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Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

Michael Ruzhansky (UGent) , Serikbol Shaimardan (UGent) and Kanat Tulenov (UGent)
(2024)
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Organization
Abstract
In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [37]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.

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Please use this url to cite or link to this publication:

MLA
Ruzhansky, Michael, et al. Sobolev Inequality and Its Applications to Nonlinear PDE on  Noncommutative Euclidean Spaces. 2024.
APA
Ruzhansky, M., Shaimardan, S., & Tulenov, K. (2024). Sobolev inequality and its applications to nonlinear PDE on  noncommutative Euclidean spaces.
Chicago author-date
Ruzhansky, Michael, Serikbol Shaimardan, and Kanat Tulenov. 2024. “Sobolev Inequality and Its Applications to Nonlinear PDE on  Noncommutative Euclidean Spaces.”
Chicago author-date (all authors)
Ruzhansky, Michael, Serikbol Shaimardan, and Kanat Tulenov. 2024. “Sobolev Inequality and Its Applications to Nonlinear PDE on  Noncommutative Euclidean Spaces.”
Vancouver
1.
Ruzhansky M, Shaimardan S, Tulenov K. Sobolev inequality and its applications to nonlinear PDE on  noncommutative Euclidean spaces. 2024;
IEEE
[1]
M. Ruzhansky, S. Shaimardan, and K. Tulenov, “Sobolev inequality and its applications to nonlinear PDE on  noncommutative Euclidean spaces,” 2024.
@article{01JHFDMGEEMFYRY1YRJWSGV865,
  abstract     = {{  In this work, we study the Sobolev inequality on noncommutative Euclidean
spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type
inequality and as its application we show global well-posedness of nonlinear
PDEs in the noncommutative Euclidean space. Moreover, we show that the
logarithmic Sobolev inequality is equivalent to the Nash inequality for
possibly different constants in this noncommutative setting by completing the
list in noncommutative Varopoulos's theorem in [37]. Finally, we present a
direct application of the Nash inequality to compute the time decay for
solutions of the heat equation in the noncommutative setting.
}},
  author       = {{Ruzhansky, Michael and Shaimardan, Serikbol and Tulenov, Kanat}},
  language     = {{und}},
  title        = {{Sobolev inequality and its applications to nonlinear PDE on
  noncommutative Euclidean spaces}},
  year         = {{2024}},
}