Existence of weak solutions to \(p\)-Navier-Stokes equations. (English) Zbl 1533.35270
Summary: We study the existence of weak solutions to the \(p\)-Navier-Stokes equations with a symmetric \(p\)-Laplacian on bounded domains. We construct a particular Schauder basis in \(W_0^{1,p}(\Omega)\) with divergence free constraint and prove existence of weak solutions using the Galerkin approximation via this basis. Meanwhile, in the proof, we establish a chain rule for the \(L^p\) integral of the weak solutions, which fixes a gap in our previous work. The equality of energy dissipation is also established for the weak solutions considered.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35D30 | Weak 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 |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
Keywords:
Navier-Stokes equations; \(p\)-Laplacian; Galerkin approximation; Leray projection; energy dissipation; Aubin-Lions lemmaReferences:
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