| _version_ | 1866901743275606016 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper investigates the fractal regularity of solutions to non-autonomous parabolic partial differential equations (PDEs) where the coefficients exhibit roughness in both space and time. We establish bounds on the Hausdorff dimension of the graphs of solutions, demonstrating how the regularity of the coefficients influences the fractal properties of the solutions. Our approach combines techniques from harmonic analysis, functional analysis, and fractal geometry to derive estimates on the Besov norms of the solutions. We consider coefficients that belong to suitable Besov spaces and demonstrate that the solutions inherit a certain degree of fractal regularity depending on the Besov regularity of the coefficients. This regularity is quantified in terms of the Hausdorff dimension of the graph of the solution. We provide examples to illustrate the sharpness of our results and discuss the implications of our findings for the numerical approximation of solutions to these PDEs. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17830182 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Fractal Regularity of Solutions to Non-Autonomous Parabolic PDEs with Rough Coefficients Revista, Zen MATH, 10 This paper investigates the fractal regularity of solutions to non-autonomous parabolic partial differential equations (PDEs) where the coefficients exhibit roughness in both space and time. We establish bounds on the Hausdorff dimension of the graphs of solutions, demonstrating how the regularity of the coefficients influences the fractal properties of the solutions. Our approach combines techniques from harmonic analysis, functional analysis, and fractal geometry to derive estimates on the Besov norms of the solutions. We consider coefficients that belong to suitable Besov spaces and demonstrate that the solutions inherit a certain degree of fractal regularity depending on the Besov regularity of the coefficients. This regularity is quantified in terms of the Hausdorff dimension of the graph of the solution. We provide examples to illustrate the sharpness of our results and discuss the implications of our findings for the numerical approximation of solutions to these PDEs. |
| title | Fractal Regularity of Solutions to Non-Autonomous Parabolic PDEs with Rough Coefficients |
| url | https://doi.org/10.5281/zenodo.17830182 |