Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,467 questions
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Use of games to approximate solutions to Partial Differential Equations
Hi there,
Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "...
- 203
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Finding constant curvature metrics on surfaces for the case of positive Euler characteristic
We approach the problem of finding a metric of constant curvature on a surface (i.e. a $C^\infty$ 2-manifold). Specifically, what we want to do is, given a surface $M$ and a metric $g_0$, show that ...
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General validity of separation of variables
Let $L$ be any differential operator (not necessarily linear).
Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:
Given a boundary ...
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Does harmonic map heat flow of a curve always fully converge to a geodesic?
Consider a smooth closed curve $u_0$ in a compact Riemannian manifold $(M,g)$. Let $u_0$ evolve by harmonic map heat flow, $\partial_tu=\nabla_{\partial_su}\partial_su$, and call the result $u(t)$.
...
- 738
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Well-posedness of Fokker-Planck equation
Consider the following equation on $[0,T]\times\mathbb{R}^n$
\begin{eqnarray}
&\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\
&\rho|_{t=0}=\rho^0,
\end{eqnarray}
where $V\in C^2(\...
- 339
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Wick rotation for Laplace and wave equations
I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...
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Propagators and PDEs
I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...
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Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
- 311
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Proving a system is nonintegrable /not solvable with Inverse Scattering Transform
Question:
Given a PDE, is there a general method to show that it is not solvable using the inverse scattering transform?
Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it ...
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Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
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How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?
Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.
For an H-J eqn, I mean a first order pde $H\circ dS=...
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Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
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Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
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Reference Request: Probability and (Nonlinear) PDEs
I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability). I'm mostly interested in things along the ...
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Linearization instability and singular points of algebraic varieties
In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
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The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon
In relation to the question on the Hardy inequality and the answer by Terry Tao, I've always been curious about the following:
Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t ...
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Rigorous treatment of Ostrogradsky's instability theorem?
The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
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Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?
Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...
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Characterize where the Dirichlet Problem for the Laplacian is always solvable
Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
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Open Questions about Wasserstein Space and PDE
While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
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Curvature of the boundary vs. normal derivative of the first eigenfunction
Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.
Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
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Evolution operator for a linear parabolic equation
Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
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Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user127411
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Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
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Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck
Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
...
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A geometrical problem in terms of a convex function
I wish to know whether the following problem has ever been investigated.
Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
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Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?
The following question is related to Singular Yang-Mills theory
The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-...
user21574
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Is there a classification of differential equations over the field of fractions of formal power series? (characteristic 0)
Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ...
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Chain rule for distributional derivative
Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
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Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?
Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the ...
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Density of restrictions of harmonic functions inside a ball
Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
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What is soliton
I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking..
In literature, i am finding following words:(Wikipedia+ others).
...
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List of generic properties of Riemannian metrics
I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
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Which PDE from physics (and geometry) are supercritical?
I am currently trying to understand the notion of criticality (as discussed, e.g., in Terence Tao's book on nonlinear dispersive equations) from a physical viewpoint. That's why i'm interested in the ...
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Newlander-Nirenberg for surfaces
Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...
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Applications of geometric evolution equations.
Hi everybody,
I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological ...
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Representing immersions from a surface into 3-space
Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$
Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping
$R:\Imm(\...
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A simple example where elliptic boundary regularity fails due to a kink in the domain
I'm seeking a simple example of where elliptic (preferably linear) boundary regularity fails due to a simple kink in the domain.
So far my gueses were to look at $-\Delta u = f$ on $[0,2\pi] \times [...
- 2,641
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Hyperbolic PDE in mathematics
Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most ...
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How to prove Liouville measure is invariant under geodesic flow?
Let $M$ be a complete n dimensional Riemannian manifold. $vol$ denotes the n dimensional Hausdorff measure. Let
$$
SM=\{(x,v)|x\in M, v\in T_xM, \|v\|=1\}
$$
be the unit tangent bundle of $M$. Then $...
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Green's formula for nonorientable manifolds
Usually in differential geometry one proves the Stokes theorem and then obtains divergence theorem and Green's formulas as corollaries. However, divergence theorem is also valid for nonorientable ...
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Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
user47808
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Calderon-Zygmund decomposition on manifolds?
The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...
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Counterexamples to weak dispersion for the Schrödinger group
Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples ...
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Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
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Rate of convergence of smooth mollifiers
How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...
- 2,243
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Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
- 133
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Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
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When does the eikonal equation $\lvert Du \rvert^2 = f$ admit a local solution?
Let $f$ be a smooth function defined on the unit disc $D \subset \mathbf{R}^2$ with
\begin{equation}
f \geq 0 \text{ in $D$ and } f(0) = 0.
\end{equation}
This is allowed to have a degenerate minimum ...
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