On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system. Research done in the University of Michigan-Ann Arbor, submission from Temasek Laboratories, National University of Singapore.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Stabilization of flows through porous media

In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope of b in c : The equilibrium is unstable if b′(c) < 0, whereas b′(c) > 0 implies exponential stability. Zhaoyong Feng: He is grateful to the DFG for financial support through the Graduiertenkolleg 615 “Interaction of Modeling, Computation Methods and Software Concepts for Scientific-Technological Problems”.     

Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation (FDE) u _t = Δu ^m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L ^p −L ^q smoothing effects of the type ∥u(t)∥ _q ≤ Ct ^−α ∥u ₀∥^γ _p , the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.      

Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

Entropy formulation for fractal conservation laws

Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L^∞ framework. We also establish a result of controled speed of propagation that generalizes the finite propagation speed result of scalar conservation laws. We finally let the non-local term vanish to approximate solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].     

A PDE PROBLEM ARISING FROM CALCULATION OF THE MODEL FOR CONTINUOUS CASTING OF STEEL

This paper deals with the relationship between a certain three-dimensional elliptic problem and a two-dimensional parabolic problem, which arises from numerical calculation in the process of the continuous casting of steel.     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

BIFURCATION OF TWO-DIMENSIONAL KURAMOTO-SIVASHINSKY EQUATION

This paper deals with the steady state bifurcation of the K-S equation in two spatial dimensions with periodic boundary value condition and of zero mean. With the increase of parameter a, the steady state bifurcation behaviour can be very complicated. For convenience, only the cases a=2 and a=5 witl be discussed. The asymptotic expressions of the steady state solutions bifurcated from the trivial solution near a=2 and a=5 are given. And the stability of thenontriviat sotutions bifurcated from a=2 is studied. Of course, the cases a=n^2 m^2,n,m∈N(a≠2,5) can be similarly discussed by the same method which is used to discussing the cases a=2 and a= 5.     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem. A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔu_tt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of u_t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u_t we use analyticity of the corresponding thermoelastic semigroup proved in [25]. The isothermal variant of this problem with additional mechanical damping term −εΔu_t , ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain.     

Ill-posedness for semilinear wave equations with very low regularity

In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on u and ∂ _t u. We prove a ill-posedness result for the “defocusing” case, and give an alternative proof for the supercritical “focusing” case, which improves the result in Fang and Wang (Chin. Ann. Math. Ser. B 26(3), 361–378, 2005). Supported by NSF of China 10571158.     

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

Local and asymptotic analysis of the flow generated by the Cahn–Hilliard–Gurtin equations

We consider the Cahn–Hilliard–Gurtin equation which corresponds, in the isotropic case, to the viscous Cahn–Hilliard equation. The convergence of its solutions toward some steady state is investigated by means of a proper generalization of the Lojasiewicz–Simon Theorem to nongradient-like flows. Furthermore, when the anisotropic coefficients are small, we prove that these steady states can be approximated by the corresponding stationary solutions of the viscous Cahn–Hilliard equation provided that the latter are local minimizers of the Ginzburg–Landau free energy. Received: April 26, 2004; revised: February 24, 2005     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

Oscillatory integrals generated by the Ostrovsky equation

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. Received: November 23, 2004; revised: March 13, 2005 Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.