Upper and lower <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo>˙</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math> estimates for solutions to parabolic equations

Guterres, Robert; Niche, César J.; Perusato, Cilon; Zingano, Paulo R.

doi:10.1016/j.jde.2023.01.036

Cited by 10 publications

(4 citation statements)

References 39 publications

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“…Given

\hspace{-0.42677pt}\mbox{\boldmath $f$}(\cdot ,t) \hspace{-0.42677pt} \in \hspace{-0.56917pt} L^{1}((0,\infty ), L^{2}_{\sigma }(\mathbb {R}^{n}))

satisfying (2.2) and, in addition,

\Vert \hspace{0.85355pt} \mbox{\boldmath $f$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\dot{H}^{m}} \!= O(1 + t)^{-\;\!\alpha \;\!-\:\!1\:\!-\;\! m/2}

for some

m \geqslant 0

, then it follows from Wiegner's theorems above and [6, Theorem 1.1] that, for

t \gg 1

\Vert \hspace{0.1424pt} D^{\ell }\hspace{0.1424pt} \mbox{\boldmath $u$}\hspace{0.1424pt}(\cdot ,t)\hspace{0.56917pt} \Vert _{L^{2}(\mathbb {R}^{n})} \hspace{-0.85355pt}= \hspace{0.28436pt}O\hspace{0.28436pt}(\hspace{0.28436pt} t^{\:\!-\;\!\alpha \hspace{1.42271pt}-\hspace{1.42271pt} \ell \hspace{0.28436pt}/2}\hspace{0.28436pt}) \hspace{-0.28436pt}

and

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

“…Remark Under the same conditions given in Remark 2.3, but assuming more strongly that

\Vert \hspace{0.28436pt} \mbox{\boldmath $f$}(\cdot ,t) \hspace{0.56917pt}\Vert _{L^{2}(\mathbb {R}^{n})} \hspace{-1.13791pt}=\hspace{0.28436pt} o\hspace{0.56917pt}(1 + t)^{-\;\!\alpha \;\!-\:\!1} \hspace{-1.42271pt}

, it follows from Wiegner's theorems and [6, Theorem 1.5] that

\begin{equation*} \nonumber \liminf _{t\,\rightarrow \,\infty } \hspace{1.42271pt} t^{\:\!\alpha \hspace{1.42271pt}+\hspace{1.42271pt} \ell \hspace{0.28436pt}/2} \;\! \Vert \hspace{0.56917pt}D^{\ell } \mbox{\boldmath $u$}\hspace{0.28436pt}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \hspace{-0.56917pt}&gt;\hspace{1.42271pt}0 \end{equation*}

for all

\hspace{0.85355pt} 0 \hspace{-0.28436pt}\leqslant \hspace{-0.42677pt} \ell \hspace{-0.56917pt}\leqslant \hspace{-0.42677pt} m + 1

, provided that one has

λ (α) \equiv {lim inf}_{t \to \infty} t^{α} false∥ boldu (...

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

“…Moreover, (cf. [6, Theorem 1.5]),

\begin{equation*} \nonumber \liminf _{t\,\rightarrow \,\infty } \hspace{1.70709pt} t^{\:\!\alpha \hspace{1.13791pt}+\hspace{1.13791pt} \ell /2} \hspace{1.42271pt} \Vert \hspace{0.56917pt} D^{\ell } \mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \hspace{-1.13791pt}\geqslant \hspace{0.28436pt} K\hspace{-0.56917pt}(\alpha ,\ell ,r) \cdot \hspace{0.56917pt} \liminf _{t\,\rightarrow \,\infty } \hspace{1.70709pt} t^{\:\!\alpha } \hspace{0.85355pt} \Vert \hspace{1.13791pt}\mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \end{equation*}

where

$ { \hspace{0.56917pt} r = \lambda (\alpha )/ \limsup _{t\,\rightarrow \,\infty } \hspace{1.42271pt} t^{\:\!\alpha } \;\! \Vert \hspace{1.42271pt}\mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\scriptstyle L...

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

See 2 more Smart Citations

On the topological size of the class of Leray solutions with algebraic decay

Brandolese,

Perusato,

Zingano

2023

Bulletin of London Math Soc

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In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions to the incompressible Navier–Stokes in : if the associated Stokes flows had their norms bounded by for some , then the same would be true of . The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form .

show abstract

“…Given

\hspace{-0.42677pt}\mbox{\boldmath $f$}(\cdot ,t) \hspace{-0.42677pt} \in \hspace{-0.56917pt} L^{1}((0,\infty ), L^{2}_{\sigma }(\mathbb {R}^{n}))

satisfying (2.2) and, in addition,

\Vert \hspace{0.85355pt} \mbox{\boldmath $f$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\dot{H}^{m}} \!= O(1 + t)^{-\;\!\alpha \;\!-\:\!1\:\!-\;\! m/2}

for some

m \geqslant 0

, then it follows from Wiegner's theorems above and [6, Theorem 1.1] that, for

t \gg 1

\Vert \hspace{0.1424pt} D^{\ell }\hspace{0.1424pt} \mbox{\boldmath $u$}\hspace{0.1424pt}(\cdot ,t)\hspace{0.56917pt} \Vert _{L^{2}(\mathbb {R}^{n})} \hspace{-0.85355pt}= \hspace{0.28436pt}O\hspace{0.28436pt}(\hspace{0.28436pt} t^{\:\!-\;\!\alpha \hspace{1.42271pt}-\hspace{1.42271pt} \ell \hspace{0.28436pt}/2}\hspace{0.28436pt}) \hspace{-0.28436pt}

and

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

“…Remark Under the same conditions given in Remark 2.3, but assuming more strongly that

\Vert \hspace{0.28436pt} \mbox{\boldmath $f$}(\cdot ,t) \hspace{0.56917pt}\Vert _{L^{2}(\mathbb {R}^{n})} \hspace{-1.13791pt}=\hspace{0.28436pt} o\hspace{0.56917pt}(1 + t)^{-\;\!\alpha \;\!-\:\!1} \hspace{-1.42271pt}

, it follows from Wiegner's theorems and [6, Theorem 1.5] that

\begin{equation*} \nonumber \liminf _{t\,\rightarrow \,\infty } \hspace{1.42271pt} t^{\:\!\alpha \hspace{1.42271pt}+\hspace{1.42271pt} \ell \hspace{0.28436pt}/2} \;\! \Vert \hspace{0.56917pt}D^{\ell } \mbox{\boldmath $u$}\hspace{0.28436pt}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \hspace{-0.56917pt}&gt;\hspace{1.42271pt}0 \end{equation*}

for all

\hspace{0.85355pt} 0 \hspace{-0.28436pt}\leqslant \hspace{-0.42677pt} \ell \hspace{-0.56917pt}\leqslant \hspace{-0.42677pt} m + 1

, provided that one has

λ (α) \equiv {lim inf}_{t \to \infty} t^{α} false∥ boldu (...

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

“…Moreover, (cf. [6, Theorem 1.5]),

\begin{equation*} \nonumber \liminf _{t\,\rightarrow \,\infty } \hspace{1.70709pt} t^{\:\!\alpha \hspace{1.13791pt}+\hspace{1.13791pt} \ell /2} \hspace{1.42271pt} \Vert \hspace{0.56917pt} D^{\ell } \mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \hspace{-1.13791pt}\geqslant \hspace{0.28436pt} K\hspace{-0.56917pt}(\alpha ,\ell ,r) \cdot \hspace{0.56917pt} \liminf _{t\,\rightarrow \,\infty } \hspace{1.70709pt} t^{\:\!\alpha } \hspace{0.85355pt} \Vert \hspace{1.13791pt}\mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\mbox{}_{\scriptstyle L^{2}(\mathbb {R}^{n})}} \end{equation*}

where

$ { \hspace{0.56917pt} r = \lambda (\alpha )/ \limsup _{t\,\rightarrow \,\infty } \hspace{1.42271pt} t^{\:\!\alpha } \;\! \Vert \hspace{1.42271pt}\mbox{\boldmath $u$}(\cdot ,t) \hspace{0.56917pt} \Vert _{\scriptstyle L...

…”

Section: Inverse Wiegner's Theorem For the Navier–stokes Equationsmentioning

confidence: 99%

See 1 more Smart Citation

On the topological size of the class of Leray solutions with algebraic decay

Brandolese,

Perusato,

Zingano

2023

Bulletin of London Math Soc

Self Cite

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show abstract

“…Equations as (1.1) that combines effects of the advective and diffusive terms have been widely studied due to their applications in several areas [2,3,5,8,9,10,12,14,15].…”

Section: Introductionmentioning

confidence: 99%