Global strong solution for 3D viscous incompressible heat conducting Navier-Stokes flows with non-negative density. (English) Zbl 1377.35227
The global existence and uniqueness theorems of a strong solution for nonsteady nonhomogeneous incompressible heat conducting Navier-Stokes fluids with vacuum are proved, improving the result of Y. Cho and H. Kim [J. Korean Math. Soc. 45, No. 3, 645–681 (2008; Zbl 1144.35307)], which only looked at local existence. A blow-up criterion is given, independent of the temperature. The Gronwall, Holder, Sobolev and Young inequalities are used. A strong solution is considered, on the maximal time interval \((0, T)\). The author extends this solution beyond \(T\), in contradiction with the maximality of \(T\). The existence of a unique global strong solution is proved, if a specific ratio containing the viscosity, the initial velocity and density is small enough. The author proved that the maximal existence time interval \((0,T)\) can be extended to \((0, \infty)\), also by contradiction. For this, some regularity results for Stokes equations and very interesting energy estimates are used, based on the maximum principle. A large list of references is given in the last part, containing recent results in the considered topic.
Reviewer: Gelu Paşa (Bucureşti)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35D35 | Strong solutions to PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B44 | Blow-up in context of PDEs |
| 35B50 | Maximum principles in context of PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Citations:
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