All Questions
7 questions
36
votes
12
answers
18k
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Open problems in PDEs, dynamical systems, mathematical physics
(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math ...
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
4
votes
1
answer
368
views
Long wavelength instability: Linear Vs nonlinear phenomenon
I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:
$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}...
- 318
3
votes
0
answers
73
views
What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
- 111
3
votes
0
answers
127
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Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?
My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...
- 2,281
1
vote
0
answers
48
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Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user530766
1
vote
0
answers
61
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Stability of Fokker plank solutions with drift not coming from potential: Lyapunov analysis
Consider the FP equation on two dimensional space:
$\dfrac{\partial{\rho(x,y,t)}}{{\partial t}}+u(x,y)\dfrac{\partial\rho}{\partial x}+v(x,y)\dfrac{\partial\rho}{\partial y}=D\Delta\rho(x,y,t)$.
It ...
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