Bootstrap Argument Research Articles

First transition dynamics of reaction–diffusion equations with higher order nonlinearity

AbstractIn this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of and . The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section.

Sums of squares III: Hypoellipticity in the infinitely degenerate regime

This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^{2,\delta} generalization of M. Christ’s smooth sum of squares theorem, and then use a bootstrap argument with the sum of squares decomposition for matrix functions, obtained in our second paper of this series, to prove a hypoellipticity theorem that generalizes some cases of the results of Christ, Hoshiro, Koike, Kusuoka and Stroock and Morimoto for sums of squares, and of Fedĭi and Kohn for degeneracies not necessarily a sum of squares.

Velocity-Vorticity Geometric Constraints for the Energy Conservation of 3D Ideal Incompressible Fluids

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.

Existence of global solution to 3D density-dependent incompressible Navier-Stokes equations

<abstract><p>In this article, we are committed to studying the three-dimensional incompressible Navier-Stokes equations, where the viscosity depends on density according to a power law. We investigate the Cauchy problem by constructing an approximated solution and bootstrap argument. Finally, we establish the existence of a global strong solution under the conditions of small initial data and the compatibility condition. Meanwhile, the algebraic decay-in-time rates for the solution are also obtained. It is worth pointing out that the degradation of viscosity is allowed.</p></abstract>

Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space–time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.

Rayleigh–Taylor instability of 3D inhomogeneous incompressible Euler equations with damping in a horizontal slab

In this paper, we consider the Rayleigh–Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. We show that if the steady density profile is non-monotonous along the height, then the Euler system with damping is nonlinearly unstable around the given steady state. In this article, we develop a new variational structure to construct the growing mode solution, and overcome the difficulty in proving the sharp exponential growth rate by exploiting the structures in linearized Euler equations. Then combined with error estimates and a standard bootstrapping argument, we finish the nonlinear instability.

The well posedness of solutions for the 2D magnetomicropolar boundary layer equations in an analytic framework

In this paper, we prove the existence and uniqueness of solutions to the 2D magnetomicropolar boundary layer equations on the half-plane by using the classical bootstrap argument in an analytic framework.

Stability of the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation

In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation. But there are only vertical kinematic viscosity, horizontal angular viscosity, and diffusivity; some nonlinear terms cannot be controlled. In addition to the classical energy functional, we introduce the energy functional to help us dominate the nonlinear parts and make the energy closed. Then based on the energy method and the bootstrapping argument, global smooth solutions are proved under the assumptions of small initial data.

On the decay property of the cubic fourth-order Schrödinger equation

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on R d \mathbb {R}^d ( 5 ≤ d ≤ 8 5 \leq d \leq 8 ) enjoys the same decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result by Pausader [J. Funct. Anal. 256 (2009), pp. 2473–2517]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

Cauchy Problem for Incompressible Neo-Hookean Materials

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

A class of three dimensional Cahn-Hilliard equation with nonlinear diffusion

In this paper, we study the well-posedness and asymptotic behavior for a class of Cahn-Hilliard equation with nonlinear diffusion in R3. In order to overcome the difficulties caused by the derivatives of multi-well potential and the nonlinear terms, we “borrow” a linear principle part from the derivatives of multi-well potential, rewrite the equation as an equivalent equation with linear principle part. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Moreover, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H4-norm of initial data is sufficiently small. In the end, based on parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also establish the space-time decay estimates of strong solutions.

On the energy scaling behaviour of singular perturbation models with prescribed dirichlet data involving higher order laminates

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplifiedm-well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by theorder of lamination of the Dirichlet data. This follows by deducing matching upper and lower scaling bounds. For theupperbound we argue by providing iterated branching constructions, and complement this with ansatz-freelowerbounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated “bootstrap arguments” based on the ideas from [A. Rüland and A. Tribuzio,Arch. Rational Mech. Anal.243(2022) 401–431]. Relying on these observations, we study models involving laminates of arbitrary order.

Sharp decay estimates for 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. By using the energy methods, we obtain that this system possesses a global solution when the initial data is small in H3(R3). In addition, the optimal decay rates of this global solution and its first-order derivatives are established by bounding two time-weighted norms and applying a bootstrapping argument.

Decay estimates of solutions to nonlinear elastic wave equations with viscoelastic terms in the framework of Lp-Sobolev spaces

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in the Lp framework. Sharp estimates of the global solutions in t for the higher order derivatives are established. The proof is based on the Gronwall type argument for 1<p<∞, the use of the difference of the propagation speed of the fundamental solutions for p=1 and the bootstrap argument for p=∞, combined with the sharp estimates for the linearized solutions.

Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators

This paper deals with direct and inverse source problems for parabolic or byperbolic PDEs containing nonlinear Volterra operators (including the variable order time-fractional derivatives). We show the well posedness (existence and uniqueness of a solution) of various settings. The proof-technique is based on the contraction mapping theorem and on a bootstrap argument, which ensures the existence of a global in time solution.

Stability and instability of Kelvin waves

The m-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with m-fold rotational symmetry for \(m\ge 2\). For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the \(L^1\) norm of the vorticity, for m-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework.

Hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes-Vlasov equations

We consider the hydrodynamic limit of weak solutions to the inhomogeneous incompressible Navier-Stokes-Vlasov equations in three-dimensional torus domain. It is shown that the global weak solutions of the inhomogeneous incompressible Navier-Stokes-Vlasov equations converge to the smooth solutions of the two-fluid model in the sense that the distribution function fϵ converges to a Dirac distribution in velocity, the fluid density ρϵ and the velocity uϵ converge to ρ and u in some sobolev space, respectively. The proofs mainly involve the relative entropy method, bootstrap argument and delicate energy estimates.

Global well-posedness for the non-linear Maxwell-Schrödinger system

In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic quantum plasma, interacting with its self-generated electromagnetic potential. One consequence of our analysis is to demonstrate that the Lorentz force associated with the electromagnetic field is well-defined for solutions slightly more regular than the finite energy class. This aspect is of fundamental importance since all the related physical models require the observability of electromagnetic effects. The well-posedness of the Lorentz force still seems to be a major open problem in the class of solutions which are only finite energy. We show the global well-posedness at high regularity for the cubic and sub-cubic case, and we provide polynomial bounds for the growth of the Sobolev norm of the solutions, for a certain range of non-linearities. An important role is played by appropriate a priori dispersive estimates, obtained by means of Koch-Tzvetkov type bounds for the non-homogeneous Schr\"odinger equation, which overcome the lack of Strichartz estimates for the magnetic Schr\"odinger flow. Because of the power-type non-linearity, the propagation of higher regularity, globally in time, cannot be achieved via a bootstrap argument as done in [44]. Our approach then exploits the analysis of a modified energy functional, combined with the a priori bounds coming from the dispersive estimates obtained previously.

Stability for the 2D MHD equations with horizontal dissipation

In this paper we consider the following 2D MHD system with horizontal dissipation in a strip domain T × R. ∂ t u + u · ∇ u + ∂ 11 u + ∇ p = b · ∇ b , ∂ t b + u · ∇ b + ∂ 11 b = b · ∇ u , ∇ · u = ∇ · b = 0 , u ( 0 ) = u 0 ( x ) , b ( 0 ) = b 0 ( x ) . A bootstrapping argument together with a more accurate energy functional is employed in order to get the stability for the above system. Moreover, using a suitable transform, we also investigate the 2D MHD system with vertical dissipation in a strip domain R × T.

Bootstrap methods in bounding discrete Radon operators

The aim of this paper is to develop bootstrap arguments to establish maximal, oscillation, variational and jump inequalities for the discrete averaging Radon operators on ℓp(Zd).