Showing papers in "Communications in Partial Differential Equations in 2006"
TL;DR: By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 2003a b) and Strain a...
Abstract: By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 2003a b) and Strain a...
207 citations
TL;DR: The article proves existence of entropic solutions to the Cauchy problem when the road network has only one junction and discusses stability in L ∞ norm of such solutions.
Abstract: The article deals with a fluid dynamic model for traffic flow on a road network. This consists of a hyperbolic system of two equations proposed in Aw and Rascle (2000). A method to solve Riemann pr...
170 citations
TL;DR: In this article, the authors prove local well-posedness of the initial-boundary value problem for the Korteweg-de Vries equation on right half line, left half line and line segment, in the low regularity setting.
Abstract: We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.
164 citations
TL;DR: For the linearized Boltzmann and Landau operators, the authors proved explicit coercivity estimates for a general class of interactions including any inverse-power law interactions, and hard spheres.
Abstract: We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.
150 citations
TL;DR: In this paper, the convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.
Abstract: The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.
137 citations
TL;DR: In this paper, a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities was studied and it was shown that existenc...
Abstract: This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existenc...
128 citations
TL;DR: In this article, the authors studied the Cauchy problem of the viscoelastic fluid system of the Oldroyd model in space dimension greater than 1 and proved that if the smooth solution (U,v) to this s...
Abstract: In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this s...
124 citations
TL;DR: In this paper, coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid were proposed, where the system consists in a Vlasov-Fokker-Planck equation to describe the micros.
Abstract: We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov–Fokker–Planck equation to describe the micros...
120 citations
TL;DR: The local L 2-mapping property of Fourier integral operators has been established in Hormander (1971) and in Eskin (1970) as discussed by the authors, and it has been extended to pseudodifferential integral operators by Asada and Fujiwara (1978) or Kumano-go (1976).
Abstract: The local L 2-mapping property of Fourier integral operators has been established in Hormander (1971) and in Eskin (1970). In this article, we treat the global L 2-boundedness for a class of operators that appears naturally in many problems. As a consequence, we improve known global results for several classes of pseudodifferential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara (1978) or Kumano-go (1976). As an application, we show a global smoothing estimate for generalized Schrodinger equations which extends the results of Ben-Artzi and Devinatz (1991) and Walther (1999); (2002).
116 citations
TL;DR: In this article, the existence of a solution for the p-Laplacian problem with homogeneous Dirichlet boundary conditions was studied, where the nonlinear term f(u) is a so-called jumping nonlinearity.
Abstract: In this work we study the existence of a solution for the problem − Δ p u = f(u) + tΦ(x) + h(x), with homogeneous Dirichlet boundary conditions. Here the nonlinear term f(u) is a so-called jumping nonlinearity. In the proofs we use topological arguments and the sub-supersolutions method, together with comparison principles for the p-Laplacian.
106 citations
TL;DR: In this article, explicit conditions for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations are presented. But the convergence of the convergence is not known.
Abstract: Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence r...
TL;DR: In this paper, Deng et al. derived improved geometric conditions on vortex filaments which can prevent finite-time blowup of the 3D incompressible Euler equation, which is the worst possible blowup scenario for velocity field due to Kelvin's circulation theorem.
Abstract: This is a follow-up of our recent article Deng et al. (2004 Deng, J.,Hou, T. Y., Yu, X. (2004). ). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D incompressible Euler equation. In this article, we derive improved geometric conditions which can be applied to the scenario when velocity blows up at the same time as vorticity and the rate of blowup of velocity is proportional to the square root of vorticity. This scenario is in some sense the worst possible blow-up scenario for velocity field due to Kelvin's circulation theorem. The improved conditions can be checked by numerical computations. This provides a sharper local geometric constraint on the finite time blowup of the 3D incompressible Euler equation.
TL;DR: In this paper, the question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated, and it is shown that no extra assumption of convexity or genuine non-linearity is required for the problem to be wellposed and prove it.
Abstract: In this paper, the question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated. We show that no extra assumption of convexity or genuine non-linearity with respect to the state variable of the flux function is required for the problem to be well-posed and prove it. The proof uses a kinetic formulation of the conservation law.
TL;DR: In this article, the authors obtained local C α, C 1,α, and C 2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.
Abstract: We obtain local C α, C 1,α, and C 2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.
TL;DR: In this article, a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrodinger operator in R n, n ≤ 3.
Abstract: We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrodinger operator in R n , n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces.
TL;DR: In this article, the uniqueness properties of solutions of Schrodinger equations of the form are studied and sufficient conditions on the asymptotic behavior of the difference of two solutio...
Abstract: In this paper we study uniqueness properties of solutions of Schrodinger equations of the form The aim is to deduce sufficient conditions on the asymptotic behavior of the difference of two solutio...
TL;DR: In this paper, the authors study the problem in general smooth bounded domains and show that it possesses nontrivial solutions provided that: f is sublinear, or f is superlinear and the equation admits a priori bounds.
Abstract: We study the equation in general smooth bounded domain Ω, and show it possesses nontrivial solutions provided that: f is sublinear, or f is superlinear and the equation admits a priori bounds. The existence result in the superlinear case is based on a new Liouville type theorem for − ℳλ,Λ +(D 2 u) = u p in a half-space.
TL;DR: In this article, the asymptotic behavior in time of small solutions for the Schrodinger equation with dissipative nonlinearity is studied, and it is shown that in the case of, there exists a unique global solution for this initial value problem which decays like (t log t)−n/2 as t→ ǫ+∞ in L ∞ for small initial data.
Abstract: The asymptotic behavior in time of small solutions for the initial value problem of the Schrodinger equation with dissipative nonlinearity is studied. The nonlinearity is −λ|u|2/n u, where λ is a complex constant such that and the space dimension n = 1, 2, or 3. This nonlinearity is critical between the short range interaction and the long range one. If , then the nonlinearity has a dissipative property. The main purpose of this article is to show that in the case of , there exists a unique global solution for this initial value problem which decays like (t log t)−n/2 as t → +∞ in L ∞ for small initial data, and to obtain the large time asymptotics of it.
TL;DR: In this article, the authors studied the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton-Jacobi Equations in the whole space ℝ N.
Abstract: We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ℝ N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λmin. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.
TL;DR: In this article, the authors investigated the global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ℝ n+1, n->-1.
Abstract: In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ℝ n+1, n > 1. We give some sufficient conditions for existence of the global weak solutions. These conditions tie together nonlinearity with the speed of propagation and with the dimension n. We also prove necessity of these (or close) conditions. In fact, we extend these necessity results to the nonlocal semilinear equations.
TL;DR: In this article, the inverse conductivity problem with partial data was considered and it was shown that knowledge of the Dirichlet-to-Neumann map measured on particular subsets of the boundary determines uniquely a conductivity with essentially 3/2 derivatives.
Abstract: In this article we consider the inverse conductivity problem with partial data. We prove that in dimensions n ≥ 3 knowledge of the Dirichlet-to-Neumann map measured on particular subsets of the boundary determines uniquely a conductivity with essentially 3/2 derivatives.
TL;DR: In this paper, it was shown that for a large class of Bernoulli problems, a free boundary which is symmetric with respect to a vertical line through an isolated singular point must necessarily have a corner at that point, and a formula for the contained angle.
Abstract: In this article we show that, for a large class of Bernoulli problems, a free boundary which is symmetric with respect to a vertical line through an isolated singular point must necessarily have a corner at that point, and we give a formula for the contained angle. The assumptions used admit the possibility of other singular points, even uncountably many, on the free boundary. This result is an extension of the first Stokes conjecture in the theory of hydrodynamic waves. We also show that, even in the presence of singularities, a geometrically simple Bernoulli free boundary is necessarily symmetric.
TL;DR: In this paper, the authors consider a shallow water equation of the Camassa-Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects and prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equations converge to discontinuous weak solutions of a scalar conservation law.
Abstract: We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.
TL;DR: For any nonnegative, quasi-convex Hamiltonian H ∞-functional F(u, ·) = H(∇ u) ∈ L ∞(·) over R n, the notion of comparison with generalized cones (abbreviated CGC) was introduced by Crandall et al..
Abstract: For any non-negative, quasi-convex Hamiltonian H ∈ C 2(R n ), we consider the L ∞-functional F(u, ·) = ‖H(∇ u)‖ L ∞(·) over . We introduce the notion of comparison with generalized cones (abbreviated CGC) and prove that CGC, viscosity solutions of the Aronsson equation, and absolute minimizers of F(·) are equivalent. This extends an earlier result by Crandall et al. (2001).
TL;DR: In this article, the authors consider the Schrodinger equation corresponding to the scattering metric and study the propagation of singularities for the solution in terms of the homomorphic singularity.
Abstract: Given a scattering metric on the Euclidean space. We consider the Schrodinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the “homo...
TL;DR: In this article, the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein-Uhlenbeck operator in ℝ N was studied.
Abstract: We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ℝ N . The long time behavior in the main results is sta...
TL;DR: In this article, the existence and uniqueness of the solutions of first-order Hamilton-Jacobi Equations arising in the theory of dislocations were proved under suitable regularity assumptions on the initial data and the velocity.
Abstract: We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions on the initial data and the velocity. These results are based on new L 1-type estimates on the viscosity solutions of first-order Hamilton–Jacobi Equations appearing in the so-called “level-sets approach”. Our work is inspired by and simplifies a recent work of Alvarez et al. (2005).
TL;DR: In this paper, the existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions was proved for both the cases when the forcing frequency is a rational number and an irrational number.
Abstract: We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases when the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.
TL;DR: In this article, a collisionless plasma is modelled by the Vlasov-Poisson system in three space dimensions, and the situation in which mobile negative ions balance the positive charge as | x | → ∞ is considered.
Abstract: A collisionless plasma is modelled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge—dependant upon only velocity—is assumed. The situation in which mobile negative ions balance the positive charge as | x | → ∞ is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large | x |, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as | x |−6. This article also establishes decay estimates for the electrostatic field and its derivatives.
TL;DR: In this article, the authors prove uniqueness for two dimensional transport across a noncharacteristic curve, under the hypothesis that the vector field is autonomous, bounded, and with bounded divergence.
Abstract: We prove uniqueness for two dimensional transport across a noncharacteristic curve, under the hypothesis that the vector field is autonomous, bounded, and with bounded divergence. We also obtain uniqueness for the Cauchy problem in under an additional condition on the local direction of the vector field.