On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

A doubly critical degenerate parabolic problem

It is shown that the Dirichlet problem for where Ω??ⁿ is critical in that it has first eigenvalue one, is globally solvable for any continuous positive initial datum vanishing at ?Ω. Moreover, for p<3 all solutions are bounded and tend to some nonnegative eigenfunction of the Laplacian as t→∞, while if p?3 then there are both bounded and unbounded solutions. Finally, it is shown that unlike the case p∈[0,1), all steady states are unstable if p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

A modified phase field approximation for mean curvature flow with conservation of the volume

This paper is concerned with the motion of a time‐dependent hypersurface ?Ω(t) in ?^d that evolves with a normal velocity where κ is the mean curvature of ?Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation where ε is an approximation parameter, W a double well potential and c_W a constant that depends only on W. We study here a modified version of this equation and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.     

A note about stabilization in Aℝ(𝔻)

It is shown that for A_?(??) functions f₁ and f₂ with and f₁ being positive on real zeros of f₂ then there exists A_?(??) functions g₂ and g₁, g₁^–1 with and This result is connected to the computation of the stable rank of the algebra A_?(??) and to Control Theory (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Local Invertible Operators in L2 (ℝ1, H)

We study operators of the form Lu = — G(t) u(t) in L²([t₀ — δ, t₀ + δ], H) with = L² ([t₀— δ, t₀ + δ], H ) in the neighbourhood [t₀— δ, t₀ + δ] of a point t₀ ∈ ℝ¹. Such problems arise in questions on local solvability of partial differential equations (see [6] and [7]). For these operators,one of the major questions is if they are invertible in a neighbourhood of a point t ∈ ℝ¹. To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization theorems for nonanalytic operator-functions we give additional conditions for the nonanalytic operator -function G(t) and show that the operator L (or ) with some boundary conditions is local invertible.     

Phase transition for potentials of high‐dimensional wells

For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N^± in , we discuss the asymptotics, as ? → 0, of minimizers u_? of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of E _? (u_?) with respect to , we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N⁺ and N^?, and the zeroth‐order term by the energy of minimizing harmonic maps into N^± both under the Dirichlet boundary condition on ?Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.     

Positive solution for asymptotically linear elliptic systems

In this paper, we show that semilinear elliptic systems of the form (1) possess at least one positive solution pair (u, v)∈H¹(?^N)×H¹(?^N), where λ and µ are nonnegative numbers, f(x, t) and g(x, t) are continuous functions on ?^N×? and asymptotically linear as t→+∞. Copyright © 2009 John Wiley & Sons, Ltd.     

Asymptotic profile of solutions to a non‐linear dissipative evolution system with conservation

We investigate the asymptotic profile to the Cauchy problem for a non‐linear dissipative evolution system with conservational form (1) provided that the initial data are small, where constants α, ν are positive satisfying ν²<4α(1 ? α), α<1. In (J. Phys. A 2005; 38 :10955–10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t → ∞ the solution and solution in the L^p sense, where G(t, x) denotes the heat kernel and is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to ?₊ and ?_? which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z. Angew Math Phys 2006; 57 : 604–614). Copyright © 2007 John Wiley & Sons, Ltd.     

Interval oscillation criteria for second order super‐half linear functional differential equations with delay and advanced arguments

Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form where ?_δ (u) = |u |^{δ –1}u; α > 0, β ≥ α, and γ ≥ α are real numbers; k, p, q, e, τ, σ are continuous real‐valued functions; τ (t) ≤ t and σ (t) ≥ t with lim_{t →∞} τ (t) = ∞. The functions p (t), q (t), and e (t) are allowed to change sign, provided that p (t) and q (t) are nonnegative on a sequence of intervals on which e (t) alternates sign. As an illustrative example we show that every solution of is oscillatory provided that either m₁ or m₂ or r₀ is sufficiently large (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

For a d‐dimensional diffusion of the form dX_t = μ(X_t)dt + σ(X_t)dW_t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ?? is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y_t = v(t, X_t), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.     

Asymptotic characterization of standing waves and of the static limit for a class of wave equations of higher order with a variable coefficient and time-independent incitation

We consider the equation (?1)^m?^m (p?^mu) + ?u = ? in ?ⁿ × (0, ∞) for arbitrary positive integers m and n and under the assumptions p ? 1, ? ? C(?ⁿ) and p > 0. Even if the differential operator (?1)^m?^m (p?^mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (?1)^m?^m (p?^mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.     

Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory (1) is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T^? < ∞ , with locally bounded final time profiles u(x, T^?) , are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T , all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) (2) created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self‐similar gradient blow‐up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2) , the NDE (1) can be written as (3) with the standard linear integral operator (?D²_x)^?1 > 0 . However, because (3) is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of (3) is increasing for BV data u₀(x) ).     

Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

An ordinary differential equation of the type with parameterξ ? IRⁿ and smooth coefficients a_j,a ? C^∞([-T,T]) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t ≠ 0 they are real and distinct. The constructions of real-valued phase functions ?pH_kl (k,l = 1., m) and of amplitude functions A_jkl such that for a given s ? [-T, T] every solution u(t, ξ) of the equation can be represented as where Ψ_j(s, ξ)= D^j_tu(s,ξ), j = 0,m-1 are given.     

Perturbation analysis for the generalized Schur complement of a positive semi‐definite matrix

Let and S=C?B^HA^?B be the generalized Schur complement of A?0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR‐95‐38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Hadamard matrices constructed from supplementary difference sets in the class ℋ︁1

In this article we give the definition of the class ??₁ and prove: (1) ??₁(v) ≠ ? for v ∈ ?? = ??₁ ∪ ??₂ ∪ ??₃ where (2) there exists 2 ? {2q²; q² ± q, q²;q² ± q} supplementary difference sets for q² ∈ ??; (3) there exists an Hadamard matrix of order 4v for v ∈ ??; (4) if t is an order of T-matrices, there exists an Hadamard matrix of order 4tv for v ∈ ??. © 1994 John Wiley & Sons, Inc.     

Stability for the beam equation with memory in non‐cylindrical domains

In this paper, we prove the exponential decay as time goes to infinity of regular solutions of the problem for the beam equation with memory and weak damping where is a non‐cylindrical domains of ?ⁿ⁺¹ (n?1) with the lateral boundary and α is a positive constant. Copyright © 2004 John Wiley & Sons, Ltd.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

A Fujita-type global existence—global non-existence theorem for a system of reaction diffusion equations with differing diffusivities

In this paper we condiser non-negative solutions of the initial value problem in ?^N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.

• (a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
• (b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
• (c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
• (d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L^∞(?^N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣^?2α and 0 < v(X, t) ? c ∣x∣^?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
• (e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H¹ (K) where K(x) ? exp(¼∣x∣²). They decay like e^{[max(α,β)-(N/2)+ε]t} for every ε > 0. These solutions are classical solutions for t > 0.
• (f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u₀, v₀) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
• (g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
• (h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
• (i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
• (j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.

We also indicate some extensions of these results to moe general systems and to othere geometries.     

The small dispersion limit of the Korteweg-de Vries equation. III

In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of u_xxx tends to zero. This formula contains n parameters β_1, …, β_n whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t<tb, n=1, and the value of β, was determined. In Section 5 we have shown that the parameters β_i satisfy a nonlinear system of partial differential equations. In Part III, Section 6 we show that for t large, n=3, and we determine the asymptotic behavior of β_1, β_2, β_3, and of u and u ², for t large. The explicit formulas show that u and u ² are O(t^?1) and O(t^-2) respectively (see formulas (6.2) and (6.24)). In Section 7 we study initial data whose value tends to zero as x→+∞, and to -1 as x→?∞. If one accepts some plausible guesses about the behavior of solutions with such initial data, we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution: . The function s(ζ) is expressible in terms of complete elliptic integrals; a similar formula is derived for U ². In Section 8 we indicate how to extend the treatment of this series of papers to multihumped (but still negative) initial data.