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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On solutions of the continuity equation
Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous continuity equation $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact ...
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What does $A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right)$ mean, exactly?
Motivation: In the theory of harmonic maps between manifold, we often see the characterization
$$
\Delta_g u = -g^{ij}A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right)
$...
- 1,245
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Improved maximum principle estimates (deleting first mode)
Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write
$$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$
where $ r=|x|$ and $ \theta = \frac{...
- 1,385
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Are there closed form solutions available for this equation below?
Solutions for the poisson equation are well known.
$$\nabla^2U=\nabla\times W
$$
If one more linear operator$$ \nabla(\nabla\cdot\ U)$$ is present,
$$
\nabla(\nabla\cdot\ U)-...
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Global Poincaré type estimate
For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
- 4,135
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Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]
Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
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Is there any solution for this PDE system?
Let $(\mathbb{R}^2,\langle .,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with
\begin{align}
J_{\...
- 11
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Reference request for the focussing example
I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. ...
user71046
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Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases}
-\...
- 266
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Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$
I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...
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Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in L^...
- 1,051
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Integrability of the Poisson integral
Maybe this is rather obvious, but I'm stuck. Let's consider the Laplace equation in the upper half plane with boundary condition $g$, $i.e.$
$$
\Delta u(x,y)=0, u(x,0)=g(x).
$$
Then the solution is ...
- 843
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Differences between parabolic operators of second order and higher order
Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly ...
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Existence of Green's function and the Dirichlet problem
Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \...
- 31
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Proof of global (in time) existence of classical solutions for 2D Euler equation in bounded domain
Anyone can explain the main idea, or recommend some paper or book on that?
For the whole space case, or the periodic case, the proofs are everywhere. But those do not seem to apply to the bounded ...
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approximating functions pointwise [closed]
If we have in certain norm
1). $g_j(x) \rightarrow h(x), j\rightarrow\infty$ and
2). $ f_{ij}(x)\rightarrow g_j(x), i\rightarrow \infty, \forall j$ ,
then we can choose a subsequence $\{f_{ij_{(i)...
- 129
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612
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Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213
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Coupled semilinear PDEs
For a surface $z(x,y)$, let $p = z_x$ and $q = z_y$. I have a pair of coupled semilinear PDEs in p and q.
PDE1: $p_x - a p_y = b q$.
PDE2: $q_x - a q_y = -b p$.
Note that $a$ and $b$ are functions ...
- 13
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From an integral equation to a differential equation
Hello,
I am wondering whether it is possible to convert the following integral equation to a partial differential equation.
Integral Equation here http://ima.epfl.ch/~lechen/images/integralEq.jpg
...
- 1,649
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Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,135
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Approximation on $H^1_0(B)$ and cut-off functions
Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that
$$
\| u - \...
- 167
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1
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$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.
Consider the problem
$$\Delta u = f \quad\text{in $\...
- 93
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Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
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Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions
We know in dimension $3$,
\begin{align}
\partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} ,
\end{align}
where $\varepsilon_{ijk}$ are Levi-Civita symbols ...
- 89
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How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
- 835
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$\|\hat{f}\|_{L^q}< \infty \implies \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty $
Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $ \int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier ...
- 325
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The derivation of Reynolds-averaged Navier-Stokes equations
The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations)
When we talk about turbulent flows we can represent the velocity of the fluid as:
$$
...
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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$?
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An inequality involving weight $|x|^\alpha$
Background:
Let $u:\mathbb{R}^n\to \mathbb{R}$. Then the paper considers the following problem
\begin{align*}
-\operatorname{div}(w(x) \nabla u) &= w(x) \text{ in } \Omega \\
...
- 537
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Explicit solutions for linear system of PDEs with constant coefficients
I've been recently trying to to solve the following system of linear 1st order PDE's:
$f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$
$\dfrac{\...
- 9
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Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
user139844
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468
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Harmonic functions in infinite domain in Euclidean space
EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
- 21.8k
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Explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)$
Do we know an explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)?$
- 537
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Log-concavity of the modified Bessel function of a second kind
I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
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Reference request and methods indication to the continuity of solutions to the problema $L_tu = F(u), ~t\in [0,1],$ and $L_t$ elliptic
Let $M$ be a closed manifold and assume that is given a family of elliptic operators $L_t,~t\in [0,1]$ and a smooth function $F :[a,b] \to \mathbb{R}$ such that for each $t$ the elliptic problem $L_tu ...
- 1,774
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Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
- 261
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Showing that a Gaussian achieves equality in a logarithmic Sobolev inequality
Consider the following Logarithmic Sobolev inequality on page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14): for $f\in H^1(\mathbb R^n),$ and $a>0$ any positive number,
$$
\frac{a^2}{...
- 1,755
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Existence of solutions of a system of first order PDEs
Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset.
Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions.
That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
- 261
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1
answer
325
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Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
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465
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How can I prove this Weitzenbock formula
Update: It is almost sure that the expression of $\kappa$ in coordinates given in the book, namely
$$\kappa(u_t)=h_{\alpha\beta}\frac{\partial u_t^\alpha}{\partial x^i}\frac{\partial u_t^\beta}{\...
- 283
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1
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180
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What can be an interesting problem of differential equations involving the definition of the Gudermannian function? [closed]
In this post I denote the Gudermannian function as $$\operatorname{gd}(x)=\int_0^x\frac{dt}{\cosh t}$$
and its inverse as $\operatorname{gd}^{-1}(x)$, please see if you need it the definitions, ...
0
votes
2
answers
289
views
Derivations of $\chi^{\infty}(M)$ which are elliptic operator
What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
- 356
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1
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787
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Estimates for Green function for fractional Laplacian
Can the Green function for the fractional Laplacian operator be estimated from above and below.
$$ \left\{\begin{aligned}
(-\Delta_x)^{s} G(x, y)+ G(x, y)&= \delta_{y}(x) &&\text{in } \...
- 407
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votes
1
answer
266
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Existence and smoothness for viscous Burgers equation?
What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?
- 51
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1
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418
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Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
2
answers
132
views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
- 823
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1
answer
495
views
Heat kernel and convergence
Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
user50806
0
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1
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160
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boundary integral estimates for elliptic pde
Consider smooth positive solutions $u_m$ of
$$-\Delta u_m(x) = u_m(x)^p \quad \mbox{ in } \Omega$$ with $u_m=0$ on $ \partial \Omega$. My interest is in obtaining some sort of global integral ...
- 1,385
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96
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$L^p$ regularity for semidisc
Suppose $B_r\subset \mathbb{R}^2$ is a hemidisc, i.e., $x^2+y^2 \leq r^2, y\geq 0$. Is there a regularity result of the type $\Vert \psi \Vert_{W^{2,p}(B_{1/2})} \leq C (\Vert \psi \Vert_{L^p(B_{1})} +...
- 3,383
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1
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109
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Lipschitz type estimate for the Green function for fractional Laplacian
Let $G(x, y)$ be the Green function of the fractional Laplacian $(-\Delta)^s$ in a bounded interval $I$ of $\mathbb{R}$ with $G(x, y)=0$ on $\mathbb{R}\setminus I$ and $s\in (0, 1).$ Is it possible ...
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