Nonlinear evolution PDEs in $\mathbb R^{+}\times \mathbb C^{d}$: existence and uniqueness of solutions, asymptotic and Borel summability properties

Costin, Ovidiu; Tanveer, S.

doi:10.1016/j.anihpc.2006.07.002

Cited by 17 publications

(19 citation statements)

References 15 publications

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“…Let (Ĥ,Ŝ)(k, p) be the solution to (17) provided by Theorem 2.1. Recall the definitions of (Ĥ,Ŝ) (a) and (Ĥ,Ŝ) (s) given by (7) and (8) and the functionals in (9) and (10). Now, let (Ĥ,Ŝ) (b) = (Ĥ,Ŝ) − (Ĥ,Ŝ) (a) .…”

Section: Improved Growth Estimates Based On Knowledge Of the Solutionmentioning

confidence: 99%

“…There has also been work in characterizing all small solutions for a generic system of ODEs [6] or difference equations [3]. There has been work on PDEs as well, starting with linear equations [19], [2] followed by general results for a class of nonlinear system of PDEs in complex sectors [7], [8]. A Borel based approach has also led to analysis of complex singularities for a specific PDE [9].…”

Section: Introductionmentioning

confidence: 99%

“…A Borel based approach has also led to analysis of complex singularities for a specific PDE [9]. Recent developments include Navier-Stokes initial value problem (see [13], [12], [11]). Recently [20], numerical schemes have been suggested for nonlinear PDEs, based on a Borel plane reformulation.…”

Section: Introductionmentioning

confidence: 99%

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Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

Rosenblatt

Tanveer

2015

Journal of Differential Equations

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Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields:We prove that an equivalent system of integral equations in the Borel variable p ∈ R + dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to (1) in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within the time interval of existence, the solution remains analytic with the same analyticity strip width as the initial data and forcing. We also determine conditions on the integral equation solution that improve the estimate for existence time.

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Section: Improved Growth Estimates Based On Knowledge Of the Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

Rosenblatt

Tanveer

2015

Journal of Differential Equations

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“…F a (β k ) will be our β k 1 f a , where f a = F ′ a . By (14) we have the following straightforward estimate 7 . We first estimate the terms in the sum and the sum itself.…”

Section: Lemma 40 Zfdc Does Not Prove the Existence Of A Set Of Realmentioning

confidence: 99%

Foundational aspects of singular integrals

Costin

Friedman

2014

Journal of Functional Analysis

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We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-L 1 element in the class, and the Hadamard finite part integral (p.f.) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs.The Lebesgue integral as well as p.f., starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if f = g on (0, a), then their integrals from 0 to x coincide for any x ∈ (0, a).We find that integrals from zero with the essential properties of p.f., plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on (0, 1] which are L 1 ((ε, 1]) for all ε > 0. However, this existence proof does not provide a satisfactory construction. Without some regularity at 0, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-L 1 element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator.Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze p.f. on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.

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“…Borel summation has been used in other contexts including nonlinear ODEs [27], [6] as well as difference equations. It has also been used in obtaining PDE solution in parts of the complex plane [19]. We will not emphasize the theoretical aspects of Borel summation, beyond pointing out that it is limited to analytic initial data and forcing.…”

Section: Introductionmentioning

confidence: 99%

An integral equation approach to smooth 3D Navier–Stokes solution

Costin¹,

Luo²,

Tanveer³

2008

Phys. Scr.

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We summarize a recently developed integral equation approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3-D Navier-Stokes equation in the context of a periodic box problem with smooth time-independent forcing and initial condition v 0. Using an inverse Laplace transform ofv(k, t) −v 0 in 1/t, we arrive at an integral equation forÛ (k, p), where p is inverse-Laplace dual to 1/t and k is the Fourier variable dual to x. The advantage of this formulation is that the solutionÛ to the integral equation is known to exist a priori for p ∈ R + and the solution is integrable and exponentially bounded at ∞. Global existence of NS solution in this formulation is reduced to an asymptotics question. If Û (., p) l 1 (Z 3 has subexponential bounds as p → ∞, then global existence to NS follows. Moreover, if f = 0, then the converse is also true in the following sense: if NS has global solution, then there exists n ≥ 1 for which the the inverse Laplace transform ofv(k, t) −v 0 in 1/t n necessarily decays as q → ∞, where q is the inverse-Laplace dual to 1/t n. We also present refined estimates of the exponential growth when the solu-tionÛ is known on a finite interval [0, p 0 ]. We also show that for analytic v [0] and f , with finitely many nonzero Fourier-coefficients, the series forÛ (k, p) in powers of p has a radius of convergence independent of initial condition and forcing; indeed the radius gets bigger for smaller viscosity. We also show that the the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations for Kida ([14]) initial conditions, though far from being optimized, and performed on a modest interval in the accelerated variable q show decay in q.

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Nonlinear evolution PDEs in $\mathbb R^{+}\times \mathbb C^{d}$: existence and uniqueness of solutions, asymptotic and Borel summability properties

Abstract: We consider a system of n-th order nonlinear quasilinear partial differential equations of the form

Cited by 17 publications

References 15 publications

Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

Foundational aspects of singular integrals

An integral equation approach to smooth 3D Navier–Stokes solution

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