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.
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How to regularize $\prod_{\mathbb{Z}^2\backslash \{(0,0) \}} (a^2 + b^2)$?
Is there any way to regularize this infinite product?
$$ \det \Delta = \prod_{a,b > 0} (a^2 + b^2) $$
I tried to say this was the derivative at $0$ of an L-function
$$ \zeta_{K}'(0) = \sum_{(a,...
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Qualitative Solution of PDE on the 2-sphere (for weather prediction)
While I was watching the news last month I realized the weather report was basically a discussion of solutions to PDE. In particular, I was paying attention to the hurricane season (which is not yet ...
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Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
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Reference request: what is the definition of $H^1(\Omega)$ with $\Omega=(0,1)\times\mathbb{T}$?
Denote the 1-D torus as $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$. Using Fourier series, one can define the Sobolev space $H^k(\mathbb{T})$ (see for instance this note from Wikipedia). On the other hand, ...
user14319
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Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
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Bound of solutions by initial value of Navier Stokes equations
For the mollified Navier Stokes equations: $$\partial_t u_{\epsilon} - \Delta u_{\epsilon} + \mathbb P \nabla \cdot((u_{\epsilon} \ast \omega_{\epsilon})\otimes u_{\epsilon})=0 $$ $$\nabla \cdot u_{\...
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Convergence of derivatives of $\sum c_{k} \phi_{k}e^{-\lambda_{k}t}$ eigenfunction expansion
Q: Is the $\sum c_{k} \phi_{k}(z)e^{-\lambda_{k}t}$ smooth over $\Omega\times (0,\infty)$ (both z and t), where $\Delta \phi_{k}+\lambda_{k}\phi_{k}=0$ for bounded domain $\Omega$ and $|c_{k}|\leq k$...
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Surfaces in isothermal coordinates and particular PDE
From Brioschi's formula, if we have a surface in isothermal coordinates were $g_{ij}=E(u,v)*\delta_{ij}$ is the metric tensor, the gaussian curvature is:
$K=-\frac{1}{2E}[\frac{\partial}{\partial u}(...
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Boundary regularity of the solution of a Poisson equation in a polyhedron
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
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$u$ satisfies Schrödinger equation implies $\mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \right) $ also?
Consider the Schrödinger equation (SE):
$i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0, (x, t)\in \mathbb R^{N}\times \mathbb R.$
$u(0,x)=\phi(x).$
Then, formally,
the solution of (SE) ...
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Composition of a negative operator and a positive one
Let $\Omega$ a bounded open set of $R^n$, $\omega$ a non-empty open subset of $\Omega$ and
$\chi_{\omega} : L^2(\Omega) \longrightarrow L^2(\omega)$
be the restriction operator to $\omega$,...
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The analyticity of distance function
Given a real analytic compact manifold $M$ with boundary $\partial M$, suppose that $M$ is embedded in an open analytic manifold $N$ which has the same dimension as $M$. Is the distance function $d(x)...
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Singular elliptic PDE: $-h\,\Delta u +\langle \nabla h,\nabla u\rangle =h$
Let $U\subset\mathbb{R}^n$ with $0\in U$. Fix $h\in L^2(U)\cap C^\infty(U)$ with $h(0)=0$. Is there some $C^1$-function $u\neq 0$ in such that $u$ is solution of
$$-h\,\Delta u +\langle \nabla h,\...
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A question on weak formulation of the p-laplacian operator
Can it be said that $$\int_{\Omega}\Delta_p u |\phi|^{p-2}\phi dx=\int_{\Omega}\Delta_p \phi |u|^{p-2}u dx\qquad\forall \phi\in C_0^2(\overline{\Omega})$$ is the generalized weak formulation of $$\...
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Domain of dependence for Hyperbolic system of PDES
In Leveque's "Numerical Methods for Conservation Laws", Ch. 3.1.1., he says that given a system of Hyperbolic PDE's and a point $(\bar{t},\bar{x} )$, its domain of dependence $D(\bar{t},\bar{x} )$ is ...
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$L^{\infty}$ norm of Integral-Differntial equation's solution
Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$
, $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE
$$
\partial_t\Phi(t,z) - \phi(0,z) - \...
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Sum of quasi-m-accretive operators
Kato's existence theory (see link) gives conditions under which a PDE has a (unique) solution. The equation of evolution he considers is
$$ \frac{dv}{dt} + A(v)v = f(v),$$
where $A(y) \in L(Y,X)$ for $...
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Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
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Regularity for a div-curl system
Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
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existence of an initial-boundary value problem with nonhomogeneous boundary conditions
Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:
$u_t=\Delta u$, for $x\in \Omega$, $t>0$;
$u(x, 0)=f(x), x\in\...
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compact injection
Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
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Isothermal coordinates
Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?
I know that this metric is a special kind ...
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Is $\Delta u+f\in (H^1(\Omega))^*$ with $u\in H^1_0(\Omega)$ and $f\in L^2(\Omega)$?
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
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Inverse trace theorem for partial trace
A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\...
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Harmonic/Subharmonic lifting of functions on an annulus
Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...
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For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?
Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that $...
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A priori estimate for diffraction problem for linear elliptic PDEs
I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation.
I looked at ...
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Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
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The monotone operator in $BV$ space
I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
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How Minimal solution is obtained as limit of approximations
I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this:
I can understand the proof, but I don't know why this solution which is obtained as a ...
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A heat equation approach to the perturbation of vector field with center
Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...
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Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
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One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
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Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
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solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
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Any reference in absorbing boundary conditions for non-abelian gauge fields?
Is there any paper on absorbing boundary conditions for non-abelian gauge fields?
Currently I only saw some on elastic wave equations and some on EM fields.
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Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
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Boundedness of heat semigroup on $L^1(\Omega)$
On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...
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inflow/outflow Boundary Conditions for flow in pipe
I have a question about boundary condition of solving Navier-Stokes equation through pipe.
When I simulate the flow in pipe using periodic boundary condition, it works good. But when I tried to change ...
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Schauder estimate on a bounded domain
We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that
$$
||\Delta u||=||{\nabla}^2u||
$$
where $||\cdot||$ is the $L^2(\...
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The minimizing problem over a sequence of shrinking balls
Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define
$$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$
where $1<q<5$. Hence we know that each $\...
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How to prove it is uniformly bounded?
Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
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Notion of solution of pde
Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in L^\infty(0,...
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
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Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A
In other words, the probability that Brownian motion stays within $A^{c}$.
What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...
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Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
$$\...
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Compatibility of initial and boundary conditions
Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \text{int}D^2, t > 0$$
where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary conditions $$\...
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Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary [closed]
I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|_{L^p} \leq ...
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$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0(...
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Estimate for an integral of a function of the solution to a PDE
Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and
$\lambda\leq \sigma_1, \sigma_2 \leq \...
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