Solutions In Critical Spaces Research Articles

Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity

Abstract In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.

Global Existence and Exponential Decay to Equilibrium for DLSS-Type Equations

In this paper, we deal with two logarithmic fourth order differential equations: the extended one-dimensional DLSS equation and its multi-dimensional analog. We show the global existence of solution in critical spaces, its convergence to equilibrium and the gain of spatial analyticity for these two equations in a unified way.

Optimal decay rate for strong solutions in critical spaces to the compressible Navier–Stokes equations

In this paper we are concerned with the convergence rates of the global strong solution to motionless state with constant density for the compressible Navier–Stokes equations in the whole space Rn for n≥3. It is proved that the perturbations decay in critical spaces, if the initial perturbations of density and velocity are small in B2,1n2(Rn)∩B˙1,∞0(Rn) and B2,1n2−1(Rn)∩B˙1,∞0(Rn), respectively.

Global existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity

We prove the global-in-time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ0,A0 ∈ L3,u0 ∈ L3 ∕ 2,u0 ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.

Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces

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 the analysis of Danchin and of the author inasmuch as we may take initial density in \(B_{p,1}^{\tfrac{N} {p}}\) 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.

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.

Global existence for the multi-dimensional compressible viscoelastic flows

The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic–parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the scaling of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the L 1 -decay of the density and deformation gradient.