On Onsager's type conjecture for the inviscid Boussinesq equations
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| Format: | Preprint |
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2024
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| _version_ | 1866911910590414848 |
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| author | Miao, Changxing Nie, Yao Ye, Weikui |
| author_facet | Miao, Changxing Nie, Yao Ye, Weikui |
| contents | In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For $1\le p\le \infty$, we show that the threshold regularity exponent for $L^p$-norm conservation of temperature of this system is $1/3$, consistent with Onsager exponent. More precisely, for $1\le p\le\infty$, every weak solution $(v,θ)\in C_tC^β_x$ to the inviscid Boussinesq equations satisfies that $\|θ(t)\|_{L^p(\mathbb{T}^3)}=\|θ_0\|_{L^p(\mathbb{T}^3)}$ if $β>\frac{1}{3}$, while if $β<\frac{1}{3}$, there exist infinitely many weak solutions $(v,θ)\in C_tC^β_x$ such that the $L^p$-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in $C_tC^β_x$ for $β<\frac{1}{3}$ displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution $(v, θ)$ of this system has at least one interval of regularity, then this weak solution $(v,θ)$ is not unique in $C_tC^β_x$ for $β<\frac{1}{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_05337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Onsager's type conjecture for the inviscid Boussinesq equations Miao, Changxing Nie, Yao Ye, Weikui Analysis of PDEs In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For $1\le p\le \infty$, we show that the threshold regularity exponent for $L^p$-norm conservation of temperature of this system is $1/3$, consistent with Onsager exponent. More precisely, for $1\le p\le\infty$, every weak solution $(v,θ)\in C_tC^β_x$ to the inviscid Boussinesq equations satisfies that $\|θ(t)\|_{L^p(\mathbb{T}^3)}=\|θ_0\|_{L^p(\mathbb{T}^3)}$ if $β>\frac{1}{3}$, while if $β<\frac{1}{3}$, there exist infinitely many weak solutions $(v,θ)\in C_tC^β_x$ such that the $L^p$-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in $C_tC^β_x$ for $β<\frac{1}{3}$ displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution $(v, θ)$ of this system has at least one interval of regularity, then this weak solution $(v,θ)$ is not unique in $C_tC^β_x$ for $β<\frac{1}{3}$. |
| title | On Onsager's type conjecture for the inviscid Boussinesq equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.05337 |