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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Complex transport equation
Consider an n dimensional Riemannian manifold with boundary.
Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...
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Dichotomy for global existence or blow up for solutions of evolution problems
Consider the problem (Nonlinear Schrödinger equation)
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u\mp u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\...
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Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?
Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...
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Surfaces with specific types of second fundamental form
Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...
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Compactly supported functions and Sobolev spaces on manifolds
It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
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PDEs on torus $\mathbb T$
(Hope this question is o.k. for MO)
I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...
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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 ...
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Solvability of quasilinear elliptic equations on closed manifolds
Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds?
In particular, I am looking for solvability condition for function $f$ of following equation
$\...
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Isothermal-related functions in higher dimensions
I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.
Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
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Minimal surfaces + Semi-Geodesic Coordinates
Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
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Equivalence of two definitions of Sobolev spaces
Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) \...
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Notion of solution of pde
Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in L^\infty(0,...
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decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
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Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1
In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
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$L^p$ stability of the Beltrami equation
Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} &...
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Measurability of solution of diffusion equation in sub sigma algebra
I want to solve the following problem:
Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$.
Let $a( x;. )$ and $f(x;.)$ be $\...
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What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?
I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi \...
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Uniqueness theorems related to Hardy Uncertainty Principle
Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that
if $f(x)=O(e^{-\alpha^2|x|^2})$, $\hat{...
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Sobolev Space, "characteristic function" for the weak derivative
Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product
$$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$
for $u\in H_0^1$ and $\mu(...
- 8,086
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Stochastic methods for solving very high-dimensional PDE
I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, ...
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Invariance of a tensor Laplacian
Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let $...
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Green's function and eigenvalues with multiplicity
Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
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Elliptic Fourier integral operators
I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...
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Reference request: Systems of linear PDES with constant coefficients
I am looking for a reference for the following statement:
Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs
\begin{align}
P_i(\partial / \partial x_1, \dots, \...
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A Liouville theorem involving an advection term
I am curious whether there are any Liouville theorems for the following pde:
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$
where $a(x)$ is a smooth bounded vector field ...
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Bogomol’nyi’s Formula for the Critical Action
I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
$(|F_A|^2+|d_A\phi|^2+|\frac12(1-|\phi|^2)|^2)...
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Reference request: Wasserstein metric spaces for non linear weights/mobility?
There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ...
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Existence of solution?
I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let $\mathcal{...
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Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE
A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
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Lens-shaped vs globally hyperbolic
The theorem of uniqueness of solutions of first order, quasilinear, symmetric hyperbolic systems is naturally formulated in terms of so-called lens-shaped domains. Roughly, a domain is lens-shaped if ...
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semi-classical Green's function
I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations.
In the two papers, he gave a semi-classical approximation of the Green's function to ...
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Equation of motion of a charged string in a twisted torus.
The flat torus background:
Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...
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What are the applications of Grillakis Shatah and Strauss paper?
I am studying the following paper.
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197.
...
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Motivation for weak solution of a PDE (initial condition)
The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ \...
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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 $V :=\{f\in L^2(X,m):\epsilon(f)&...
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Probability that d-Brownian Motion ,$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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Wave equation with linear coefficients
The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...
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Properties of solutions of Parabolic type equations
Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation
$$
\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\
u(0)=u_{0}.
$$
with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...
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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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The complex heat kernel on a Riemann manifold
There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial u}{\...
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Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
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Textbook for Partial Differential Equations with a viewpoint towards Geometry
I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here.
Though similar questions have been asked at https://...
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Sobolev spaces of maps between manifolds and the Palais-Smale Condition
I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
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A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable
First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
$$\left[\left(\cos\phi\partial_{z}+\...
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Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]
Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
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Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{...
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a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs?
Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am ...
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Monge-Ampere type PDE
NB: I have edited this question to clarify what the OP is asking – Robert Bryant
Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = ...
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Log of heat kernel for positive time
A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and $p(t,x,...
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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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