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Decay of dissipative equations and negative Sobolev spaces
Yan Guo,Yanjin Wang +1 more
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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.read more
Citations
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Journal ArticleDOI
Decay of the Navier-Stokes-Poisson equations
TL;DR: In this article, the authors established the time decay rates of the solution to the Cauchy problem for the compressible Navier-Stokes-Poisson system via a refined pure energy method.
Journal ArticleDOI
The Boltzmann equation, Besov spaces, and optimal time decay rates in Rxn
Vedran Sohinger,Robert M. Strain +1 more
TL;DR: In this article, it was shown that k-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) converge in large time to the global Maxwellian with the optimal decay rate of O( t − 1 2 (k + ϱ + n 2 − n r ) ) in the L x r (L v 2 ) -norm for any 2 ≤ r ≤ ∞.
Journal ArticleDOI
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis
Zhi-An Wang,Zhaoyin Xiang,Pei Yu +2 more
TL;DR: In this paper, the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated.
Journal ArticleDOI
Global well-posedness and large-time decay for the 2D micropolar equations
TL;DR: In this article, the authors studied the global (in time) regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation.
Journal ArticleDOI
Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework
Raphaël Danchin,Jiang Xu +1 more
TL;DR: In this article, it was shown that the global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago.
References
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Book
Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Book
Fourier Analysis and Nonlinear Partial Differential Equations
TL;DR: In this paper, the compressible Navier-Stokes system was proposed to solve semilinear dispersive equations, and the smoothing effect in quasileinear wave equations was analyzed.
Book ChapterDOI
On elliptic partial differential equations
TL;DR: In this paper, a series of lectures on the theory of elliptic differential equations is presented, including the Hilbert space approach to the Dirichlet problem for strongly elliptic systems, and various inequalities.
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