tricky ingredient is to establish the normal component of higher derivatives controlled by the horizontal component of higher derivatives of solutions in the neighborhood at any boundary point, which is achieved by comparing the solution under consideration with that for some reference problems.

AbstractThis paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $\mathbb{R}^{n}$ R n . Since the corresponding equation includes the degenerate term $\operatorname{div}\{a(x)\nabla u\}$ div { a ( x ) ∇ u } , it requires us to give appropriate assumptions about the weight function $a(x)$ a ( x ) for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ p − 1 ($p\geq 2$ p ≥ 2 ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.

Abstract In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {\int }_{{{\mathbb{R}}}^{N}}| u{| }^{r}{\rm{d}}x=c for given c > 0 . c\gt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.

AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

<p style='text-indent:20px;'>In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping <inline-formula><tex-math id="M1">\begin{document}$ |u|^{r-1}u, r\in[1, \infty) $\end{document}</tex-math></inline-formula> in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any <inline-formula><tex-math id="M2">\begin{document}$ r\geq 1, $\end{document}</tex-math></inline-formula> the existence and uniqueness of a weak solution is discussed when the domain <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is periodic/bounded in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb R^3 $\end{document}</tex-math></inline-formula> while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.</p>

<p style='text-indent:20px;'>We consider the conserved phase-field system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE111"> \begin{document}$\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula> is a relaxation time, <inline-formula><tex-math id="M2">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> is the viscosity parameter, <inline-formula><tex-math id="M3">\begin{document}$ \epsilon\in (0,1] $\end{document}</tex-math></inline-formula> is the heat capacity, <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> is the order parameter, <inline-formula><tex-math id="M5">\begin{document}$ u $\end{document}</tex-math></inline-formula> is the absolute temperature, the Laplace operator <inline-formula><tex-math id="M6">\begin{document}$ N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega) $\end{document}</tex-math></inline-formula> is subject to either Neumann boundary conditions (in which case <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset{\mathbb R}^d $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary) or periodic boundary conditions (in which case <inline-formula><tex-math id="M8">\begin{document}$ \Omega = \Pi_{i = 1}^d(0,L_i), $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M9">\begin{document}$ L_i&gt;0 $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M10">\begin{document}$ d = 1,2 $\end{document}</tex-math></inline-formula> or 3, and <inline-formula><tex-math id="M11">\begin{document}$ G(\phi) = \int_0^\phi g(\sigma)d\sigma $\end{document}</tex-math></inline-formula> is a double-well potential. Let <inline-formula><tex-math id="M12">\begin{document}$ j = 1 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M13">\begin{document}$ d = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ j = 2 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> or 3. We assume that <inline-formula><tex-math id="M16">\begin{document}$ g\in{\mathcal C}^{j+1}(\mathbb R) $\end{document}</tex-math></inline-formula> and satisfies the conditions <inline-formula><tex-math id="M17">\begin{document}$ g'(\phi)\geq -{\mathscr C}_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ G(\phi)\ge -{\mathscr C}_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ (\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi)) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M20">\begin{document}$ {\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M21">\begin{document}$ l = 3,4 $\end{document}</tex-math></inline-formula>, whenever <inline-formula><tex-math id="M22">\begin{document}$ |m(\phi)|\le \varrho $\end{document}</tex-math></inline-formula>), where <inline-formula><tex-math id="M23">\begin{document}$ \varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M24">\begin{document}$ {\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M25">\begin{document}$ m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx $\end{document}</tex-math></inline-formula>. For instance, <inline-formula><tex-math id="M26">\begin{document}$ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M27">\begin{document}$ p\in{\mathbb N}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M28">\begin{document}$ p\ge 2, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M29">\begin{document}$ a_{2p-1}&gt;0, $\end{document}</tex-math></inline-formula> satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space <inline-formula><tex-math id="M30">\begin{document}$ {\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2}) $\end{document}</tex-math></inline-formula> equipped with the norm <inline-formula><tex-math id="M31">\begin{document}$ \|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2} $\end{document}</tex-math></inline-formula>. Moreover, we demonstrate that the global attractor is upper semicontinuous at <inline-formula><tex-math id="M32">\begin{document}$ \epsilon = 0 $\end{document}</tex-math></inline-formula> in the metric induced by the norm <inline-formula><tex-math id="M33">\begin{document}$ \|.\|_{{\mathcal V}_{j+1}} $\end{document}</tex-math></inline-formula>. In addition, the exponential attractors are proven to be Hölder continuous at <inline-formula><tex-math id="M34">\begin{document}$ \epsilon = 0 $\end{document}</tex-math></inline-formula> in the metric induced by the norm <inline-formula><tex-math id="M35">\begin{document}$ \|.\|_{{\mathcal V}_{j}} $\end{document}</tex-math></inline-formula>. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition <inline-formula><tex-math id="M36">\begin{document}$ |g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right), $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M37">\begin{document}$ {\mathscr C}_7&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M38">\begin{document}$ p&gt;0 $\end{document}</tex-math></inline-formula> is arbitrary when <inline-formula><tex-math id="M39">\begin{document}$ d = 1, 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M40">\begin{document}$ p\in [0,3] $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M41">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, was required, preventing <inline-formula><tex-math id="M42">\begin{document}$ g $\end{document}</tex-math></inline-formula> to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.</p>

Introduction The decline of marriage in the West has been extensively researched over the last three decades (Carmichael and Whittaker; de Vaus; Coontz; Beck-Gernshein). Indeed, it was fears that the institution would be further eroded by the legalisation of same sex unions internationally that provided the impetus for the Australian government to amend the Marriage Act (1961). These amendments in 2004 sought to strengthen marriage by explicitly defining, for the first time, marriage as a legal partnership between one man and one woman. The subsequent heated debates over the discriminatory nature of this definition have been illuminating, particularly in the way they have highlighted the ongoing social significance of marriage, even at a time it is seen to be in decline. Demographic research about partnering practices (Carmichael and Whittaker; Simons; Parker; Penman) indicates that contemporary marriages are more temporary, fragile and uncertain than in previous generations. Modern marriages are now less about a permanent and “inescapable” union between a dominant man and a submissive female for the purposes of authorised sex, legal progeny and financial security, and more about a commitment between two social equals for the mutual exchange of affection and companionship (Croome). Less research is available, however, about how couples themselves reconcile the inherited constructions of romantic love as selfless and unending, with trends that clearly indicate that romantic love is not forever, ideal or exclusive. Civil marriage ceremonies provide one source of data about representations of love. Civil unions constituted almost 70 per cent of all marriages in Australia in 2010, according to the Australian Bureau of Statistics. The civil marriage ceremony has both a legal and symbolic role. It is a legal contract insofar as it prescribes a legal arrangement with certain rights and responsibilities between two consenting adults and outlines an expectation that marriage is voluntarily entered into for life. The ceremony is also a public ritual that requires couples to take what are usually private feelings for each other and turn them into a public performance as a way of legitimating their relationship. Consistent with the conventions of performance, couples generally customise the rest of the ceremony by telling the story of their courtship, and in so doing they often draw upon the language and imagery of the Western Romantic tradition to convey the personal meaning and social significance of their decision. This paper explores how couples construct the idea of love in their relationship, first by examining the western history of romantic love and then by looking at how this discourse is invoked by Australians in the course of developing civil marriage ceremonies in collaboration with the author. A History of Romantic Love There are many definitions of romantic love, but all share similar elements including an intense emotional and physical attraction, an idealisation of each other, and a desire for an enduring and unending commitment that can overcome all obstacles (Gottschall and Nordlund; Janowiak and Fischer). Romantic love has historically been associated with heightened passions and intense almost irrational or adolescent feelings. Charles Lindholm’s list of clichés that accompany the idea of romantic love include: “love is blind, love overwhelms, a life without love is not worth living, marriage should be for love alone and anything less is worthless and a sham” (5). These elements, which invoke love as sacred, unending and unique, perpetuate past cultural associations of the term. Romantic love was first documented in Ancient Rome where intense feelings were seen as highly suspect and a threat to the stability of the family, which was the primary economic, social and political unit. Roman historian Plutarch viewed romantic love based upon strong personal attraction as disruptive to the family, and he expressed a fear that romantic love would become the norm for Romans (Lantz 352). During the Middle Ages romantic love emerged as courtly love and, once again, the conventions that shaped its expression grew out of an effort to control excessive emotions and sublimate sexual desire, which were seen as threats to social stability. Courtly love, according to Marilyn Yalom, was seen as an “irresistible and inexhaustible passion; a fatal love that overcomes suffering and even death” (66). Feudal social structures had grounded marriage in property, while the Catholic Church had declared marriage a sacrament and a ceremony through which God’s grace could be obtained. In this context courtly love emerged as a way of dealing with the conflict between the individual and family choices over the martial partner. Courtly love is about a pure ideal of love in which the knight serves his unattainable lady, and, by carrying out feats in her honour, reaches spiritual perfection. The focus on the aesthetic ideal was a way to fulfil male and female emotional needs outside of marriage, while avoiding adultery. Romantic love re-appeared again in the mid-eighteenth century, but this time it was associated with marriage. Intellectuals and writers led the trend normalising romantic love in marriage as a reaction to the Enlightenment’s valorisation of reason, science and materialism over emotion. Romantics objected to the pragmatism and functionality induced by industrialisation, which they felt destroyed the idea of the mysterious and transcendental nature of love, which could operate as a form of secular salvation. Love could not be bought or sold, argued the Romantics, “it is mysterious, true and deep, spontaneous and compelling” (Lindholm 5). Romantic love also emerged as an expression of the personal autonomy and individualisation that accompanied the rise of industrial society. As Lanz suggests, romantic love was part of the critical reflexivity of the Enlightenment and a growing belief that individuals could find self actualisation through the expression and expansion of their “emotional and intellectual capacities in union with another” (354). Thus it was romantic love, which privileges the feelings and wishes of an individual in mate selection, that came to be seen as a bid for freedom by the offspring of the growing middle classes coerced into marriage for financial or property reasons. Throughout the 19th century romantic love was seen as a solution to the dehumanising forces of industrialisation and urbanisation. The growth of the competitive workplace—which required men to operate in a restrained and rational manner—saw an increase in the search for emotional support and intimacy within the domestic domain. It has been argued that “love was the central preoccupation of middle class men from the 1830s until the end of the 19th century” (Stearns and Knapp 771). However, the idealisation of the aesthetic and purity of love impacted marriage relations by casting the wife as pure and marital sex as a duty. As a result, husbands pursued sexual and romantic relationships outside marriage. It should be noted that even though love became cemented as the basis for marriage in the 19th century, romantic love was still viewed suspiciously by religious groups who saw strong affection between couples as an erosion of the fundamental role of the husband in disciplining his wife. During the late 19th and early 20th centuries romantic love was further impacted by urbanisation and migration, which undermined the emotional support provided by extended families. According to Stephanie Coontz, it was the growing independence and mobility of couples that saw romantic love in marriage consolidated as the place in which an individual’s emotional and social needs could be fully satisfied. Coontz says that the idea that women could only be fulfilled through marriage, and that men needed women to organise their social life, reached its heights in the 1950s (25-30). Changes occurred to the structure of marriage in the 1960s when control over fertility meant that sex was available outside of marriage. Education, equality and feminism also saw women reject marriage as their only option for fulfilment. Changes to Family Law Acts in western jurisdictions in the 1970s provided for no-fault divorce, and as divorce lost its stigma it became acceptable for women to leave failing marriages. These social shifts removed institutional controls on marriage and uncoupled the original sexual, emotional and financial benefits packaged into marriage. The resulting individualisation of personal lifestyle choices for men and women disrupted romantic conventions, and according to James Dowd romantic love came to be seen as an “investment” in the “future” that must be “approached carefully and rationally” (552). It therefore became increasingly difficult to sustain the idea of love as a powerful, mysterious and divine force beyond reason. Methodology In seeking to understand how contemporary partnering practices are reconstituting romantic love, I draw upon anecdotal data gathered over a nine-year period from my experiences as a marriage celebrant. In the course of personalising marriage ceremonies, I pose a series of questions designed to assist couples to explain the significance of their relationship. I generally ask brides and grooms why they love their fiancé, why they want to legalise their relationship, what they most treasure about their partner, and how their lives have been changed by their relationship. These questions help couples to reflexively interrogate their own relationship, and by talking about their commitment in concrete terms, they produce the images and descriptions that can be used to describe for guests the internal motivations and sentiments that have led to their decision to marry. I have had couples, when prompted to explain how they know the other person loves them say, in effect: “I know that he loves me because he brings me a cup of coffee every morning” or “I know that she loves me because she takes care of me so well.” These responses are grounded in a realism that helps to convey a sense of sincerity and authenticity about the relationship to the couple’s guests. This realism also helps to address the cynicism about the plausibility of enduring love. The brides and grooms in this sample of 300 couples were a socially, culturally and economically diverse group, and they provided a wide variety of responses ranging from deeply nuanced insights into the nature of their relationship, to admissions that their feelings were so private and deeply felt that words were insufficient to convey their significance. Reoccurring themes, however, emerged across the cases, and it is evident that even as marriage partnerships may be entered into for a variety of reasons, romantic love remains the mechanism by which couples talk of their feelings for each other. Australian Love and Marriage Australians' attitudes to romantic love and marriage have, understandably, been shaped by western understandings of romantic love. It is evident, however, that the demands of late modern capitalist society, with its increased literacy, economic independence and sexual equality between men and women, have produced marriage as a negotiable contract between social equals. For some, like Carol Pateman, this sense of equality within marriage may be illusory. Nonetheless, the drive for individual self-fulfilment by both the bride and groom produces a raft of challenges to traditional ideas of marriage as couples struggle to find a balance between independence and intimacy; between family and career; and between pursuing personal goals and the goals of their partners. This shift in the nature of marriage has implications for the “quest for undying romantic love,” which according to Anthony Giddens has been replaced by other forms of relationship, "each entered into for its own sake, for what can be derived by each person from a sustained association with another; and which is continued only in so far as it is thought by both parties to deliver enough satisfactions for each individual to stay within it” (qtd. in Lindholm 6). The impact of these social changes on the nature of romantic love in marriage is evident in how couples talk about their relationship in the course of preparing a ceremony. Many couples describe the person they are marrying as their best friend, and friendship is central to their commitment. This description supports research by V.K. Oppenheimer which indicates that many contemporary couples have a more “egalitarian collaborative approach to marriage” (qtd. in Carmichael and Whittaker 25). It is also standard for couples to note in ceremonies that they make each other happy and contented, with many commenting upon how their partners have helped to bring focus and perspective to their work-oriented lives. These comments tend to invoke marriage as a refuge from the isolation, competition, and dehumanising elements of workplaces. Since emotional support is central to the marriage contract, it is not surprising that care for each other is another reoccurring theme in ceremonies. Many brides and grooms not only explicitly say they are well taken care of by their partner, but also express admiration for their partner’s treatment of their families and friends. This behaviour appears to be seen as an indicator of the individual’s capacity for support and commitment to family values. Many couples admire partner’s kindness, generosity and level of personal self-sacrifice in maintaining the relationship. It is also not uncommon for brides and grooms to say they have been changed by their love: become kinder, more considerate and more tolerant. Honesty, communication skills and persistence are also attributes that are valued. Brides and grooms who have strong communication skills are also praised. This may refer to interpersonal competency and the willingness to acquire the skills necessary to negotiate the endless compromises in contemporary marriage now that individualisation has undermined established rules, rituals and roles. Persistence and the ability not to be discouraged by setbacks is also a reoccurring theme, and this connects with the idea that marriage is work. Many couples promise to grow together in their marriage and to both take responsibility for the health of their relationship. This promise implies awareness that marriage is not the fantasy of happily ever after produced in romantic popular culture, but rather an arrangement that requires hard work and conscious commitment, particularly in building a union amidst many competing options and distractions. Many couples talk about their relationship in terms of companionship and shared interests, values and goals. It is also not uncommon for couples to say that they admire their partner for supporting them to achieve their life goals or for exposing them to a wider array of lifestyle choices and options like travel or study. These examples of interdependence appear to make explicit that couples still see marriage as a vehicle for personal freedom and self-realisation. The death of love is also alluded to in marriage ceremonies. Couples talk of failed past relationships, but these are produced positively as a mechanism that enables the couple to know that they have now found an enduring relationship. It is also evident that for many couples the decision to marry is seen as the formalisation of a preexisting commitment rather than the gateway to a new life. This is consistent with figures that show that 72 per cent of Australian couples chose to cohabit before marriage (Simons 48), and that cohabitation has become the “normative pathway to marriage” (Penman 26). References to children also feature in marriage ceremonies, and for the couples I have worked with marriage is generally seen as the pre-requisite for children. Couples also often talk about “being ready” for marriage. This seems to refer to being financially prepared. Robyn Parker citing the research of K. Edin concludes that for many modern couples “rushing into marriage before being ‘set’ is irresponsible—marrying well (in the sense of being well prepared) is the way to avoid divorce” (qtd. in Parker 81). From this overview of reoccurring themes in the production of Australian ceremonies it is clear that romantic love continues to be associated with marriage. However, couples describe a more grounded and companionable attachment. These more practical and personalised sentiments serve to meet both the public expectation that romantic love is a precondition for marriage, while also avoiding the production of romantic love in the ceremony as an empty cliché. Grounded descriptions of love reveal that attraction does not have to be overwhelming and unconquerable. Indeed, couples who have lived together and are intimately acquainted with each other’s habits and disposition, appear to be most comfortable expressing their commitment to each other in more temperate, but no less deeply felt, terms. Conclusion This paper has considered how brides and grooms constitute romantic love within the shifting partnering practices of contemporary Australia. It is evident “in the midst of significant social and economic change and at a time when individual rights and freedom of choice are important cultural values” marriage remains socially significant (Simons 50). This significance is partially conveyed through the language of romantic love, which, while freighted with an array of cultural and historical associations, remains the lingua franca of marriage, perhaps because as Roberto Unger observes, romantic love is “the most influential mode of moral vision in our culture” (qtd. in Lindholm 5). It is thus possible to conclude, that while marriage may be declining and becoming more fragile and impermanent, the institution remains important to couples in contemporary Australia. Moreover, the language and imagery of romantic love, which publicly conveys this importance, remains the primary mode of expressing care, affection and hope for a partnership, even though the changed partnering practices of late modern capitalist society have exposed the utopian quality of romantic love and produced a cynicism about the viability of its longevity. It is evident in the marriage ceremonies prepared by the author that while the language of romantic love has come to signify a broader range of more practical associations consistent with the individualised nature of modern marriage and demystification of romantic love, it also remains the best way to express what Dowd and Pallotta describe as a fundamental human “yearning for communion with and acceptance by another human being” (571). 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