Timeline for $L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
Current License: CC BY-SA 4.0
13 events
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| Nov 20 at 11:22 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting (formula hyperlinking)
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| Jul 31 at 8:46 | vote | accept | Garou Garou | ||
| Jul 30 at 15:36 | comment | added | Jochen Glueck | Ah, I see - I overlooked the $x$ after $f(t)$. The non-autonomous equation will not be governed by a $C_0$-semigroup, but probably by a non-autonomous evolution family. The problem I see is rather that typical convergence theorems for $C_0$-semigroups don't apply to non-autonomous evolution families (even on finite dimensional spaces). | |
| Jul 30 at 14:37 | comment | added | Garou Garou | Thank you very much for your answer! Concerning the more complex problem, the mass is preserved since it is $f(t) \times x $ that vanishes at $x=0$. Do you think that a $C_0$-semigroup still exists for this more complex problem (let's call it $(\star \star \star)$)? I am also wondering about the validity of a Duhamel formula with the semigroup $T$, seeing $(f(t)xu)_x$ as a source term --- this could require a priori estimates on this source term. | |
| Jul 30 at 14:21 | comment | added | Jochen Glueck | (Just for the sake of completeness I wrote down the semigroup argument in the autonomous case in an answer.) | |
| Jul 30 at 14:18 | answer | added | Jochen Glueck | timeline score: 4 | |
| Jul 30 at 13:58 | comment | added | Jochen Glueck | Good question. Semigroup theory can certainly be used to study some non-autonomous problems. But the methods that I have in mind for treating the long-term behaviour might not work in this case. I'll have to think about it. By the way, does $f$ depend on $x$? It seems that $f$ destroys the mass perservation property, unless $f(t)$ vanishes at $x=0$ for all $t$. | |
| Jul 30 at 13:20 | comment | added | Garou Garou | In the more complex problem, first line of $(\star)$ becomes $$ u_t = u_{xx} + [(1+f(t) x) u]_x, $$ where f is a smooth function that vanishes exponentially fast as $t$ goes to $\infty$. Do you still think that the semigroup theory could work for this kind of non-autonomous problem? | |
| Jul 30 at 13:00 | comment | added | Jochen Glueck | Thanks for your reply! Is your more complex problem linear or nonlinear? In the linear case, one can often also use operator semigroup methods to show the convergence without explicitly computing the solution. (For instance, this also works for the equation in the question; but whether it works in your more complex problem depends on the problem, of course.) | |
| Jul 30 at 12:53 | comment | added | Garou Garou | Yes, actually continuation arguments may be used to reach an explicit solution and then the $L^{1}$ convergence follows. $(\star)$ arises from a more complex problem for which explicit solutions cannot be obtained. That's why I want to completely understand $(\star)$ with entropy methods. | |
| Jul 30 at 12:33 | comment | added | Jochen Glueck | You wrote "by using entropy methods" in bold. Why is it important for you to use entropy methods for this problem? | |
| S Jul 30 at 12:03 | review | First questions | |||
| Jul 30 at 15:13 | |||||
| S Jul 30 at 12:03 | history | asked | Garou Garou | CC BY-SA 4.0 |