It is known that the one-dimensional nonlinear heat equation ut ： f（u）x1x1, f＇（u） 〉 0, u（±∞, t） ： u, u＋ ≠ u- has a unique self-similar solution u（x1/√1＋t）. In multi-dimensional space, （x1/√1＋t...It is known that the one-dimensional nonlinear heat equation ut ： f（u）x1x1, f＇（u） 〉 0, u（±∞, t） ： u, u＋ ≠ u- has a unique self-similar solution u（x1/√1＋t）. In multi-dimensional space, （x1/√1＋t） is called a planar diffusion wave. In the first part of the present paper, it is shown that under some smallness conditions, such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation： ut -△f（u） = 0, x ∈ R^n. The optimal time decay rate is obtained. In the second part of this paper, it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping： utt ＋ ut - △f（u） = 0, x ∈ R^n. The time decay rate is also obtained. The proofs are given by an elementary energy method.展开更多
In this paper, we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d ≥ 3. Moreover, u has a uniq...In this paper, we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d ≥ 3. Moreover, u has a unique trajectory which is HSlder continuous with respect to the space variables.展开更多
This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≥ 2. We address the question of well-posedness for large data having critical Besov regularity. Our result improves N the...This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≥ 2. We address the question of well-posedness for large data having critical Besov regularity. Our result improves N the analysis of Danchin and of the author inasmuch as we may take initial density in BN/p p,1 with 1 ≤ p 〈 ＋∞. Our result relies on a new a priori estimate for the velocity, where we introduce a new unknown called effective velocity to weaken one of the couplings between the density and the velocity. In particular, our result is the first in which we obtain uniqueness without imposing hypothesis on the gradient of the density.展开更多
Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey pa...Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey paper, we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space. We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics.展开更多
We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. In the case of free surface incompressible flows, the FS-model is formally obtained, using form...We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. In the case of free surface incompressible flows, the FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the P-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope with the transition between a free surface state and a pressured （i.e., compressible） state, we propose a mixed model, the PFS-model, taking into account changes of section and slope variation.展开更多
For the focusing mass-critical NLS iu, ＋ △u = --｜u｜u, it is conjectured that the only global nonscattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, w...For the focusing mass-critical NLS iu, ＋ △u = --｜u｜u, it is conjectured that the only global nonscattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H1 initial data in dimensions d = 2, 3 with spherical symmetry and d ≥ 4 with certain splitting-spherically symmetric initial data.展开更多
We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations. We address the Cau...We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations. We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves, also termed anchored waves. We further discuss related models which have a similar hyperbolic-parabolic structure, notably the Saint- Venant system with a temperature equation as well as the equations governing chemical equilibrium flows. We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure. We finally discuss numerical algorithms specifically devoted to complex chemistry flows, in particular the evaluation of multicomponent transport properties, as well as the impact of multicomponent transport.展开更多
The H@braud-Lequeux model is a model describing the flow of soft glassy material in a simple shear flow configuration. It is given by a kinetic/Fokker-Planck-type equation whose coefficients depend on the shear rate o...The H@braud-Lequeux model is a model describing the flow of soft glassy material in a simple shear flow configuration. It is given by a kinetic/Fokker-Planck-type equation whose coefficients depend on the shear rate of the experiment. In this paper we want to study what happens to the stationary solutions of this model when the shear rate is asymptotically large. In order to do that, we expand the solution of the equation using singular perturbation tools. In the end, we rigorously prove the estimate of Hebraud and Lequeux that the material asymptotically behaves as a Newtonian fluid.展开更多
In this note, we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity. The derivation of such equations is based on formal a...In this note, we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity. The derivation of such equations is based on formal asymptotic expansions of solutions of Cauchy momentum equations in the shallow water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q.展开更多
基金Acknowledgements He＇s research is supported in part by National Basic Research Program of China （Grant No. 2006CB805902）. Huang＇ research is supported in part by National Natural Science Foundation of China for Distinguished Youth Scholar （Grant No. 10825102）, NSFC-NSAF （Grant No. 10676037） and National Basic Research Program of China （Grant No. 2006CB805902）.
文摘It is known that the one-dimensional nonlinear heat equation ut ： f（u）x1x1, f＇（u） 〉 0, u（±∞, t） ： u, u＋ ≠ u- has a unique self-similar solution u（x1/√1＋t）. In multi-dimensional space, （x1/√1＋t） is called a planar diffusion wave. In the first part of the present paper, it is shown that under some smallness conditions, such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation： ut -△f（u） = 0, x ∈ R^n. The optimal time decay rate is obtained. In the second part of this paper, it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping： utt ＋ ut - △f（u） = 0, x ∈ R^n. The time decay rate is also obtained. The proofs are given by an elementary energy method.
基金Acknowledgements We are very grateful to the referee＇s suggestions and comments on the improvement of the paper. Part of this work was done when we were visiting Morningside Center of Mathematics, Chinese Academy of Sciences, in the summer of 2010. We appreciate the hospitality and the financial support from the center. P. Zhang is partially supported by National Natural Science Foundation of China （Grant Nos. 10421101 and 10931007）, and the innovation grant from the Chinese Academy of Sciences （Grant No. GJHZ200829）. T. Zhang is partially supported by the Program for New Century Excellent Talents in University, National Natural Science Foundation of China （Grant Nos. 10871175, 10931007 and 10901137）, and the Zhejiang Provincial Natural Science Foundation of China （Grant No. Z6100217）.
文摘In this paper, we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d ≥ 3. Moreover, u has a unique trajectory which is HSlder continuous with respect to the space variables.
文摘This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≥ 2. We address the question of well-posedness for large data having critical Besov regularity. Our result improves N the analysis of Danchin and of the author inasmuch as we may take initial density in BN/p p,1 with 1 ≤ p 〈 ＋∞. Our result relies on a new a priori estimate for the velocity, where we introduce a new unknown called effective velocity to weaken one of the couplings between the density and the velocity. In particular, our result is the first in which we obtain uniqueness without imposing hypothesis on the gradient of the density.
文摘Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey paper, we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space. We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics.
文摘We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. In the case of free surface incompressible flows, the FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the P-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope with the transition between a free surface state and a pressured （i.e., compressible） state, we propose a mixed model, the PFS-model, taking into account changes of section and slope variation.
基金Acknowledgements The first author is supported in part by National Science Foundation （Grant No. 0908032） and a start up fund in UBC. The second author is supported by an Alfred P. Sloan fellowship.
文摘For the focusing mass-critical NLS iu, ＋ △u = --｜u｜u, it is conjectured that the only global nonscattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H1 initial data in dimensions d = 2, 3 with spherical symmetry and d ≥ 4 with certain splitting-spherically symmetric initial data.
文摘We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations. We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves, also termed anchored waves. We further discuss related models which have a similar hyperbolic-parabolic structure, notably the Saint- Venant system with a temperature equation as well as the equations governing chemical equilibrium flows. We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure. We finally discuss numerical algorithms specifically devoted to complex chemistry flows, in particular the evaluation of multicomponent transport properties, as well as the impact of multicomponent transport.
文摘The H@braud-Lequeux model is a model describing the flow of soft glassy material in a simple shear flow configuration. It is given by a kinetic/Fokker-Planck-type equation whose coefficients depend on the shear rate of the experiment. In this paper we want to study what happens to the stationary solutions of this model when the shear rate is asymptotically large. In order to do that, we expand the solution of the equation using singular perturbation tools. In the end, we rigorously prove the estimate of Hebraud and Lequeux that the material asymptotically behaves as a Newtonian fluid.
文摘In this note, we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity. The derivation of such equations is based on formal asymptotic expansions of solutions of Cauchy momentum equations in the shallow water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q.
