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Daniele Tampieri
Daniele Tampieri
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Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer analysis of a model?

For example,in in https://arxiv.org/pdf/2006.03894.pdf - the authors consider the following problem: let us consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ and a positive number $T$, then in $\Omega \times (0,T)$ the locally bounded solutions of the inhomogeneous porusporous medium equation

$u_{t} - div(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r}, m>1$

are $$ u_{t} - \operatorname{div}(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r},\quad m>1 $$ are locally Hölder continuous, with exponent

$\gamma = min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big)$

where $$ \gamma = \min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big) $$ where $\iota^{-}$ means that we can select all $s \in (0,\iota)$ and $L^{q,r}$ is a Lebesgue space with mixed norm (see the paper above for more informations)

So, why find this $\alpha$ which is a sharp exponent is so important and what applications we can obtain from the quantitative Holder regularity perspective?

Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer analysis of a model?

For example,in https://arxiv.org/pdf/2006.03894.pdf - the authors consider the following problem: let us consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ and a positive number $T$, then in $\Omega \times (0,T)$ the locally bounded solutions of the inhomogeneous porus medium equation

$u_{t} - div(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r}, m>1$

are locally Hölder continuous, with exponent

$\gamma = min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big)$

where $\iota^{-}$ means that we can select all $s \in (0,\iota)$ and $L^{q,r}$ is a Lebesgue space with mixed norm (see the paper above for more informations)

So, why find this $\alpha$ which is a sharp exponent is so important and what applications we can obtain from the quantitative Holder regularity perspective?

Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer analysis of a model?

For example, in https://arxiv.org/pdf/2006.03894.pdf - the authors consider the following problem: let us consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ and a positive number $T$, then in $\Omega \times (0,T)$ the locally bounded solutions of the inhomogeneous porous medium equation $$ u_{t} - \operatorname{div}(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r},\quad m>1 $$ are locally Hölder continuous, with exponent $$ \gamma = \min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big) $$ where $\iota^{-}$ means that we can select all $s \in (0,\iota)$ and $L^{q,r}$ is a Lebesgue space with mixed norm (see the paper above for more informations)

So, why find this $\alpha$ which is a sharp exponent is so important and what applications we can obtain from the quantitative Holder regularity perspective?

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Cézar Bezerr
Cézar Bezerr
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Hölder regularity in a quantitative manner

Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer analysis of a model?

For example,in https://arxiv.org/pdf/2006.03894.pdf - the authors consider the following problem: let us consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ and a positive number $T$, then in $\Omega \times (0,T)$ the locally bounded solutions of the inhomogeneous porus medium equation

$u_{t} - div(m\lvert u \rvert^{m-1}\nabla u) = f \in L^{q,r}, m>1$

are locally Hölder continuous, with exponent

$\gamma = min \Big(\frac{\alpha^{-}_{0}}{m}, \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \Big)$

where $\iota^{-}$ means that we can select all $s \in (0,\iota)$ and $L^{q,r}$ is a Lebesgue space with mixed norm (see the paper above for more informations)

So, why find this $\alpha$ which is a sharp exponent is so important and what applications we can obtain from the quantitative Holder regularity perspective?