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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Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?
Let $U$ be an open subset of $\mathbb{R}^n$ such that
$\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice''
(for simplicity you can assume piecewise smooth). I also want to allow the ...
- 3,245
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2
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Existence of solution to quasilinear parabolic PDEs
Hello.
I want to prove the existence of a weak solution to:
Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that
$$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$
...
- 45
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2
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A hyperbolic PDE with infinite boundary conditions
Given known functions $a(x,y)$ and $b(x,y)$, I have a linear, second-order, hyperbolic PDE for a surface $z(x,y)$. The PDE is of the form:
$z_{xx} - a^2 z_{yy} + ab z_x - b z_y = 0$.
The PDE does not ...
- 13
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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 $\...
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Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$
Note that rescaling $2^{j} s\mapsto s$ shows that
$$J(j)\leq 2^{-j(1+a)}\int_{0}^...
- 852
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Why is this integrability condition needed for uniqueness in the continuity equation?
I am reading about the uniqueness problem for the continuity equation $\partial_t \mu_t + div_x (b \mu_t)=0$ in the lecture notes by Ambrosio (here: https://warwick.ac.uk/fac/sci/maths/research/events/...
- 697
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170
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Vorticity equation for incompressible 2D fluid dynamics [closed]
I want to ask what advantage of using vorticity equations in fluid dynamics.
Does it help to find large curls? Does it have singularities connected to presence of curls?
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What functions are equal to their symmetric decreasing rearrangement?
I am trying to understand the set
$$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$
where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if ...
- 537
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A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with ...
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140
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Physical significance of a fractional operator
Is there any physical significance of the operator $(-\Delta)^s\pm \Delta$ when $0<s<1.$ I would like to know if there is any real life applications other than in pure mathematics.
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Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?
Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...
- 1,774
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Bilinear Strichartz estimates for the Schrodinger equation
Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
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A basic stability question
Let $u_i \in C^1(\Omega)$ with $|\nabla u_i|>0$ in a simply connected region $\Omega$ with connected boundary, and $u_1=u_2$ on $\partial \Omega$. Assume
$$ \nabla u_i(x) \cdot V_i (x)=|\nabla u_i(...
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547
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$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...
- 363
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2
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119
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Uniqueness problem for an elliptic system
I want to prove the uniqueness of the solution of the following problem:
$$\eqalign{
& - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr
& u > 0 \text{ in } \Omega \cr
& \...
- 617
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387
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Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
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381
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Converse of Lax-Milgram theorem [closed]
Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’...
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Solution of Poisson equation vanishing at the boundary of any order
Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
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In the proof of the existence of weak solutions to the NSE
In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made:
Let $u_m$ converges weakly to $u$ in $L^2(0,...
user14319
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$H_0^1(\Omega)$ in the study of the Navier-Stokes Equations
This is cross-posted on MSE: https://math.stackexchange.com/q/1584519/9464
Let $\mathcal{V}$ be the space (without topology)
$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\...
user14319
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2
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222
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Operator on a Sobolev space [closed]
I'm studying Sobolev spaces using Evans' PDE book.
I can't figure out this simple fact.
Let $L$ be an operator in this form:
$$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$
I can't understand why ...
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2
answers
177
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Separation of variables for a particular PDE
Given the partial differential equation
\begin{equation}
(1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0
\...
- 343
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658
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Implicit function theorem for elliptic partial differential equations
Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...
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Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?
Should I expect the following Friedrichs/Poincare inequality to hold for $u \in C^\infty(S_n \times (0,\infty))$ with $u(x,0) = 0$:
$$\int_{S_n \times (0,\infty)}|u|^2 \leq C\int_{S_n \times (0,\infty)...
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257
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Frobenius Condition for a specific first order pde
I would appreciate it if Someone would be kind enough to share some insights about the following question:
Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
- 4,145
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Wave equation v.s.Schrödinger equation
The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$
From the above that a wave operator can be ...
- 1,644
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1k
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Maximum principle for heat equation on infinite domain
Let $u(x, t)$ be a solution of $u_t=u_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in ...
- 167
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846
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Units of time in the gradient flow equation?
From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow equation?...
- 167
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118
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Nodal domain theorem for clamped plate equation
Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the clamped plate equation in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...
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Multi-variate Picard-Lindelöf? Convergence of analytic PDEs (w/ commutative partial derivatives & value at a base point)
I am looking for a theorem to give existence and uniqueness of solutions to PDEs of the following form.
Find an analytic $u : \mathbb{R}^n \to \mathbb{R}^m$ satisfying the equations $\partial_{x_i}u =...
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1
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172
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Harmonic functions and monotonic decay
I have a general question surrounding certain harmonic functions.
I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
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Perturbation methods for stochastic/partial differential equations
I'm asking for a good reference on perturbation methods for stochastic and/or partial differential equations.
Something like this: Perturbation of a stochastic differential equation
I'm familiar with ...
- 131
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334
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On the weak derivative of $|u|^{(p-2)/2}u$
Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$.
How can I show that
$$
D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1}
$$
or how can I show that, ...
- 163
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How to understand the unique continuation result
Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm
$$
\|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}.
$$
Suppose $K(x) \in C^1\left(\mathbf{R}^...
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A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
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Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
The inequality (2.3) in this ...
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232
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An estimate of the gradient of heat kernel
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I have already proved that
$$
\...
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Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
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157
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Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law
Consider the scalar conservation law
$$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>...
user139844
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1
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121
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Inequality involving the fractional Laplacian
Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
- 205
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128
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Estimating singular double integral
How can I estimate
$$\int_{(0,1) \setminus B_{\delta}(1/2)} \int_{B_\delta(1/2)} \frac{u(y)v(y)}{|x-y|^{\alpha +1}} \, dy \, dx$$ in terms of a positive power of $\delta$ and suitable norms of $u$ ...
- 217
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344
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Is this PDE solvable?
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$:
$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text{...
- 969
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1
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185
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"Arc" length parametrization for surfaces
If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that:
$|\nabla d(x,y)|=1,\ \...
- 1,759
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2
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274
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The uniqueness of Barycenters in the Wasserstein space
I am reading the paper Barycenters in the Wasserstein space by Martial Agueh and Guillaume Carlier. In Proposition 3.5, they prove the existence and uniqueness of
$$\nu \mapsto \sum_{i=1}^p \frac{\...
- 288
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148
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Is it a sufficient condition for linearity?
Edit: According to the comment by LSpice we come back to the initial version of this question
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the ...
- 356
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4
answers
1k
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Does the Leibniz (product) rule hold for the spectral fractional Laplacian?
Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
user139845
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1
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Compatible solution of PDE
Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
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2
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153
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Reference request for fractional Poincare inequality
Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
- 83
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Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?
Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...
- 7,789
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438
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Regularity of Laplace equation with Dirichlet data on a part of the boundary
From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system
\begin{align}
-\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\
u &= g &\text{on}\ \...
- 363