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Results tagged with ap.analysis-of-pdes
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user 30684
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
23
votes
Looking for an interesting result on the Navier-Stokes equations
Beale Kato Majda criteria: Beale, J. Thomas, Tosio Kato, and Andrew Majda. "Remarks on the breakdown of smooth solutions for the 3-D Euler equations." Communications in Mathematical Physics 94.1 (1984 …
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3
votes
Applications of PDE in mathematical subjects other than geometry & topology
Probability: PDEs are all over the place in problems related to optimal filtering problems. For example, the Kushner-Stratanovich equation of nonlinear filtering.
Several of the optimal filtering typ …
0
votes
Non-linear Basis for PDE's
A natural way to think about this is via Koopman (or composition) operator, corresponding to the nonlinear operator. This linear koopman operator acts on the $\it{functions}$ of state, and hence is ne …
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4
votes
Practical applications of Sobolev spaces
Formulating optimization/control problems in Sobolev spaces often lead to better numerically conditioned problems, and more practically implementable solutions.
E.g: Consider the problem of devising …
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4
votes
1
answer
251
views
Boundary flux maximizing drift (velocity) vector fields for 2D heat equation
Looking for literature / known results on the following class of problems:
Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar fie …
- 2,281
1
vote
Accepted
Role of the divergence of the vector field in transport equations: mass concentration?
It is instructive to think about 1 dimensional case. Take $a(x)=b-\alpha x$, ($\alpha\geq 0$) then the divergence is simply $-\alpha$.
Case 1: $b=0,\alpha\geq 0$: A trajectory starting at $x(0)=x_0$ …
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3
votes
Accepted
PDE-oriented textbook on probability and random processes?
Stochastic processes and application by Pavliotis is a good one.
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3
votes
0
answers
125
views
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 …
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1
vote
1
answer
519
views
Reference request: Spectral analysis of advection diffusion PDE
As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the foc …
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9
votes
Open problems in PDEs, dynamical systems, mathematical physics
Dynamical systems is a huge field, with at least 3 (or more) subdisciplines which often interact with each other, but also have self-contained advances. Ergodic theory, topological dynamical systems, …
0
votes
Link between controllability of ODEs and controllability of transport equations
Here's another references:Elamvazhuthi, Karthik, et al. "Bilinear controllability of a class of advection–diffusion–reaction systems." IEEE Transactions on Automatic Control 64.6 (2018): 2282-2297.
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0
votes
Reference request for spectral theory of elliptic operators
Mathematical Methods in Quantum Mechanics
With Applications to Schrödinger Operators
Gerald Teschl
- 2,281
2
votes
1
answer
222
views
When is a stationary measure of a Markov chain "exponentially localized"?
Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes.
Some intuition can gained by thinking about a diffusion process, …
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10
votes
Book Recommendation - PDE's for geometricians / topologists
Since you claim that you know "nothing about PDEs", I think it would be very hard to appreciate the topological/geometric applications of PDEs without at least a basic familiarity with the theory of P …