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We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.
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On the fast spreading scenario
We study two types of divergence-free fluid flows on unbounded domains in two and three dimensions—hyperbolic and shear flows—and their influence on chemotaxis and combustion. We show that fast spreading by these flows, when they are strong enough, can suppress growth of solutions to PDE modeling these phenomena. This includes prevention of singularity formation and global regularity of solutions to advective Patlak-Keller-Segel equations on R 2 \mathbb {R}^2 and R 3 \mathbb {R}^3 , confirming numerical observations by Khan, Johnson, Cartee, and Yao [Involve 9 (2016), pp. 119–131], as well as quenching in advection-reaction-diffusion equations.
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- PAR ID:
- 10358327
- Date Published:
- Journal Name:
- Communications of the American Mathematical Society
- Volume:
- 2
- Issue:
- 4
- ISSN:
- 2692-3688
- Page Range / eLocation ID:
- 149 to 171
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/cams/6
