Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

On Low Mach Number Preconditioning of Finite Volume Schemes

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments, as well as by analysis, that the preconditioned scheme yields a physically corrected pressure distribution and combined with an explicit time integrator it is stable if the time step Δt satisfies the requirement to be 𝒪(M ²) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Δt = 𝒪(M ),M → 0, though producing unphysical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptives quadratures and fractional IDEs: Applications in image processing

In this paper we show adaptive time discretizations of a fractional integro–differential equation ∂^α_tu = Δu + f, where A is a linear operator in a complex Banach space X and ∂^α_t stands for the fractional time derivative, for 1 < α < 2. Some numerical illustrations are provided showing practical applications where the computational cost is one of drawbacks, e.g., some problems related to images processing. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive and maximal positive solutions of singular mixed boundary value problem

The paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.     

Error Estimates for Finite Element Approximations of a Viscous Wave Equation

We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on the energy method is developed. Optimal error estimates in both time and space are obtained. For sufficiently smooth solutions, it is demonstrated that the maximal error in the L ²-norm over a finite time interval converges optimally as O(h ^p+1 + Δt ^s ), where p denotes the polynomial degree, s = 1 or 2, h the mesh size, and Δt the time step.     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Critical Exponents for a System of Heat Equations Coupled in a Non-linear Boundary Condition

In this paper we consider a system of heat equations u_t = Δu, v_t = Δv in an unbounded domain Ω⊂ℝ^N coupled through the Neumann boundary conditions u_v = v^p, v_v = u^q, where p>0, q>0, pq>1 and ν is the exterior unit normal on ∂Ω. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.     

Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

In this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L² initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space H_M and small time step Δt² on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space H_m(m = O(M^1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H²‐stability and the L²‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

The Effect of Surface Tension on the Moore Singularity of Vortex Sheet Dynamics

We investigate the regularization of Moore’s singularities by surface tension in the evolution of vortex sheets and its dependence on the Weber number (which is inversely proportional to surface tension coefficient). The curvature of the vortex sheet, instead of blowing up at finite time t ₀, grows exponentially fast up to a O(We) limiting value close to t ₀. We describe the analytic structure of the solutions and their self-similar features and characteristic scales in terms of the Weber number in a O(We⁻¹) neighborhood of the time at which, in absence of surface tension effects, Moore’s singularity would appear. Our arguments rely on asymptotic techniques and are supported by full numerical simulations of the PDEs describing the evolution of vortex sheets.     

A parametric study on creep-fatigue endurance of welded joints

This work is devoted to parametric study on creep-fatigue endurance of the steel AISI type 316N(L) weldments defined as type 3 according to R5 Vol. 2/3 procedure at 550°C. The study is implemented using a novel direct method known as the Linear Matching Method (LMM) and based upon the creep-fatigue evaluation procedure considering time fraction rule for creep-damage assessment. Seven geometrical configurations of the weldment, which are characterised by individual values of a geometrical parameter ρ, are proposed. Parameter ρ, which represents different grades of TIG dressing, is a ratio between the radius of the fillet of the remelted metal on a weld toe and the thickness of welded plates. For each configuration, the total number of cycles to failure N^* in creep-fatigue conditions is assessed numerically for different loading cases including normalised bending moment and dwell period Δt. The obtained set of N^* is extrapolated by the analytic function dependent on , Δt and ρ. Proposed function for N^* shows good agreement with numerical results obtained by the LMM. It is used for the identification of Fatigue Strength Reduction Factors (FSRFs) effected by creep and dependent on Δt and ρ. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Fast Implicit Numerical Method for Time Dependent Viscous Flows

The iterative method used by Esch (1964) and Pearson (1964, 1965a, b), for the solution of an implicit finite difference approximation to the Navier Stokes equation, is analysed. A more general iteration method is suggested that may require many iteration parameters, and it is shown how these parameters can be computed. It was found that when the non-dimensional number vΔt/2L² is small, a single optimum iteration parameter exists (v being the kinematic viscosity, Δt the time step and L a characteristic length). An approximate expression for the “best” parameter is developed, and a procedure is described for improving that estimate. With the improved estimate and extrapolation in time, convergence is achieved in one or two iterations per time step on the average. In some cases the time step used was 200 times bigger than the time step required for stability of explicit schemes.     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.     

On the convergence and superconvergence for a class of two-dimensional time fractional reaction-subdiffusion equations

This paper analyzes a class of two-dimensional (2-D) time fractional reaction-subdiffusion equations with variable coefficients. The high-order L2-1_σ time-stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L²-norm error estimation and H¹-norm superclose results are rigorously proved. The superconvergence result in the H¹-norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

The Behavior of Solutions of Multidimensional Aggregation Equations with Mildly Singular Interaction Kernels

The authors consider the multidimensional aggregation equation tp-div（p K＊ ρ） = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin （Lipschitz or better）, and review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. The problem with bounded initial data, data in L^p ∩ L^1, and measure solutions are also considered.     

Singular boundary value problems for the one-dimension p-Laplacian

The singular boundary value problem

where φ(s)=|s|^p−2s, p>1, is studied in this paper. The singularity may appear at u=0, t=0 and t=1, and the function g may change sign. The existence of solutions is obtained via an upper and lower solution method.     

Global singularity structures of weak solutions to 4-D semilinear dispersive wave equations

In this paper, we are concerned with the global singularity structures of weak solutions to 4-D semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere. Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions are C²−regular away from the focusing cone surface |x|=|t−1| and the outgoing cone surface |x|=t+1. This research was supported by the National Natural Science Foundation of China and the Doctoral Foundation of NEM of China.