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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Literature on Lateral cauchy problem
Victor Isakov's Inverse Problems for Partial Differential Equations has a quite extensive treatment of the Cauchy problem with lateral boundary data, see chapters 3, 7, 8.
- 188k
1
vote
Accepted
Exactly solvable examples of diffusion equation with variable diffusivity?
This would be the Schrödinger equation with a position dependent mass. Some exactly solvable examples are presented in
Analytic results in the position-dependent mass Schrödinger problem
Explicit s …
- 188k
1
vote
Finite speed of propagation of wave equation
The wave propagation speed can be infinite if your boundaries have a fractal shape, because of the different scaling of space and time. This idea is due to Strichartz, see Laplacians on fractals (2005 …
- 188k
5
votes
Scaling invariance for Hartree equation
substitution of $u_\mu(t,x)=\text{constant}\times u(\mu^\alpha t,\mu x)$ in the fractional GHE shows that this is a solution if $\text{constant}=\mu^{\frac{d-\gamma+\alpha}{2(p-1)}}$, so the scale inv …
- 188k
4
votes
Accepted
WKB expansion for NLS
A. Note that the time derivative $\partial u^\epsilon/\partial t$ and the spatial derivative $\partial u^\epsilon/\partial x$ are both of order $1/\epsilon$, since $u^\epsilon\propto e^{i\phi(t,x)/\ep …
- 188k
3
votes
How to find the associated conservation law from a given symmetry
You can find an overview of methods to obtain conservation laws from a wave equation in On the structure of conservation laws of (3+1)-dimensional wave equation. Noether's method requires that the PDE …
- 188k
4
votes
Accepted
What Morrey and Campanato space characterize
The lecture notes by Melanie Rupflin answer the question "What is a Morrey Space? What is a Campanato Space?"
The Morrey space $L^{p,\lambda}$ is a subset of $L^p$ containing functions $f$ on a domai …
- 188k
3
votes
Nondimensionalization of Navier Stokes Equations
I think the count is off, because the unit of mass does not appear as an independent degree of freedom in the Navier-Stokes equation (unless you include gravitational effects). You have the independen …
- 188k
2
votes
Does there exist always an $L^2$ threshold below (or above) which a traveling waves of a non...
This addresses the case of a traveling wave (not a solitonic wave).
If I consider the one-dimensional nonlinear Schrödinger equation
$$i\partial_t\Psi+\partial_x^2\Psi+\Psi(f(|\Psi|^2)=0,$$
a travelli …
- 188k
2
votes
Regularity of stochastic heat equation
The presence of noise allows to exchange ("trade") time regularity for space regularity. For a simple example, consider the map
$$(t,x)\mapsto \int_0^t b(s,x+W_s)\,ds,$$
with Gaussian noise $W_s$. If …
- 188k
3
votes
Conserved quantities for the Cauchy momentum equation
1) conserved quantity for incompressible flow:
$$\frac{d\rho}{dt}=\frac{\partial\rho}{\partial t}+\bar{v}\cdot\nabla\rho=0$$
so if the flow is stationary, $\partial\rho/\partial t=0$, the density $\ …
- 188k
6
votes
Relativistic Control Theory
Maximum
Mass of a Neutron Star (1974)
Control
theory and singular Riemannian geometry (1982)
The
Condition of Hydrostatic Equilibrium of Stellar Models Using Optimal
Control (2002)
Investigating a …
- 188k
4
votes
What to read for many-body problems in 3D Schrodinger equation
A question along these lines was asked recently to two eminent mathematical physicists, Mel Levy and Elliott Lieb, and here is their wish list of open problems in many-electron theory.
- 188k
2
votes
Accepted
semi-classical Green's function
The difference between the semiclassical approximations of the full Green's function and the trace is whether or not you restrict the sum over paths to closed orbits; for a treatment of the semiclassi …
- 188k
5
votes
Accepted
Stochastic methods for solving very high-dimensional PDE
It seems to me this question "has not received enough attention" because of the conflation of two issues: dimensional reduction of a high-dimensional PDE and stochastic (Monte Carlo) integration of th …
- 188k