Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients
We study the well-posed ness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit (t,x)-dependence: ∂tu + ∑i=1d ∂xifi(u,t,x) = ∑i,j=1d ∂xj (aij (u,t,x)∂xi u), where a(u,t,x) = (aij(u,t,x)) = σa(u,t,x) σa(u,t,x)T is nonnegative definite and each x...
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| Format: | Journal article |
| Language: | English |
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2005
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| _version_ | 1797099435171250176 |
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| author | Chen, G Karlsen, K |
| author_facet | Chen, G Karlsen, K |
| author_sort | Chen, G |
| collection | OXFORD |
| description | We study the well-posed ness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit (t,x)-dependence: ∂tu + ∑i=1d ∂xifi(u,t,x) = ∑i,j=1d ∂xj (aij (u,t,x)∂xi u), where a(u,t,x) = (aij(u,t,x)) = σa(u,t,x) σa(u,t,x)T is nonnegative definite and each x → fi(u,t,x) is Lipschitz continuous. We establish a well-posedness theory for the Cauchy problem for such degenerate parabolic equations via Kružkov's device of doubling variables, provided σa(u,t, ·) ∈ W2,∞ for the general case and the weaker condition σa(u,t,·) ∈ W1,∞ for the case that a is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions. |
| first_indexed | 2024-03-07T05:23:40Z |
| format | Journal article |
| id | oxford-uuid:dfcc97a7-4e02-427b-89b2-469532934bb4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:23:40Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:dfcc97a7-4e02-427b-89b2-469532934bb42022-03-27T09:41:58ZQuasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficientsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:dfcc97a7-4e02-427b-89b2-469532934bb4EnglishSymplectic Elements at Oxford2005Chen, GKarlsen, KWe study the well-posed ness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit (t,x)-dependence: ∂tu + ∑i=1d ∂xifi(u,t,x) = ∑i,j=1d ∂xj (aij (u,t,x)∂xi u), where a(u,t,x) = (aij(u,t,x)) = σa(u,t,x) σa(u,t,x)T is nonnegative definite and each x → fi(u,t,x) is Lipschitz continuous. We establish a well-posedness theory for the Cauchy problem for such degenerate parabolic equations via Kružkov's device of doubling variables, provided σa(u,t, ·) ∈ W2,∞ for the general case and the weaker condition σa(u,t,·) ∈ W1,∞ for the case that a is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions. |
| spellingShingle | Chen, G Karlsen, K Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title_full | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title_fullStr | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title_full_unstemmed | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title_short | Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients |
| title_sort | quasilinear anisotropic degenerate parabolic equations with time space dependent diffusion coefficients |
| work_keys_str_mv | AT cheng quasilinearanisotropicdegenerateparabolicequationswithtimespacedependentdiffusioncoefficients AT karlsenk quasilinearanisotropicdegenerateparabolicequationswithtimespacedependentdiffusioncoefficients |