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.
4,468 questions
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Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
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Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
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increasing property of the heat equation on the interval
$Q_T^l=\{(x,t); 0<x<l, 0<t\leq T\}$,
$u_l\in C^{2,1}(Q_T^l)\cap C(\bar Q_T^l)$ is the solution of $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...
- 119
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Viscous stress equation in Newtonian fluid
In this Wikipedia entry, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is
$$
\tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\...
- 9
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136
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Fractional Laplacian of smooth cut off functions
Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$
...
- 537
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129
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Sturm-Liouville result
Suppose $n \ge 2$ an integer and consider finding the first eigenvalue of
$$ -\partial_\theta \left( \omega(\theta) \psi'(\theta) \right) = \mu_1 \omega(\theta) \psi(\theta)$$ for $ 0<\theta<\...
- 1,385
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A Grönwall-type inequality for $u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s$?
Quick question: Are we able to show a Gronwall-type inequality assuming that $$u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s,$$ where $\alpha$ is nondecreasing (or constant) and $\beta$ is ...
- 167
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How to prove this estimate involving the Stein Derivative?
Recall the Stein Derivative,
$$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$
I want to show that,
$$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \...
- 537
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Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
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126
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Hessian estimates of eigenfunctions without Bochner
let $\Omega$ be a bounded domain in a Riemannian manifold $(M,g)$. Consider the Dirichlet eigenvalues and eigenfunctions of Laplacian on $\Omega$, that are, the $\lambda_i>0$ and $\phi_i\in H^{1}_0(...
- 193
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Convergence of a infinite sum in Besov space
Suppose we have an annulus $A \subset \mathbb R^n$, which is the set $\{x|0<r \leqslant\|x\| \leqslant R\}$, $\alpha \in \mathbb R$ and $\{ u_j\}_{j\geqslant-1} $ be a seqence of smooth functions ...
- 253
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419
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Sobolev inequality on the sphere derivation
I am reading the following paper (preprint here) and the author starts by stating the Sobolev inequality on the Sphere $\mathbb{S}^d$
$$\frac{p-2}{d}\int |\nabla u|^2 + \int |u|^2 \geq \left(\int |u|^...
- 537
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Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
- 2,533
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Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
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Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
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129
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Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
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Does such a vector field exist?
Does there exist a velocity field $\mathbf{u}(x,t)\in \mathbb{R}^3$ such that $$\text{Div}\begin{bmatrix}
\mathbf{u}\cdot \nabla w_1\\
\mathbf{u}\cdot \nabla w_2\\
\mathbf{u}\cdot \nabla w_3\\
\end{...
- 317
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Has this form of the heat equation been solved for the radiation boundary condition
Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
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Is the time of solution shorter as the initial data increases?
I'm reading the book superlinear parabolic problems and I came across the following situation twice: given two initial data $u_0$ and $\underline{u_0}$ with $u_0\geq \underline{u_0}$, $u_0\neq \...
- 677
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Is Poisson formula valid for the weak solution of Laplacian?
In the book "Regularity Theory for Elliptic PDE", here is a theorem as follows
Theorem(Harnack's inequality). Assume $ u\in H^1(B_1) $ is a non-negative, is the weak solution for the ...
- 1,219
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131
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Green kernel vs fundamental solution
Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...
- 650
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83
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Difference between two fractional Schrödinger equations
Let us consider the fractional Schrödinger equation with periodic boundary conditions
$$
\begin{cases}
iu_t\mathbf{+}(-\Delta)^{\alpha}u= \pm |u|^2u,\; x \in \mathbb{T}, t \in \mathbb{R}_+\\
u(x,0)=...
- 205
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92
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Results about Schrödinger equations
Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
- 677
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127
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Is every minimal graph smooth?
The following result was taken from the book of Gilbarg-Trudinger:
In particular, if the graph is minimal, then $u$ is smooth.
Now comes my question: does the same conclusion hold for graphs over ...
- 2,095
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Solve $(A(x).\nabla)u+cu=0$
ِDoes the equation
$y\partial_x u(x,y)-x\partial_y u(x,y)+cu=0$
have complex-valued compact-supported or vanishing-at-infinity $C^1$ solution defined on the whole plane without any singularity? Here $...
- 107
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173
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Lemma 3.10 of paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain
I am reading a paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain.
And I have a questions in the proof of lemma 3.10.
Please click the paper title for the link.
The ...
- 239
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77
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On global solutions of a semilinear elliptic PDE
Consider the semilinear elliptic PDE
$$-\Delta \phi + \phi +\phi^3=f$$
for a real-valued function $\phi$ on $\mathbb{R}^n$, where $f$ is a fixed $C^2$ function. What is known about the existence and ...
- 505
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263
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Solving Fokker–Planck equation
Consider the Fokker–Planck equation:
$${\displaystyle {\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(...
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Holder continuity relative to Rellic-Kondrachov compactness via the nonlinear Aronsson operator
Connected to the question,
Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ ...
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981
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Weak $H^1$ convergence implies strong $L^p_{\text{loc}}$ convergence
On page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14), there is a theorem stating that if $f_j \to f$ weakly in $H^1(\mathbb R^n)$ then $\mathbf 1_A f_j \to \mathbf 1_A f$ strongly in $...
- 1,755
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93
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Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
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69
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Reference about the backward heat flow on closed manifold
On the paper: Bruce Kleiner, John Lott, Notes on Perelman's papers,
Geom. Topol. 12(5): 2587-2855 (2008). DOI: 10.2140/gt.2008.12.2587
The authors wrote the following statement:
(Here $-\Delta$ is ...
- 1,915
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164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
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239
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Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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184
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Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$.
Consider the Laplace equation with a Neumann boundary condition
$$
-\Delta u = 0 \quad\mbox{in } \Omega, \qquad
\frac{\...
- 11
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104
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Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
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150
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Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
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153
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Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$
How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
- 99
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Sufficient conditions for an asymptotic compactness
This question relates a theory of Mosco convergence.
Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$.
A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
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158
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Help finding an article (now solved)
I'm searching for this article:
"Positive solutions for some quasilinear elliptic equations"
Communications on Applied Nonlinear Analysis 3 (1996), no. 4, 89–98.
by Michel Artola and Lucio ...
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142
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Variational formulation for an elliptic boundary value problem
I'm trying to determine the variational formulation of
$$
\begin{cases}
-\Delta u(\mathbf{x})=1, & \mathbf{x}\in (0,1)\times (0,1) \\
-\partial_{x}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=0, & ...
user165558
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216
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Signed distance function
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function:
$d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
- 1,759
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121
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Positivity of solution for Fisher-Kolmogorov Equation
How can we prove that if $y=y(t,x)$ is the solution of the problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y(t,x)=r(x)y(t,x)-\rho(x) y^2(t,x),\ (t,x)\in (0,T)\times \Omega \\ \...
- 1,759
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61
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$L^p$ estimate for perturbed heat equation
Let us consider the heat equation
$$
\begin{cases}
u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\
u(t,-1) = a(t), \\
u(t,1) = b(t), \\
u(0,x) = u_0(x)
\end{cases}
$$
where $f \in C^\...
- 131
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152
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Metric obstructions for area-preserving diffeomorphisms with constant singular values
Let $\mathbb{T}^2$ be the topological $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Let $g$ be an arbitrary smooth Riemannian metric on $\mathbb{T}^2$.
...
- 6,741
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330
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Comparison of the spectrum decomposition
In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
- 1
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37
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Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 943
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101
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Equation $u_t - u_{tx} - u_{xx} = 0$
Consider the following heat equation with a perturbation given by a second order mixed derivative:
$$u_t - u_{tx} - u_{xx} = 0$$
Does this equation have a name? How can one prove a wellposedness ...
- 839
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62
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How to prove this integral inequality in a 2-D region?
Let $\Omega$ be a 2D region. Now we have a partial differential equation system describing the characteristics of the region:
\begin{align*} \nabla \cdot (h_0^3 P_0 \nabla P_0) &= 0 \\ \nabla \...
- 29