Initialization of the Bootstrap Argument

This paper is devoted to understanding the global stability of perturbations near a background magnetic field of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. We establish the global stability for the solutions of the nonlinear MHD system by the bootstrap argument.

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LONG-TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR REACTION DIFFUSION EQUATIONS INVOLVING L1 DATA

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500071 ◽

2012 ◽

Vol 14 (01) ◽

pp. 1250007 ◽

Cited By ~ 5

Author(s):

CHENGKUI ZHONG ◽

WEISHENG NIU

Keyword(s):

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Time Behavior ◽

Long Time Behavior ◽

Smooth Bounded Domain ◽

Bootstrap Argument ◽

Nonlinear Reaction ◽

L1 Data ◽

Long Time

In this paper we consider the long-time behavior of solutions to nonlinear reaction diffusion equations involving L1 data, [Formula: see text] where Ω is a smooth bounded domain and u0, g ∈ L1(Ω). Using a decomposition technique combined with a bootstrap argument we establish some uniform regularity results on the solutions, by which we prove that the solution semigroup generated by the problem above possesses a global attractor [Formula: see text] in L1(Ω). Moreover, we obtain that the attractor is actually invariant, compact in [Formula: see text], q < max {N/(N-1), (2p-2)/p}, and attracts every bounded subset of L1(Ω) in the norm of [Formula: see text], 1 ≤ r < ∞.

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Global well-posedness of the Cauchy problem for the 3D Jordan–Moore–Gibson–Thompson equation

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500698 ◽

2020 ◽

pp. 2050069

Author(s):

Reinhard Racke ◽

Belkacem Said-Houari

Keyword(s):

Cauchy Problem ◽

Existence Result ◽

Local Existence ◽

Decay Rates ◽

Polynomial Decay ◽

Small Data ◽

Third Order ◽

Bootstrap Argument ◽

The Cauchy Problem ◽

Linear Decay

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained.

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Revisiting the characteristic initial value problem for the vacuum Einstein field equations

General Relativity and Gravitation ◽

10.1007/s10714-020-02747-2 ◽

2020 ◽

Vol 52 (10) ◽

Author(s):

David Hilditch ◽

Juan A. Valiente Kroon ◽

Peng Zhao

Keyword(s):

Initial Value Problem ◽

Existence Of Solutions ◽

Local Existence ◽

Einstein Field Equations ◽

Field Equations ◽

Initial Value ◽

Bootstrap Argument ◽

Null Hypersurfaces ◽

Characteristic Initial Value Problem ◽

Local Existence Of Solutions

AbstractUsing the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

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Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

Advances in Mathematical Physics ◽

10.1155/2011/238138 ◽

2011 ◽

Vol 2011 ◽

pp. 1-39

Author(s):

Kyoko Tomoeda

Keyword(s):

Kdv Equation ◽

Type Equation ◽

Smoothing Effect ◽

Initial Value ◽

Time And Space ◽

Third Order ◽

Bootstrap Argument ◽

Real Analytic ◽

Fifth Order ◽

Bourgain Space

We consider the initial value problem for the reduced fifth-order KdV-type equation: , , , . This equation is obtained by removing the nonlinear term from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data satisfies the condition , for some constant . Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

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Rebooting the bootstrap argument: Two puzzles for bootstrap theories of concept development

Behavioral and Brain Sciences ◽

10.1017/s0140525x10002190 ◽

2011 ◽

Vol 34 (3) ◽

pp. 145-146 ◽

Cited By ~ 3

Author(s):

Lance J. Rips ◽

Susan J. Hespos

Keyword(s):

Belief Change ◽

Concept Development ◽

Bootstrap Argument ◽

Meaning Change

AbstractThe Origin of Concepts sets out an impressive defense of the view that children construct entirely new systems of concepts. We offer here two questions about this theory. First, why doesn't the bootstrapping process provide a pattern for translating between the old and new systems, contradicting their claimed incommensurability? Second, can the bootstrapping process properly distinguish meaning change from belief change?

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EXTENSION OF THE MORTAR FINITE ELEMENT METHOD TO A VARIATIONAL INEQUALITY MODELING UNILATERAL CONTACT

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202599000154 ◽

1999 ◽

Vol 09 (02) ◽

pp. 287-303 ◽

Cited By ~ 55

Author(s):

FAKER BEN BELGACEM ◽

PATRICK HILD ◽

PATRICK LABORDE

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Variational Inequality ◽

Unilateral Contact ◽

Contact Conditions ◽

Bootstrap Argument ◽

Deformable Bodies ◽

Mortar Finite Element Method ◽

The One ◽

Element Method

The purpose of this paper is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite element meshes which do not fit on the contact zone. The mortar technique allows one to match these independent discretizations of each solid and takes into account the unilateral contact conditions in a convenient way. By using an adaptation of Falk's lemma and a bootstrap argument, we give an upper bound of the convergence rate similar to the one already obtained for compatible meshes.

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GLOBAL NEARLY-PLANE-SYMMETRIC SOLUTIONS TO THE MEMBRANE EQUATION

Forum of Mathematics Pi ◽

10.1017/fmp.2020.10 ◽

2020 ◽

Vol 8 ◽

Author(s):

LEONARDO ABBRESCIA ◽

WILLIE WAI YEUNG WONG

Keyword(s):

Wave Solution ◽

Spatial Dimension ◽

Travelling Wave ◽

Higher Order ◽

Perturbation Equation ◽

Nonlinear Interactions ◽

Travelling Wave Solution ◽

Plane Symmetric ◽

Bootstrap Argument ◽

Infinite Energy

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.

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REGULARITY OF VERY WEAK SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATION

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500120 ◽

2013 ◽

Vol 15 (04) ◽

pp. 1350012 ◽

Cited By ~ 3

Author(s):

WEI ZHANG ◽

JIGUANG BAO

Keyword(s):

Weak Solution ◽

Elliptic Equation ◽

Weak Solutions ◽

Local Regularity ◽

Difference Quotient ◽

Very Weak Solutions ◽

Very Weak Solution ◽

Bootstrap Argument ◽

The Difference ◽

Higher Regularity

In this paper, we study the local regularity of very weak solution [Formula: see text] of the elliptic equation Dj(aij(x)Diu) = f - Digi. Using the bootstrap argument and the difference quotient method, we obtain that if [Formula: see text], [Formula: see text] and [Formula: see text] with 1 < p < ∞, then [Formula: see text]. Furthermore, we consider the higher regularity of u.

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Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system

Kinetic and Related Models ◽

10.3934/krm.2021042 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Dayton Preissl ◽

Christophe Cheverry ◽

Slim Ibrahim

Keyword(s):

Blow Up ◽

Large Data ◽

Classical Solutions ◽

Time Averaging ◽

Maxwell System ◽

Confined Plasmas ◽

Bootstrap Argument ◽

Norm Estimates ◽

Magnetically Confined ◽

Magnetically Confined Plasmas

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id="M1">\begin{document}$ f(t,\cdot) $\end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field <inline-formula><tex-math id="M2">\begin{document}$ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields <inline-formula><tex-math id="M4">\begin{document}$ (E,B) $\end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id="M5">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id="M6">\begin{document}$ \epsilon^{-1} $\end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id="M7">\begin{document}$ T_\epsilon $\end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id="M10">\begin{document}$ 0<T<T_\epsilon $\end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id="M12">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.

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