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A Convection-Diffusion model on a star shaped graph
Analysis of PDEs · Submitted 17 Apr 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure.
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In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $q=2$, the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. Explicit representations of the limit profiles are obtained.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. | contribution 0.73 | 0.77 | 0.04 | asserted 0.86 |
| s1 | First, we prove the global well-posedness of the solutions of the system under consideration. | result 0.77 | 0.96 | 0.12 | asserted 1.00 |
| s2 | Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. | contribution 0.91 | 0.90 | 0.08 | asserted 0.94 |
| s3 | For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. | result 1.00 | 0.95 | 0.29 | asserted 1.00 |
| s4 | In the case $q=2$, the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. | result 1.00 | 0.93 | 0.16 | asserted 1.00 |
| s5 | Explicit representations of the limit profiles are obtained. | result 0.94 | 0.94 | 0.14 | asserted 1.00 |
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Original abstract
In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $q=2$, the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. Explicit representations of the limit profiles are obtained.
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Uses the method or tool of 1 paper
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For the second estimate, by choosing $ρ(s)=|s|^p-1(s)$, $p>1$ we obtain that [equation] Using Nash-like inequalities for half-lines (see for example [ref] in the case of graphs having some infinite edges or simple use even extensions and use the classical ones for the real line) and we can follow the arguments in [ref] to obtain the desired decay in any $L^p$-norm.
Heat kernel estimates and related inequalities on metric graphs · The first term in the asymptotic behaviour · 0.96
Cites as background 2 papers
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The asymptotic expansion of the solutions for theses linear problems has been considered for general graphs in [ref] .
Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length · Main results · 0.89
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The decay of solutions of linear diffusion problems on graphs has been considered previously in [ref] for compact graphs and in [ref] in the case when some infinite edges are attached to the compact part of the graph.
Heat kernel estimates and related inequalities on metric graphs · Main results · 0.88
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