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The global well-posedness of the Navier-Stokes-Korteweg equations. (English) Zbl 1464.35249

The present paper deals with the global wellposedness of the Navier–Stokes–Korteweg system for small initial data \((\rho_0,\mathbf{u}_0)\) belonging to the Besov space \(\bigcap_{i=1}^2 \left( B^{3-2/p}_{q_i,p}(\mathbb{R}^N)\times B^{2(1-1/p)}_{q_i,p}(\mathbb{R}^N)^N\right) \) and to the Sobolev space \(W^{1,0}_{q_1/2}(\mathbb{R}^N)^{N+1}\), with \(3\leq N\leq 7\), \(2<p<\infty\), \(2<q_1\leq 4\) and \(q_1<N< q_2\) such that \(2/p+N/q_2<1\) and \(1/q_1 = 1/q_2+1/N\). The fluid viscosities and capillary coefficients are assumed constant, while the pressure is a \(C^\infty\) function only dependent on the density. Firstly, the local existence and uniqueness of solution is proved by the Banach contraction mapping theorem. Also the proof of the global existence and uniqueness relies on the Banach contraction mapping theorem, but now making recourse to the decay properties of solutions to the linearized problem.

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B65 Smoothness and regularity of 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
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