Mathematical and numerical study of a system of conservation laws

The system of equations (f (u))_t − (a(u)v + b(u))_x = 0 and u_t − (c(u)v + d(u))_x = 0, where the unknowns u and v are functions depending on , arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of u_t − ((cg + d)(u))_x = 0 and simultaneously w = f (u) is the entropy solution of w_t − ((ag + b)(f⁽⁻¹⁾(w)))_x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.     

UNIQUENESS OF RECOVERY OF SOME QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS

We prove uniqueness of the term c(u,p) of partial differential equations ?Δu + c(u, ?u) = 0 and ? _t u ? Δu + c(u, ?u) = 0 with the Dirichlet-to-Neumann map given on a part of the lateral boundary. We use a linearization method and singular solutions in the boundary reconstruction of the linearized equations     

A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on \Bbb R^N×\Bbb R{\Bbb R}^N\times{\Bbb R} , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where c ? \Bbb Rc\in{\Bbb R} is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

A second-order,chaos-free,explicit method for the numerical solution of a cubic reaction problem in neurophysiology

A second order explicit method is developed for the numerical solution of the initialvalue problem w′(t) ≡ dw(t)/dt = ?(w), t > 0, w(0) = W⁰, in which the function ?(w) = αw(1 ? w) (w ? a), with α and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first-order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. Soc. (London) A 430 , 541–576 (1990)]. In addition to being chaos free and of higher order, the method is seen to converge to one of the correct steady-state solutions at w = 0 or w = 1 for any positive value of α. Convergence is monotonic or oscillatory depending on W⁰, α, a, and l, the parameter in the discretization of the independent variable t. The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial-value boundary-value problem ?u/?t = κ?²u/?x² + αu(1 ? u)(u ? a) in which u = u(x,t). The numerical method so developed obtained the solution by solving a single linear algebraic system at each time step. © 1993 John Wiley & Sons, Inc.     

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.     

Travelling front solution for a class of competition-diffusion system with high-order singular point

In this paper, the existence of travelling front solution for a class of competition-diffusion system with high-order singular point

An algorithm for tracing the boundary curves of mushy regions in some degenerate diffusion problems

We present an algorithm for approximating the solution of the degenerate diffusion problem u_t = (?(u))_xx in (0,1) × R⁺ (with zero Dirichlet boundary conditions, and nonnegative initial datum u₀), where ?(u) = min {ku1} for some ? > 0. The algorithm also provides an approximation for the interface curves which represent the boundary of the Mushy Region ?? = {(x, t): ? (u(x, t)) = 1}. The convergence of the algorithm is proved.     

On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u⁰₋(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰₋(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰₋ and u⁰₊, a global shock front weak solution u(x, t) = u₋(x, t) for x < ϕ(t), u(x, t) = u₊(x, t) for x > ϕ(t), where u₋ and u₊ are the strong solutions corresponding (respectively) to u⁰₋ and u⁰₊ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u₋(ϕ(t), t) + u₊(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Convexity and all‐time C∞‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with Q_T = ?ⁿ × (0, T). Under the condition that Ω_o is convex and log u_o is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < T_c, with T_c denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Lp Decay problem for the dissipative wave equation in even dimensions

We study L^p decay estimates of the solution to the Cauchy problem for the dissipative wave equation in even dimensions: (□+?_t)u=0 in ?^N × (0,∞) for even N=2n?2 with initial data (u,?_tu)∣_t=0 =(u₀,u₁). The representation formulas of the solution u(t)=?_tS(t)u₀ + S(t)(u₀+u₁) provide the sharp estimates on L^p norms with p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

Fisher–KPP equation with advection on the half‐line

We consider the Fisher–KPP equation with advection: u_t=u_xx?βu_x+f(u) on the half‐line x∈(0,∞), with no‐flux boundary condition u_x?βu = 0 at x = 0. We study the influence of the advection coefficient ?β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c₀), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c₀,∞), the solution converges locally uniformly to 0, here c₀ is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c₀)t + o(t), and (ii) when β≥c₀, the asymptotic positions of the level sets on the left side are (β ? c₀)t + o(t). Copyright © 2016 John Wiley & Sons, Ltd.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u ₀(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to ? ∞. On the one hand, we give some conditions on u ₀ which guarantee that, for some λ > 0, the quantity c _λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim  _t→+∞ u(t, ct) = 1 (the stable steady state) if c < c _λ and lim  _t→+∞ u(t, ct) = 0 if c > c _λ. These conditions are fulfilled in particular when u ₀(x) e ^λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u ₀ for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ? and for all speeds c belonging to a given interval (γ₁, γ₂) with large enough γ₁ < γ₂ ≤ + ∞.     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

The stability of rarefaction wave for Navier–Stokes equations in the half‐line

This paper studies the stability of the rarefaction wave for Navier–Stokes equations in the half‐line without any smallness condition. When the boundary value is given for velocity u∥_{x = 0} = u_? and the initial data have the state (v₊, u₊) at x→ + ∞, if u_?<u₊, it is excepted that there exists a solution of Navier–Stokes equations in the half‐line, which behaves as a 2‐rarefaction wave as t→ + ∞. Matsumura–Nishihara have proved it for barotropic viscous flow (Quart. Appl. Math. 2000; 58:69–83). Here, we generalize it to the isentropic flow with more general pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.