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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Existence of the solution of a Dirichlet type differential equation
I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question:
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method for global existance for the NLS
We consider the nonlinar Schr\"odinger equation(NLS):
$$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$
where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...
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is this a known method for solving PDEs
I recently posed a system of PDEs to solve on MSE at https://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions.
However, in this post, I'd like to show here ...
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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
$$\...
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Uniqueness of weak solution L[u]=0
Suppose L is a partial differential operator of arbitrary order with constant coefficients.
If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the ...
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variation norm of a Fourier transform
Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in (...
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An interpolation type inequality
Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$
$$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le \...
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Elliptic regularity on bounded domains
I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on $\mathbb{R}^n$...
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Integrability of Continuous Tangent Subbundles
Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...
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Changing the space of test functions in PDEs
Let $V \subset H \subset V'$ be a Hilbert triple.
We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\...
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Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?
Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$.
Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in L^2(X,m):\epsilon(f)&...
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Subordination identity and heat operator
I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...
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Positive inuequality of Laplacian with a variable coefficient [closed]
Let $0 < a_0 \leq a(x)$ be a smooth function on $\mathbb{T}=[0,2\pi]$, $a(0)=a(2\pi)$, whether it holds that
$$
\int_\mathbb{T} a(x)|\partial_x \varphi|^2 \geq \int_\mathbb{T}\frac{\partial^2_x a }...
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The asymptotic distribution of a subset of Bessel function zeroes
For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J_n(\alpha)=0 \textrm{ ...
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LINEAR Parabolic equations. Smooth dependence from initial data
I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data.
More specifically I have the following problem:
CONSIDER spaces $P:=\mathbb{R}^k$ ("...
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A "gradient" weak Harnack inequality for quasilinear elliptic equations
Suppose we are in the following loosely described setting:
we have a non-negative supersolution $h$ of the following elliptic equation:
\begin{equation}
\Delta h + \|\nabla h\|^2 + f(x) \geq 0
\end{...
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
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A bound in Sobolev spaces of negative order
Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$.
I wonder if the following bound is true:
$$
\|f g_{x_1}\|_{H^{-0.5}(U)}\leq C(\...
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Laplacian mapping on various function spaces
I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< \infty$...
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Functionals continuous with respect to weak convergence
It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...
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Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform
Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
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Reference for existence results for 2D forced viscous Burgers equation
I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
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What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''?
What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to
$Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1.
This ...
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Regularity theory for "nice" differential equations
My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or ...
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Asymptotics of the TBA equation
The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is
$$\epsilon(\beta)=R\cosh\beta-\int\frac{...
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Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles
I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
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Notion of solution of pde
Let's consider the following Schrodinger equation
$$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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Distributional limits concerning the regularity of Maxwells equations
This question is related to my previous question about the regularity of the Maxwell equations.
Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...
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persistence of regularity for nonlinear Klein-Gordon equation
I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" (...
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Coarea formula in a subelliptic context
Consider smooth vector fields $X_1,..,X_k$ in ${\mathbb R}^n$, satisfying the H\"ormander condition, i.e. for all $x$, the Lie algebra generated by $X_1(x),...,X_k(x)$ is ${\mathbb R}^n$. Do you know ...
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Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
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Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?
EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful.
Let $V \subset H \...
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Pohozaev result for equations with weights
I am interested in nonnegative solutions of
$-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$.
Or instead the equation $ -\Delta u + \...
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Quantitative estimate of heat dispersion - off diagonal estimates
Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
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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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non-negativity to positivity
Let $(M,g)$ be a closed Riemannian manifold of dimension $n>7$. In this setting I have been able to prove that the Green's function of a positive Paneitz-Branson operator is non-negative. ...
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Hölder estimates for the Complex Monge-Ampere equation
If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{...
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What we mean by positive solution and radial solution of any partial differential equation
I am not getting the most of the article concern with the existence and uniqueness of positive solution and radial solution.
I just want to know how a positive solution and radial solution of ...
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fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
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Tensor analysis/Differential forms outside physics
There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...
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Density of restrictions of $p$-harmonic functions on a hypersurface
Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
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Do boundary conditions for elliptic PDE need to be homogenous to use spectral theory?
Question 1: It appears that when studying an elliptic equation $Lu=f$ in $\Omega$ with $u = g$ on $\partial \Omega$ we need to have $g=0$ in order that the inverse operator, $K=L^{-1}$ is linear. ...
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The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.
I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u_t = (\ln u)_{xx}$$
which is to run on an interval $ a \leq ...
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How to show this integral on boundary of Lipschitz domain is finite?
Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int_{\partial\Omega} \frac{1}{|y|^{...
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Existence of cut-off functions in metric spaces
Let $X_1,\ldots,X_m$ be Lipchitz continuous vector fields in an open set $\Omega \subset \Re^n$, Let $d(\cdot, \cdot)$ denote the control distance associated to $X$. With respect to this control ...
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Existence of solution for this parabolic PDE
The parabolic PDE
$$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$
has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form (...
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Solvability for constant-coefficient partial differential operators
Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
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weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions
I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...
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Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain
Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
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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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