Page 220 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 220
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
[2] D. Berti, A. Corli, L. Malaguti. Wavefronts for degenerate diffusion-convection reaction
equations with sign-changing diffusivity. Submitted, 2021.
[3] D. Berti, A. Corli, L. Malaguti. Diffusion-convection reaction equations with sign-
changing diffusivity and bi-stable reaction term. In preparation, 2021.
A Topological Approach to Nonlocal Differential Equations with
Convolution Coefficients
Christopher Goodrich, c.goodrich@unsw.edu.au
UNSW Sydney, Australia
I will consider the nonlocal problem
− A b ∗ (g ◦ u) (1) u (t) = λf t, u(t) , t ∈ (0, 1),
where ∗ denotes a finite convolution and b and g are given functions. By means of a nonstandard
cone, together with a specially tailored open set, I will demonstrate the existence of at least one
positive solution to this class of problem under given boundary conditions. It will be shown that
this approach improves results which rely on a more standard cone.
Quasilinear conservation laws with discontinuous flux as singular limit of
semilinear parabolic equations
Graziano Guerra, graziano.guerra@unimib.it
Università degli Studi di Milano-Bicocca, Italy
Coauthors: Alberto Bressan, Wen Shen
Scalar conservation laws with a discontinuous flux with respect to the space variable arise in
many applications where the conservation laws describe physical models in rough non homo-
geneous media. For example, traffic flows with rough road conditions and polymer flooding in
porous media. We are interested in solutions to this type of equations obtained by approximat-
ing them with solutions to semilinear parabolic equations. We show that the Crandall Liggett
theory [3] of nonlinear semigroups provides a very elegant framework for proving existence
and uniqueness of solutions to the Cauchy problem for the semilinear parabolic equation
ut + f (x, u)x = εuxx (1)
where the function f is only L∞ regular with respect to x. Then, when the flux has a single
discontinuity at x = 0, we show that the Brezis & Pazy convergence theorem [2] can be used to
show existence and uniqueness of the singular limit, as ε → 0, of solutions to (1) [4]. The limits
obtained in this way are the unique vanishing viscosity solutions to the quasilinear conservation
law (1) with ε = 0.
The convergence result can be generalised to fluxes less regular than BV [1] but counterex-
amples show that it is not true for the simple L∞ regularity.
218
[2] D. Berti, A. Corli, L. Malaguti. Wavefronts for degenerate diffusion-convection reaction
equations with sign-changing diffusivity. Submitted, 2021.
[3] D. Berti, A. Corli, L. Malaguti. Diffusion-convection reaction equations with sign-
changing diffusivity and bi-stable reaction term. In preparation, 2021.
A Topological Approach to Nonlocal Differential Equations with
Convolution Coefficients
Christopher Goodrich, c.goodrich@unsw.edu.au
UNSW Sydney, Australia
I will consider the nonlocal problem
− A b ∗ (g ◦ u) (1) u (t) = λf t, u(t) , t ∈ (0, 1),
where ∗ denotes a finite convolution and b and g are given functions. By means of a nonstandard
cone, together with a specially tailored open set, I will demonstrate the existence of at least one
positive solution to this class of problem under given boundary conditions. It will be shown that
this approach improves results which rely on a more standard cone.
Quasilinear conservation laws with discontinuous flux as singular limit of
semilinear parabolic equations
Graziano Guerra, graziano.guerra@unimib.it
Università degli Studi di Milano-Bicocca, Italy
Coauthors: Alberto Bressan, Wen Shen
Scalar conservation laws with a discontinuous flux with respect to the space variable arise in
many applications where the conservation laws describe physical models in rough non homo-
geneous media. For example, traffic flows with rough road conditions and polymer flooding in
porous media. We are interested in solutions to this type of equations obtained by approximat-
ing them with solutions to semilinear parabolic equations. We show that the Crandall Liggett
theory [3] of nonlinear semigroups provides a very elegant framework for proving existence
and uniqueness of solutions to the Cauchy problem for the semilinear parabolic equation
ut + f (x, u)x = εuxx (1)
where the function f is only L∞ regular with respect to x. Then, when the flux has a single
discontinuity at x = 0, we show that the Brezis & Pazy convergence theorem [2] can be used to
show existence and uniqueness of the singular limit, as ε → 0, of solutions to (1) [4]. The limits
obtained in this way are the unique vanishing viscosity solutions to the quasilinear conservation
law (1) with ε = 0.
The convergence result can be generalised to fluxes less regular than BV [1] but counterex-
amples show that it is not true for the simple L∞ regularity.
218
