Abstract. 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. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension d ≥ 2 has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of [7] but also in the more general L p critical framework of [3,6,14]. More precisely, we show that under a mild additional decay assumption that is satisfied if the low frequencies of the initial data are in e.g. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics.

This work is concerned with (N -component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the ShizutaKawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local wellposedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.

We give a new decay framework for general dissipative hyperbolic system and hyperbolic-parabolic composite system, which allow us to pay less attention on the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish the optimal decay estimates on the framework of spatially critical Besov spaces for degenerately dissipative hyperbolic system of balance laws. Based on the L p (R n ) embedding and improved Gagliardo-Nirenberg inequality, the optimal L p (R n )-L 2 (R n )(1 ≤ p < 2) decay rates and L p (R n )-L q (R n )(1 ≤ p < 2 ≤ q ≤ ∞) decay rates are further shown. Finally, as a direct application, the optimal decay rates for 3D damped compressible Euler equations are also obtained.

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In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the highand low-frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood-Paley decomposition and Bony's para-product formula..

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The global existence issue in critical spaces for compressible Navier-Stokes equations, was addressed by Danchin (2000) and then developed by Charve-Danchin (2010), Chen-Miao-Zhang (2010) and Hasport (2011) in more general L p setting. Our aim is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical framework. As we known, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed in usual L 1 space by Matsumura and Nishida [30]. In this paper, we claim a new low-frequency assumption for barotropic compressible Navier-Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov spaceḂ −σ 1 2,∞ with the regularity σ1 ∈ (1−d/2, 2d/p−d/2] , then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are also exhibited. The proof mainly depends on tricky and non classical Besov estimates. As a by-product, those optimal time-decay rates of L q-L r type are well captured in the critical regularity framework.

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