Showing papers in "Communications in Partial Differential Equations in 2012"
TL;DR: In this paper, coupled chemotaxis (Navier and Stokes) systems generalizing the prototype have been proposed to describe the collective effects arising in bacterial suspensions in fluid drops, and they have been applied to the model of collective effects of bacterial suspensions.
Abstract: In the modeling of collective effects arising in bacterial suspensions in fluid drops, coupled chemotaxis-(Navier–)Stokes systems generalizing the prototype have been proposed to describe the spont
523 citations
TL;DR: In this article, the authors developed a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space, which is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation.
Abstract: We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.
262 citations
TL;DR: It is proved that in the framework of the model, chemotaxis plays a crucial role and there is a rigid limit to how much the fertilization efficiency can be enhanced if there is no Chemotaxis but only advection and diffusion.
Abstract: Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. However, to the best of our knowledge, the question of reaction enhancement by chemotaxis has not yet received extensive treatment either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and dif...
118 citations
TL;DR: In this article, the authors established a blow up criterion for the short time classical solution of the nematic liquid crystal flow, a simplified version of the Ericksen-Leslie system, in dimensions two and three.
Abstract: In this paper, we establish a blow up criterion for the short time classical solution of the nematic liquid crystal flow, a simplified version of Ericksen–Leslie system modeling the hydrodynamic evolution of nematic liquid crystals, in dimensions two and three. More precisely, 0 < T * < + ∞ is the maximal time interval iff (i) for n = 3, ; and (ii) for n = 2, .
94 citations
TL;DR: In this article, the authors gave a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation −div(|Du| p−2 Du) = 0 coincide.
Abstract: In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation −div(|Du| p−2 Du) = 0 coincide Our proof is more direct and transparent than the original proof of Juutinen et al [8], which relied on the full uniqueness machinery of the theory of viscosity solutions We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation
76 citations
TL;DR: In this paper, the authors prove a comparison theorem on the modulus of continuity of the solution of a heat equation with a drifting term on Bakry-Emery manifolds.
Abstract: We prove a comparison theorem on the modulus of continuity of the solution of a heat equation with a drifting term on Bakry-Emery manifolds. A direct consequence of the result is an alternate proof of an eigenvalue comparison result of Bakry-Qian. Examples are given to show that the estimate is sharp. Discussions on an explicit lower estimate for the corresponding ODE and an application to the diameter lower bound for gradient shrinking solitons are also included.
72 citations
TL;DR: In this article, the analysis of the Keller-Segel system over Ω d, d ≤ 3 is presented, and the main results in the parabolic-elliptic case are: local existence without smallness assumption on the initial density and a quantified blow-up rate.
Abstract: This article is devoted to the analysis of the classical Keller–Segel system over ℝ d , d ≥ 3. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results in the parabolic-elliptic case are: local existence without smallness assumption on the initial density and a quantified blow-up rate, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon for the fully parabolic case is also given.
64 citations
TL;DR: The existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity, was proved in this paper.
Abstract: We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.
63 citations
TL;DR: In this article, weak solutions to the Cauchy problem for the three dimensional Vlasov-Poisson system of equations were considered and the authors obtained a propagation result for any velocity moment of order 2 as well as a uniqueness statement in ℝ3.
Abstract: We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ℝ3. In the periodic case, we show that velocity moments of order > 14/3 are propagated.
61 citations
TL;DR: In this article, the cubic nonlinear Schrodinger equation was studied on a Riemannian manifold and the authors proved that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.
Abstract: We consider the cubic nonlinear Schrodinger equation, posed on ℝ n × M, where M is a compact Riemannian manifold and n ≥ 2. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.
60 citations
TL;DR: In this article, the stability problem for standing waves of nonlinear Dirac models under a suitable definition of linear stability, and under some restriction on the spectrum, was considered and it was proved at the same time orbital and asymptotic stability.
Abstract: We consider the stability problem for standing waves of nonlinear Dirac models Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability We are not able to get the full result proved in [24] for the nonlinear Schrodinger equation, because of the strong indefiniteness of the energy
TL;DR: In this article, it was shown that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally Ω(mathcal C}^{0,\alpha} ).
Abstract: We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally ${\mathcal C}^{0,\alpha}$. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.
TL;DR: In this article, the authors used new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data.
Abstract: In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations, recently established by De Philippis and Figalli [14], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.
TL;DR: In this article, the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time is proved.
Abstract: We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.
TL;DR: In this paper, the large time behavior of solutions to a nematic liquid crystal system in the whole space ℝ3 was studied, where the fluid under consideration has constant density and small initial data.
Abstract: In this paper we study the large time behavior of solutions to a nematic liquid crystal system in the whole space ℝ3. The fluid under consideration has constant density and small initial data.
TL;DR: In this paper, the authors studied 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.
Abstract: This paper is concerned with the study of the large-time behaviour 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. 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 ∈ R and for all speeds c belonging to a given interval (γ 1 , γ 2) with large enough γ 1 < γ 2 ≤ +∞.
TL;DR: In this article, a generalized version of the Calderon problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension n ≥ 2.
Abstract: A generalized variant of the Calderon problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension n ≥ 2. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on this subset.
TL;DR: In this article, the authors studied a version of the stochastic "tug-of-war" game, played on graphs and smooth domains, with the empty set of terminal states, and proved that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops.
Abstract: We study a version of the stochastic “tug-of-war” game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.
TL;DR: In this article, the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting is considered.
Abstract: We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface tension is either taken into account at both free boundaries or neglected at both. We are concerned with the Rayleigh–Taylor instability, so we assume that the upper fluid is heavier than the lower fluid. When the surface tension at the free internal interface is below a critical value, which we identify, we establish that the problem under consideration is nonlinearly unstable.
TL;DR: In this article, the authors studied global in time solutions of the Higgs equation in the Minkowski and in the de Sitter spacetimes and gave sufficient conditions for the existence of the zeros of global solutions in the interior of their supports.
Abstract: In this article we study global in time (not necessarily small) solutions of the equation for the Higgs boson in the Minkowski and in the de Sitter spacetimes. We reveal some qualitative behavior of the global solutions. In particular, we formulate sufficient conditions for the existence of the zeros of global solutions in the interior of their supports, and, consequently, for the creation of the so-called bubbles, which have been studied in particle physics and inflationary cosmology. We also give some sufficient conditions for the global solution to be oscillatory in time.
TL;DR: In this paper, the existence of modified wave operators or wave operators under some mass conditions was proved under a system of nonlinear Schrodinger equations with quadratic nonlinearities in two space dimensions.
Abstract: We consider a system of nonlinear Schrodinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.
TL;DR: In this paper, the authors studied the magnetic Schrodinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive.
Abstract: We continue our study of a magnetic Schrodinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.
TL;DR: In this paper, the authors consider a reaction-diffusion system which models a fast reversible reaction of type C 1+C 2⇌C 3 between mobile reactants inside an isolated vessel.
Abstract: We consider a reaction-diffusion system which models a fast reversible reaction of type C 1 + C 2⇌C 3 between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L 2-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.
TL;DR: In this paper, the authors consider the defocusing cubic non-linear Schrodinger equation on general closed Riemannian surfaces and extend the range of global well-posedness to.
Abstract: We consider the defocusing cubic non-linear Schrodinger equation on general closed (compact without boundary) Riemannian surfaces. The problem was shown to be locally well-posed in H s (M) for in [8]. Global well-posedness for s ≥ 1 follows easily from conservation of energy and standard arguments. In this work, we extend the range of global well-posedness to . This generalizes, without any loss in regularity, the results in [6, 18], where the same result is proved for the torus 𝕋2. The proof relies on the I-method of Colliander et al. [17] a semi-classical bilinear Strichartz estimate proved by the author in [22], and spectral localization estimates for products of eigenfunctions, which is essential to develop multilinear spectral analysis on general compact manifolds.
TL;DR: In this paper, the authors studied the Fisher-KPP equation with a fractional Laplacian of order α ∈ (0, 1) and showed that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α Ω( √ 0, 1).
Abstract: We study the Fisher-KPP equation with a fractional Laplacian of order α ∈ (0, 1). We know that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α ∈ (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order -ln (1 − α), the propagation is linear, and becomes exponential as soon as the time exceeds a large multiple of -ln (1 − α).
TL;DR: In this paper, the authors study a Fokker-planck equation with linear diffusion and super-linear drift introduced by Kaniadakis and Quarati to describe the evolution of a gas of Bose-Einstein particles.
Abstract: We study a Fokker–Planck equation with linear diffusion and super-linear drift introduced by Kaniadakis and Quarati [12, 13] to describe the evolution of a gas of Bose–Einstein particles. For kinetic equation of this type it is well-known that, in the physical space ℝ3, the structure of the equilibrium Bose–Einstein distribution depends upon a parameter m*, the critical mass. We are able to describe the time-evolution of the solution in two different situations, which correspond to m ≪ m* and m ≫ m* respectively. In the former case, it is shown that the solution remains regular, while in the latter we prove that the solution starts to blow up at some finite time t c , for which we give an upper bound in terms of the initial mass. The results are in favor of the validation of the model, which, in the supercritical regime, could produce in finite time a transition from a normal fluid to one with a condensate component.
TL;DR: In this article, it was shown that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution.
Abstract: We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the initial elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of P and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.
TL;DR: In this article, a notion of weak isospectrality for continuous deformations was introduced for the Laplace-Beltrami operator on a compact Riemannian manifold with Robin boundary conditions.
Abstract: We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (ℤ/2ℤ)2 group of symmetries.
TL;DR: In this paper, a class of linear kinetic Fokker-planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered.
Abstract: A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a...
TL;DR: In this paper, the global in time well-posedness of the non-local diffusion equation with α ∈ (0, 2/3) was proved, and the initial condition u 0 is positive, radial, and non-increasing with u 0 ∈ L 1 ∩ L 2+δ(ℝ3) for some small δ > 0.
Abstract: In this paper we prove the global in time well-posedness of the following non-local diffusion equation with α ∈ (0, 2/3): The initial condition u 0 is positive, radial, and non-increasing with u 0 ∈ L 1 ∩ L 2+δ(ℝ3) for some small δ > 0. There is no size restriction on u 0. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: u t = ▵ u + αu 2.