Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in \(\mathbb{R}^3\). (English) Zbl 1534.35330
Summary: The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with [Y. Chen et al., Discrete Contin. Dyn. Syst. 36, No. 10, 5287–5307 (2016; Zbl 1353.35222)]. Under further assumptions that the stationary solution \((\rho_s (x),0,0)^T\) is in a small neighborhood of the constant state \((\bar{\rho}, 0,0)^T\) at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear \(L^p\)-\(L^q\) decay estimates.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A09 | Classical solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Navier-Stokes-Smoluchowski equations; global classical solution; decay estimates; energy methodCitations:
Zbl 1353.35222References:
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