Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces \(L^ p_ s({\mathbb{R}}^ 2)\).

*(English)*Zbl 0615.35078This paper is a sequel to the authors’ earlier papers [the first author, Proc. Symp. Pure Math. 45, Pt. 2, 1-7 (1986; Zbl 0598.35093) and the second author, Commun. Partial Differ. Equations 11, 483-511 (1986; Zbl 0594.35077)] and is concerned with the well-posedness (i.e., existence, uniqueness, and persistence) of the viscous and nonviscous solutions of the fluid dynamical equations as well as with the convergence of the viscous solutions as the viscosity tends to zero. It is shown that the Navier-Stokes (NS) and Euler equations for incompressible fluids possess well-posedness in the function space \(X=PL^ p_ s\) with \(s>1+2/p\), \(1<p<\infty\) and P is a pseudo-differential operator.

In particular, starting from the basic initial value problem for the NS equations, the following results have been established.

(i) The NS equation in \({\mathbb{R}}^ m\) is locally (in time) well-posed in the Lebesgue spaces \(PL^ p_ s({\mathbb{R}}^ m)\), \(1<p<\infty\), \(s>1+m/p.\)

(ii) The NS equation is globally well-posed in \(PL^ p_ s({\mathbb{R}}^ 2)\) (p,s, as before) with uniform estimates for the viscosity \(\nu >0.\)

(iii) The Euler equation is globally well-posed in \(PL^ p_ s({\mathbb{R}}^ 2).\)

(iv) As \(\nu\to 0\), the NS solution \(u^{\nu}(t)\) converges to the Euler solution u(t) weakly in \(PL^ p_ s\) for all \(t\geq 0\), locally uniformly in t.

The results (ii)-(iv) have been established with the help of the vorticity equation derived from the Navier-Stokes equations and its properties.

In particular, starting from the basic initial value problem for the NS equations, the following results have been established.

(i) The NS equation in \({\mathbb{R}}^ m\) is locally (in time) well-posed in the Lebesgue spaces \(PL^ p_ s({\mathbb{R}}^ m)\), \(1<p<\infty\), \(s>1+m/p.\)

(ii) The NS equation is globally well-posed in \(PL^ p_ s({\mathbb{R}}^ 2)\) (p,s, as before) with uniform estimates for the viscosity \(\nu >0.\)

(iii) The Euler equation is globally well-posed in \(PL^ p_ s({\mathbb{R}}^ 2).\)

(iv) As \(\nu\to 0\), the NS solution \(u^{\nu}(t)\) converges to the Euler solution u(t) weakly in \(PL^ p_ s\) for all \(t\geq 0\), locally uniformly in t.

The results (ii)-(iv) have been established with the help of the vorticity equation derived from the Navier-Stokes equations and its properties.

Reviewer: P.Chandran

##### MSC:

35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |

35Q30 | Navier-Stokes equations |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

58D30 | Applications of manifolds of mappings to the sciences |

46N99 | Miscellaneous applications of functional analysis |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |