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.
4,466 questions
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Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
- 101
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55
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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem
This is the equation given ($n\geq2$)
$$
\begin{cases}
u_{tt}=a^{2}\left(\Delta u\right), \\
\left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\
\left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) .
\end{...
- 1
2
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2
answers
277
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Characterization of locality in Fourier multiplier
Obviously Laplacian $-\Delta$ is a local operator, and it can be written as a Fourier multiplier $|\xi|^2$. Similarly fractional Laplacian $(-\Delta)^s$ is with Fourier multiplier $|\xi|^{2s}$ and is ...
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52
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Continuity of the constant in maximal Sobolev regularity
Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
- 313
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111
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Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 825
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60
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Galerkin’s Method for hyperbolic PDEs: proving convergence without using compactness
Lawrence Evan's PDE book prove the existence of solution to the following problem where $L$ is an elliptic operator:
$$
\begin{cases}
u_{tt} = -Lu+f,\\
u|_{t=0} = u_0,\\
u|_{\partial U} = 0
\end{cases}...
- 1,755
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1
answer
191
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What is kth vortex formula?
I want to study the kth vorticity equation. The NS equation is provided as
\begin{align}\label{eq1}
&\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \right)\textbf{u} = ...
- 317
3
votes
0
answers
80
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Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
- 167
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Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
- 123
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219
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Heat equation with nonlocal boundary condition
$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
- 5,380
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votes
3
answers
704
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Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
- 1,350
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34
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A question about the eigenfunction method and the notion of solution - distributional solution
I have a question about how a passage was made in the calculation of passage (2.5) in the calculation below. To introduce context, the author in the paper (full work) is trying to demonstrate that ...
- 677
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84
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About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
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If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?
Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What ...
- 93
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74
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Elliptic regularity for Dirichlet problem
Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective ...
- 577
1
vote
1
answer
99
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How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let
$$
\begin{matrix}
F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\
(x,z,p) \mapsto F(...
- 217
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1
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283
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Integral identity for critical points of the Ginzburg-Landau functional
I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional
$E_\epsilon(v) = \frac{1}{2}...
- 5,048
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votes
1
answer
147
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Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
- 41
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200
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Why $-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}$ type PDE is called 'mean-field equation'?
Why $$
-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}
$$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical ...
- 809
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vote
1
answer
115
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Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator
I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time.
Consider the following ...
- 187
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vote
1
answer
166
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Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$
I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$.
LEMMA 1.1 (...
- 809
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votes
1
answer
392
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Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$
I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
- 73
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votes
0
answers
47
views
Growth/Decay of conformal Killing fields in cone metrics
Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric
$$g= dr^2 + r^2 \gamma$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing ...
- 969
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509
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Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
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vote
2
answers
260
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
- 2,818
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votes
1
answer
131
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Controlling convolutions with maximal functions
For $f\in L^1(\mathbb R^n),$ let $Mf$ be the (Edited: changed the type of maximal function) Stein spherical maximal function. Let $\varphi\in C_c^\infty.$ Then, can we have a pointwise estimate of the ...
- 1,755
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Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
- 515
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answers
203
views
Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 167
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Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
- 300
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votes
0
answers
162
views
Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?
Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
- 1,759
3
votes
1
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176
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Stochastic representation of Laplace equation with Neumann boundary condition
Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...
- 1,904
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votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
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56
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Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
- 93
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votes
2
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941
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What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
- 185
-1
votes
1
answer
110
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Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 437
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112
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Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 593
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108
views
Mixture of Ornstein-Uhlenbeck operators
Consider a set of isotropic, multivariate Gaussian densities with different centers $\mu_i\in \mathbb{R}^d $, $i\in\{1, \ldots, K\}$, which are denoted $\phi_i(x)$. They all have the same variance ...
- 555
2
votes
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114
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Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
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272
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Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
- 1,904
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votes
0
answers
171
views
Extreme confusion over the tautology of the solution for PDEs
Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth divergence free vector field. Then, it is well-known from the theory of Navier-Stokes equation that there exists some $T \in (0,\infty]$ and a ...
- 3,477
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vote
1
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138
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
- 1,759
47
votes
3
answers
3k
views
Why is the Vandermonde determinant harmonic?
It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where ...
- 763
1
vote
0
answers
37
views
Inequality for function on Spinor bundle
I have a function $H(x,\psi)$ defined on the spinor bundle $\mathbb{S}$ with $H_\psi$ being the continuous derivative in fiber direction having the following properties:
(H-1) There exists $0<\...
- 11
2
votes
2
answers
558
views
Solution of a linear hyperbolic PDE
I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE
$$
\begin{cases}
u_{xy} + k(u_x+u_y) + (k^2 - \sigma^2 P(x-y))u = f(x,y) \\
u(x,0) = 0 \\
u(0,y) ...
- 71
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votes
0
answers
255
views
Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
0
votes
0
answers
135
views
Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 317
3
votes
1
answer
269
views
Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?
It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting.
It is known that if $L$ is a uniformly elliptic operator, with ...
- 1,755
5
votes
0
answers
360
views
Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
2
votes
1
answer
624
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
- 101
1
vote
1
answer
149
views
Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
- 3,477