The existence of Leray–Hopf–Masuda weak solutions with linear strain

Autor: Ryôhei Kakizawa
Rok vydání: 2020
Předmět:
Zdroj: Rendiconti del Circolo Matematico di Palermo Series 2. 70:1163-1173
ISSN: 1973-4409
0009-725X
DOI: 10.1007/s12215-020-00550-1
Popis: We are concerned with the global existence and regularity of the initial value problem for the Navier–Stokes equations perturbed from linear strain of the form A(t)x in $${\mathbb {R}}^{n}$$ ( $$n \in {\mathbb {Z}}$$ , $$n\ge 2$$ ). More precisely, we construct the Leray-Hopf weak solution which satisfies not only the Navier–Stokes equations but also the energy inequality via the Galerkin approximation provided that $$A \in C^{1}([0,T];M_{n}({\mathbb {R}}^{n}))$$ . Since any eigenvalue of the symmetric part of A is continuous with respect to t on [0, T], the Gronwall–Bellman inequality admits the uniform boundedness of the approximate solution. As for the regularity of the weak solutions, the $$L^{2}$$ -Sobolev space equipped with a weight $$w(x):=\sqrt{1+|x|^{2}}$$ is considered as the class of test functions.
Databáze: OpenAIRE
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