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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Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
- 809
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55
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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem
This is the equation given ($n\geq2$)
$$
\begin{cases}
u_{tt}=a^{2}\left(\Delta u\right), \\
\left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\
\left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) .
\end{...
- 1
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74
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Elliptic regularity for Dirichlet problem
Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective ...
- 577
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79
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Convergence of metric implies convergence of eigenvalues?
Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:
Does $g_\varepsilon$ converge to the flat metric on ...
- 537
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65
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
- 809
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28
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Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
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112
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Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 593
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72
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Some questions about the concept of stable solution of elliptic PDE
For $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is stable if
$$
Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
- 809
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Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
- 1,334
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46
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Conservation law for generic linear hyperbolic PDEs?
Consider the wave equation:
$$
u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T].
$$
To prove the only solution for the zero initial condition is zero, we only need to ...
- 1,755
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67
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Examples of symmetry-breaking solitons which retain a subgroup symmetry
There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions.
However, all symmetry breaking soliton examples I have seen go from the ...
- 49
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80
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What is the PDE corresponding to this weak formulation?
Consider a flow $(\mu_t)_{t\ge 0}$ such that
every $\mu_t$ is a probability on $\mathbb R_+$;
$\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
- 1,334
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Reference request heat equation with moving interface
Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$:
$$
\begin{cases}
u_t = \sigma_1^2u_{xx} ~~~~\...
- 393
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108
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Mixture of Ornstein-Uhlenbeck operators
Consider a set of isotropic, multivariate Gaussian densities with different centers $\mu_i\in \mathbb{R}^d $, $i\in\{1, \ldots, K\}$, which are denoted $\phi_i(x)$. They all have the same variance ...
- 555
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Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
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56
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Godunov splitting convergence research
The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
- 101
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32
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On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
- 840
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Advection diffusion equation with position dependent advection
I am trying to solve a equation that looks like $$u_t=f(x)u_x+Du_{xx}$$
any insight to whether an solution might exist would be very much appreciated!
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Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
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170
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Extreme confusion over the tautology of the solution for PDEs
Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth divergence free vector field. Then, it is well-known from the theory of Navier-Stokes equation that there exists some $T \in (0,\infty]$ and a ...
- 3,477
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87
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
- 1,759
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135
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Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 317
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111
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Reversing heat transfer with respect to time
Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...
- 119
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102
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Spectrum of Laplace-Beltrami operator on tensors
Let $(M, g)$ be a complete Riemannian manifold diffeomorphic to $\mathbb{R}^n$. Under appropriate geometric assumptions concerning the geometry near infinity, but without any curvature sign ...
- 419
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115
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Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
- 471
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Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
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52
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Coupled Kazdan-Warner type equation
Famous work of Kazdan and Warner shows that given $u\geq 0$ and a constant $c>0,$ the following equation in $f$ has a unique solution:
\begin{align*}
\Delta f+ u e^f=c
\end{align*}
I am interested ...
- 954
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92
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Solving a Catalan-like recursion of polynomials, related to the KdV energies
I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are ...
- 203
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102
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Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 371
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75
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Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets
Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 61
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81
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Is it possible to continuously embed $C^\infty(\mathbb{T}^n)$ as a vector space into $\mathcal{D}(\mathbb{R}^n)$ by some "inverse" of periodization?
Let $\mathbb{T}^n$ be the $n-$dimensional torus and $C^\infty(\mathbb{T}^n)$ be the Frechet space of smooth periodic functions on $\mathbb{R}^n$.
According to p.298 of Folland "Real Analysis"...
- 3,477
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113
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Solving $\frac{\partial}{\partial t}f = h f + h \int h f$
Is there a closed form solution to the following differential equation?
$$\frac{\partial}{\partial t}f(i, t) = a h(i) f(i, t) + b h(i) \int \mathrm{d}i\ h(i) f(i, t)$$
Where $h(i)=C (i+1)^{-p}$ with $...
- 2,442
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77
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$ \int_{\mathbb{R}^n} f R_1 f=0$ if $f\in L^2$ Riesz transform
How can I see that? It seems that it has to do with the adjoint of the Riesz transform $R_1^*=-R_1$, but here we do not have the complex $L^2$ scalar product.
- 1
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Difficulty in understanding one step in Ciraolo-Figalli-Maggi paper
I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.
Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\...
- 337
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142
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
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170
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A few questions on the paper named "On a Sobolev inequality with remainder terms"
I was reading a paper of Lu-Wei named "On a Sobolev inequality with remainder terms" (link at AMS site).
and I have quite a few questions regarding that.
(1) In page 78 they got a system of ...
- 337
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230
views
Solving PDE $u_t+f(u)u_x=0$ using its physical interpretation
Let's be this PDE:
$\begin{cases}u_t+f(u)u_x=0\\
u(x,0)=\varphi(x) \end{cases}$
and $f\in 1-1$.
I have these thoughts:
We can imagine $x'x$ having sticky particles. As we know $\frac{dx}{dt}=f(u)$. ...
- 189
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57
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If a Dirichlet problem is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
For example, $B$ is the unit ball in $R^{n}$, the ...
- 809
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34
views
Inequalities for generalized variance
Let $(X, \mu)$ be a measured space with $\mu(X) = 1$.
Given $\phi \in L^\infty(X, \mu)$, $\phi > 0$, let me define, for $\alpha \geq 1$, $\beta > 0$, the quantity
$$
I(\alpha, \beta) = \left(\...
- 1,263
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161
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Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
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111
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Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator
I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
- 101
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167
views
How does one make sense of singular solutions to constant mean curvature equation?
Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
- 537
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55
views
Weyl's law for hyperbolic operators
Let $\Omega \subset \mathbb{R}^n$ be a smooth, bounded domain and consider the operator $T: L^2([0,1]\times\Omega) \to L^2([0,1]\times\Omega)$ so that $Tf = v$ if
$$
\begin{cases}
\frac{d}{dt}v - \...
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75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
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78
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Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality:
$$\...
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85
views
Solution to the differential equation $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle =-\frac{\partial g(X)}{\partial x_{1}} $?
If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $...
- 912
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98
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
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66
views
Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
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54
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Perturbation of first eigenvalue
Let $\Omega$ be a bounded domain in $R^N$ and suppose $\gamma$ is some smooth bounded function in $\Omega$. Let $ -\Delta \phi_0 = \lambda_0 \phi_0 $ in $ \Omega$ with $ \phi_0=0$ on $ \partial \...
- 1,385
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171
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Explanation of a step in a preprinted work
I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...
- 159