The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$
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| Format: | Preprint |
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2017
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| _version_ | 1866916631203020800 |
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| author | Giga, Yoshikazu Gries, Mathis Hieber, Matthias Hussein, Amru Kashiwabara, Takahito |
| author_facet | Giga, Yoshikazu Gries, Mathis Hieber, Matthias Hussein, Amru Kashiwabara, Takahito |
| contents | Consider the primitive equations on $\R^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1 \in BUC_σ(\R^2;L^1(z_0,z_1))$ and $a_2 \in L^\infty_σ(\R^2;L^1(z_0,z_1))$ and where $BUC_σ(L^1)$ and $L^\infty_σ(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on $\R^2$, respectively, which take values in $L^1(z_0,z_1)$. These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if $\|a_2\|_{L^\infty_σ(L^1)}$ is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition $a$ is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $L^\infty(L^1)$-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1710_04434 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$ Giga, Yoshikazu Gries, Mathis Hieber, Matthias Hussein, Amru Kashiwabara, Takahito Analysis of PDEs 35Q35 (Primary), 76D03, 47D06, 86A05 (Secondary) Consider the primitive equations on $\R^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1 \in BUC_σ(\R^2;L^1(z_0,z_1))$ and $a_2 \in L^\infty_σ(\R^2;L^1(z_0,z_1))$ and where $BUC_σ(L^1)$ and $L^\infty_σ(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on $\R^2$, respectively, which take values in $L^1(z_0,z_1)$. These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if $\|a_2\|_{L^\infty_σ(L^1)}$ is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition $a$ is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $L^\infty(L^1)$-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations. |
| title | The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$ |
| topic | Analysis of PDEs 35Q35 (Primary), 76D03, 47D06, 86A05 (Secondary) |
| url | https://arxiv.org/abs/1710.04434 |