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.
3,753
questions
4
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Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 3,857
1
vote
0
answers
57
views
Uniqueness of global solution
I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$
\begin{align*}
\mathrm{d} \...
- 71
0
votes
0
answers
32
views
Can stochastic PDE theory be applied to elliptic PDE to get the solvability when it's hard to use traditional prior estimate or flow approach
Recently I'm interested in stochastic PDE on manifold and I knew that for some famous geometry theorems such as Atiyah-Singer index theorem there is proof by stochastic analysis approach, I wonder ...
- 349
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votes
0
answers
34
views
Boundary conditions for first-order nonlinear system of PDEs
Consider the following system of PDEs for the dependent variables $x=x(u,v)$ and $y=y(u,v)$,
\begin{align}
E(u,v)\:x_v^2-2F(u,v)\: x_vx_u+G\:x_u^2&=\Delta^2\\
E(u,v)\:y_v^2-2F(u,v)\: y_vy_u+G\:y_u^...
- 255
1
vote
0
answers
63
views
Change of variables for obtaining a unitary group
Consider the following NLS:
$$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$
where $F(u):=(u + \bar{u} + |u|^2)u.$
In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
- 87
2
votes
2
answers
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views
What is standard continuity argument for well-posedness?
Motivation: I'm trying to understand the proof of Theorem 3.1 in Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management. Though in the following I'...
- 121
2
votes
1
answer
69
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Representing solutions of $-\Delta u+au=f$ when $a\leq 0$
Let $\Omega=[0,1]\times [0,1]$ be the square. We say a function $f\in H^1(\Omega)$ is periodic on $\Omega$ if $f(x,0)=f(x,1)$ and $f(0,y)=f(1,y)$ (in the sense of traces of course). Now consider the ...
- 133
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vote
0
answers
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+50
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 703
0
votes
1
answer
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views
What are the solutions to this nonlinear equation?
Besides the constant solutions what are the solutions to:
$\dot{u}=u \Delta u$
where $u_0$ is defined on a domain $\Omega \subset \mathbb{R}^n$?
9
votes
1
answer
275
views
Propagators and PDEs
I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...
- 101
1
vote
0
answers
41
views
Precise decay of solution fo fractional Schroedinger equations
Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$.
...
- 429
3
votes
0
answers
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A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
- 79
3
votes
1
answer
292
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Equivalence between two fractional Sobolev spaces
For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...
- 121
4
votes
1
answer
304
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...
- 3,857
1
vote
1
answer
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views
Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
1
vote
0
answers
59
views
On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
- 1,339
1
vote
1
answer
99
views
Bott-Chern cohomology for singular complex spaces
I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic ...
- 227
1
vote
0
answers
189
views
Has anyone studied the PDE generalization of Teichmüller Space?
We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).
Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
- 847
5
votes
2
answers
559
views
Forcing the uniqueness of a solution of an ODE
For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and
$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$
Let $y_n$ be the ...
- 471
0
votes
0
answers
42
views
Highy non-linear PDE involving directional derivative
Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
- 47
6
votes
0
answers
123
views
Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
- 255
3
votes
0
answers
48
views
On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
4
votes
3
answers
246
views
Reference or proof of a lemma in PDE
I am looking for a reference or proof of a lemma (if it's true) or a counter-example otherwise. It goes as follows:
Let $B_1$ and $B_2$ are two concentric balls of radius $1$ and $2$ in some $n$-...
- 515
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votes
0
answers
60
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 87
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vote
0
answers
63
views
A basic question on analytic wave front set
Suppose $u$ is a smooth function on the closed unit disk centered at the origin in $\mathbb R^2$. Let us denote by $e_1$ and $e_2$ the unit vectors in the direction of $x$ and $y$ axis respectively. ...
- 3,087
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votes
0
answers
47
views
What do power series solutions of ordinary/partial differential equations tell us about all solutions?
I guess many people are familiar with the power series method of ordinary or partial differential equations (if not, please have a look here). In essence, we obtain for both ordinary and partial ...
- 637
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votes
0
answers
68
views
What does a Lipschitz barrier imply about boundary regularity of a domain?
Consider the Dirichlet problem for Laplace's equation in a bounded domain $\Omega \subset \mathbb R^n$:
$$
-\Delta u = 0, \quad x \in \Omega,
$$
with $u = \phi$ on $\partial\Omega$, and $\phi$ is ...
- 21
2
votes
1
answer
70
views
$C^2$-solution of Lane-Emden equation with positive frequency
Consider the Lane-Emden equation
$$-\Delta u=u^{\frac{d+2}{d-2}} $$
in $\mathbb{R}^d$ with $d\geq 3$ and $u>0$ a positive $C^2$-solution. It is well-known, due to [Caffarelli et al., CPAM '89] that ...
- 251
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votes
1
answer
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views
Sobolev space is spanned by distributions supported on half-lines?
I asked this question on Mathematics Stack Exchange previously.
This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it.
For any $s \leq 1/2$,
$$H^s(\mathbb{...
- 86
4
votes
1
answer
177
views
A text about Schwartz distributions in vector bundles
If $M$ is a smooth manifold, one may talk about the space of test functions $\mathcal D (M)$ and its topological dual $\mathcal D ' (M)$ - the space of Schwartz distributions on $M$.
Now, if $E \to M$ ...
- 4,770
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votes
0
answers
52
views
Characteristic of Sobolev space generated by Hörmander vector fields
Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
- 243
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votes
0
answers
52
views
Examples of chaotic self-similar blowup in PDEs
When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference:
Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in ...
- 97
3
votes
1
answer
117
views
General solution to a n-dimensional partial differential equation
$$
\begin{split}
\frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\...
- 131
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votes
0
answers
53
views
Explicit constants for elliptic a priori estimates
Let $V$, $W$ be vector bundles over a compact Riemannian manifold $M$ and let $F$ be a smooth elliptic operator of order $k$ from $V$ to $W$.
"Standard elliptic theory" then gives us the ...
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votes
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45
views
Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality:
$$\...
6
votes
1
answer
196
views
Geometric evolution of convex surfaces to a round sphere
Let $𝑀 = 𝑀^2$ be an embedded convex surface in $\mathbb R^3$ and let $𝑁 ∶ 𝑀 → 𝕊^2$ be the Gauss map for $𝑀.$ Let $𝑉_𝑀$ be the area measure on $𝑀$ and $𝑁_∗𝑉_𝑀$ the corresponding pushforward ...
- 161
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votes
1
answer
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views
Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics?
Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics?
Tried my luck with Google's search engine, didn't show much info.
I guess you can try to use these features ...
- 1,487
4
votes
0
answers
75
views
Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
2
votes
0
answers
45
views
Semilinear elliptic equations in complex plane
Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
- 3,087
2
votes
1
answer
101
views
Calculating the eigenvalues of the Laplacian numerically
I am trying to find the eigenvalues of the Laplacian operator, or in other words, solve the Helmholtz equation
$\nabla^2f=\lambda f$
on a compact 2D domain (comes from a quantum mechanics particle-in-...
- 3,512
3
votes
1
answer
91
views
Solvability of general linear PDE with constant coefficients
Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any ...
- 19.3k
1
vote
0
answers
37
views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 703
2
votes
0
answers
51
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t))
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads
$$...
- 703
5
votes
1
answer
375
views
The principal symbol as an element in the K-theory
This line
The symbol may naturally be thought of as an element in the K-theory
of X
appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...
- 729
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votes
0
answers
32
views
What is the deep logic for the resonance function of dispersive nonlinear PDEs
I have been studying some nonlinear dispersive PDEs since some months and I was able to understand some results related to well-posedness. However, I do not feel like I am fully understand the logic ...
- 87
0
votes
0
answers
30
views
Characterization of the extreme eigenvalue of Wishart-Laguerre/Jacobi-MANOVA ensemble
The Tracy-Widom distribution gives the limiting distribution of the rescaled largest eigenvalue of a random matrix taken from an appropriate symmetry class. According to Bloemendal, the deformed Tracy-...
0
votes
0
answers
16
views
Time localization Estimate in $X^{s,b}$ spaces
I am steadying the proof of the following Lemma:
Let $-\frac{1}{2} < b' \le b < \frac{1}{2}$, then for any $0 < T <1$ we have
$$\left\lVert \eta\left(\frac{t}{T}\right) u \right\rVert_{X^{...
- 87
0
votes
0
answers
58
views
solution of equivalent problem Kantorovich for case squared distance function
We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to
$$
\inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
5
votes
2
answers
207
views
Morse index in PDEs
I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came ...
1
vote
0
answers
157
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 87