Local Blow-up Research Articles

Quasianalytic multiparameter perturbation of polynomials and normal matrices

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$ of $C^\infty$-functions. We require that $\mathcal C$ includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy--Carleman classes, in particular, the class of real analytic functions $C^\omega$. We show that there exists a locally finite covering $\{\pi_k\}$ of the parameter space, where each $\pi_k$ is a composite of finitely many $\mathcal C$-mappings each of which is either a local blow-up with smooth center or a local power substitution (in coordinates given by $x \mapsto (\pm x_1^{\gamma_1},...,\pm x_q^{\gamma_q})$, $\gamma_i \in \mathbb N_{>0}$), such that, for each $k$, the family of polynomials $P {\o}\pi_k$ admits a $\mathcal C$-parameterization of its roots. If $P$ is hyperbolic (all roots real), then local blow-ups suffice. Using this desingularization result, we prove that the roots of $P$ can be parameterized by $SBV_{loc}$-functions whose classical gradients exist almost everywhere and belong to $L^1_{loc}$. In general the roots cannot have gradients in $L^p_{loc}$ for any $1 < p \le \infty$. Neither can the roots be in $W_{loc}^{1,1}$ or $VMO$. We obtain the same regularity properties for the eigenvalues and the eigenvectors of $\mathcal C$-families of normal matrices. A further consequence is that every continuous subanalytic function belongs to $SBV_{loc}$.

The boundary value problem for wave equations with nonlinear dissipative and source terms

The boundary value problem for wave equations with nonlinear dissipative and source terms in an angular domain is considered. The questions of existence and non-existence of global and blow-up solutions of considering problem are investigated for some values of parameters in the nonlinear terms and the right-hand side of given equations. The uniqueness, local and global existence and blow-up of solutions of the problem mentioned are investigated.

Well-posedness for the 2-D damped wave equations with exponential source terms

In this paper we study well-posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ℝ2; ω⩾0, ωλ1+µ>0 with λ1 being the first eigenvalue of −Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.

Berezin transform in polynomial bergman spaces

AbstractFix a smooth weight function Q in the plane, subject to a growth condition from below. Let Km,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n − 1 of finite L2‐norm with respect to the measure e−mQ dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure ${\rm d}B_{m,n}^{\langle z_0\rangle}(z)=K_{m,n}(z_0,z_0)^{-1}|K_{m,n}(z,z_0)|^2{\rm e}^{-mQ(z)}\,{\rm d}A(z)$ for the point z0 is a probability measure that defines the (polynomial) Berezin transform for continuous $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}f\in L^\infty(\C)$. We analyze the semiclassical limit of the Berezin measure (and transform) as m → + ∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z0 converges weak‐star to the unit point mass at the point z0 provided that Δ Q(z0) &gt; 0 and that z0 is contained in the interior of a compact set ${\cal S}_\tau$, defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}z_0\in \C\setminus{\cal S}_\tau$, the Berezin measure cannot converge to the point mass at z0. In the model case Q(z) = |z|2, when ${\cal S}_\tau$ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z0 relative to $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\C\setminus{\cal S}_\tau$.Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L2‐estimates for the equation ${\bar \partial} u=f$ when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc.

Local Well-Posedness and Blowup Criterion of the Boussinesq Equations in Critical Besov Spaces

We consider the existence of the 2D inviscid Boussinesq equations in critical Besov spaces and obtain some blowup criteria.

Lattice Boltzmann method with selective viscosity filter

For high-Reynolds number flows, the lattice Boltzmann method suffers from numerical instabilities that can induce local blowup of the computation. The von Neumann stability analysis applied to the LBE-BGK and LBE-MRT models shows that numerical instabilities occur in the high wavenumber range and are due to the interplay between acoustic modes and some other modes. As it is done in the LBE-MRT model, an increase of the bulk viscosity is an efficient way of damping spurious oscillations. However, this stabilization method induces an over-damping of acoustic waves. Some selective spatial filters can be used in order to eliminate the spurious small spatial scales without affecting the large scale physical modes. Three different lattice Boltzmann algorithms based on filtering step are proposed: the fully filtered LBE, the LBE with filtered macroscopic quantities and the LBE with filtered collision operator. The behavior of several explicit filter stencils is studied in the Fourier space. For a given filter stencil, the filtered collision operator approach leads to the highest cut-off wavenumber. In this case, the theoretical wavenumber-dependent viscosity is ν ( k ) = c s 2 ( τ / [ 1 - σ f ( k ) ] - 1 / 2 ) where f ( k ) is the filter shape and σ the filter strength. Under-resolved simulations (high Reynolds number) are performed on the case of the doubly periodic shear layers. The performance of the three filtered LBE is found to be the same as the MRT model for stability control. Propagation of acoustic plane waves is also simulated with the three filtering algorithms. The measured dissipation of acoustic wave compares well with the theoretical results.

Some properties of solutions to the weakly dissipative Degasperis–Procesi equation

In this paper, we consider the weakly dissipative Degasperis–Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis–Procesi equation are also investigated.

Explosive behavior in spatially discrete reaction-diffusion systems

Explosive instabilities in spatially discrete reaction-diffusion systems are studied. We identify classes of initial data developing singularities in finite time and obtain predictions of the blow-up times, whose accuracy is checked by comparison with numerical solutions. We present averaged and local blow-up estimates. Local blow-up results show that it is possible to have blow-up after blow-up. Conditions excluding or implying blow-up at space infinity are discussed.

Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms

In this paper, we consider the local existence and a blow-up criterion for smooth solutions to the 2-D isentropic compressible Boussinesq equations, and obtain some new commutator estimates. In particular, we show that the time integral of the spatial maximum of gradient of velocity controls the breakdown of smooth solutions.

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L p -spaces instead of the standard L p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in $${\mathbb{R}}^n , n \geq 3$$ . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces.

Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every $$u_0 \in L^1 (\mathbb {R}^2)$$ . The local existence time is characterized for $$u_0 \in L^1 \cup L^{q*}(\mathbb {R}^2)$$ with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption $$||u_0||_{L^{1}} > 8 \pi$$ and $$||x|^2u_0||{L^1} < \frac {1}{\gamma}.g (||u_0||{L^1}/8\pi)$$ , where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces

We consider the local existence and blowup criterion of the 3D Lagrangian averaged Euler equations in Besov spaces and obtain the existence and blowup criterion.

Stability and stabilization of the lattice Boltzmann method

We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface--the invariant film (up to second order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free flight. The main instability mechanisms are identified. The simplest recipes for stabilization add no artificial dissipation (up to second order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re approximately 20,000.

Local well‐posedness and blow‐up criteria of solutions for a rod equation

AbstractIn this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Flattening and analytic continuation of affinoid morphisms: remarks on a paper of Gardener and Schoutens

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ?Flattening and subanalytic sets in rigid analytic geometry?, Proc. London Math. Soc (3) 83 (2001) 681?707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. Theorem 3.12), that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.

Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations

Given a bounded domain Ω we consider local weak blow-up solutions to the equation Δ p u= g( x) f( u) on Ω. The non-linearity f is a non-negative non-decreasing function and the weight g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω. We show that if Δ p w=− g( x) in the weak sense for some w∈W 1,p 0(Ω) and f satisfies a generalized Keller–Osserman condition, then the equation Δ p u= g( x) f( u) admits a non-negative local weak solution u∈W 1,p loc (Ω)∩C(Ω) such that u( x)→∞ as x→∂Ω. Asymptotic boundary estimates of such blow-up solutions will also be investigated.

Compressible dynamics of magnetic field lines for incompressible magnetohydrodynamic flows

It is demonstrated that the deformation of magnetic field lines in incompressible magnetohydrodynamic flows results from a compressible mapping associated with the transverse motion of fluid particles. Appearance of zeros for the Jacobian of this mapping corresponds to the breaking of magnetic field lines and the local blowup of the magnetic field intensity. The occurrence of such events is found to be unlikely in two dimensions but possible in three dimensions.

On the Structure of Solutions to the Periodic Hunter--Saxton Equation

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

Local Existence and Blow-up Criterion of the Inhomogeneous Euler Equations

We prove local in time existence and uniqueness of solution of the inhomogeneous Euler equations in the critical Besov spaces. We also obtain the finite time blow-up criterion of the local solution.

A potential well theory for the wave equation with nonlinear source and boundary damping terms

The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is\cases{u_{tt}-\Delta u=|u|^{p-2}u \hfill &amp; \rm{~}OPEN~7~in &lt;hsp sp=0.25&gt;[0,\rm{inf}ty )\times \Omega ,}\hfill \cr u=0 \hfill &amp; \rm{~}OPEN~6~on &lt;hsp sp=0.25&gt;[0,\rm{inf}ty )\times \Gamma _0,}\hfill \cr \frac {\partial u}{\partial \nu }=-\alpha (x)|u_t|^{m-2}u_t \hfill &amp; \rm{~}OPEN~2~on &lt;hsp sp=0.25&gt;[0,\rm{inf}ty )\times \Gamma _1,}\cr u(0,x)=u_0(x),u_t(0,x)=u_1(x) &amp; \rm{~}OPEN~1~on&lt;hsp sp=0.25&gt;\Omega ,}\hfill }where \Omega \subset ℝ^n (n\ge 1) is a regular and bounded domain, \partial \Omega =\Gamma _0\cup \Gamma _1, \lambda _{n-1}(\Gamma _0)&gt;&gt;;0, 2&lt;&gt;;p\le 2(n-1)/(n-2) (when n\ge 3), m&gt;&gt;;1, \alpha \in L^\rm{inf}ty (\Gamma _1), \alpha \ge 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of \Omega, to this problem.