All Questions
Tagged with ap.analysis-of-pdes fa.functional-analysis
1,304 questions
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Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?
We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:
$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$
where $U(t)= e^{it\Delta} $(free ...
- 1,051
0
votes
1
answer
272
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Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result
Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)
Let $\Omega \...
- 266
0
votes
1
answer
241
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Nonlocal (parabolic) PDEs in the Sobolev space setting
Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev ...
- 155
0
votes
2
answers
664
views
Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
0
votes
1
answer
1k
views
Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
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votes
1
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488
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Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
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votes
2
answers
818
views
Application of inverse function theorem to get short time existence
I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): https://i.sstatic.net/...
- 45
0
votes
1
answer
338
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The part of an operator as an analytic generator
Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
Let $Y$ be another Banach space embedded in $X$. We consider$A_Y$, the part of $A$ in $Y$, defined as the ...
- 271
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55
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Compactness and Leray-Schauder degree
What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
- 471
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80
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Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
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36
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Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
- 109
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votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
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45
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Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
- 41
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0
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73
views
Operator globally hypoelliptic
An operateor $T$ is globally hypoelliptic if :
$u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$.
My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$.
where $\...
- 531
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55
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Time regularity vs space regularity for parabolic PDE
Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
- 311
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0
answers
36
views
Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
- 101
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votes
0
answers
28
views
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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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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52
views
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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113
views
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)$$
Where $h(i)=C (i+1)^{-p}$ with $...
- 2,252
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143
views
Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
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votes
1
answer
161
views
Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
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votes
0
answers
111
views
Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator
I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
- 101
0
votes
0
answers
55
views
Weyl's law for hyperbolic operators
Let $\Omega \subset \mathbb{R}^n$ be a smooth, bounded domain and consider the operator $T: L^2([0,1]\times\Omega) \to L^2([0,1]\times\Omega)$ so that $Tf = v$ if
$$
\begin{cases}
\frac{d}{dt}v - \...
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0
answers
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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votes
0
answers
98
views
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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0
answers
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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votes
0
answers
171
views
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
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votes
0
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83
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
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0
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57
views
Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
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votes
0
answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
0
votes
1
answer
162
views
Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
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votes
0
answers
129
views
Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
0
votes
0
answers
93
views
Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
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0
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164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
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votes
0
answers
239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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0
answers
53
views
Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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0
answers
104
views
Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
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0
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150
views
Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
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0
answers
153
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Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$
How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
- 99
0
votes
1
answer
414
views
Sufficient conditions for an asymptotic compactness
This question relates a theory of Mosco convergence.
Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$.
A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
- 721
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votes
0
answers
121
views
Positivity of solution for Fisher-Kolmogorov Equation
How can we prove that if $y=y(t,x)$ is the solution of the problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y(t,x)=r(x)y(t,x)-\rho(x) y^2(t,x),\ (t,x)\in (0,T)\times \Omega \\ \...
- 1,759
0
votes
0
answers
61
views
$L^p$ estimate for perturbed heat equation
Let us consider the heat equation
$$
\begin{cases}
u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\
u(t,-1) = a(t), \\
u(t,1) = b(t), \\
u(0,x) = u_0(x)
\end{cases}
$$
where $f \in C^\...
- 131
0
votes
0
answers
330
views
Comparison of the spectrum decomposition
In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
- 1
0
votes
0
answers
37
views
Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 943
0
votes
0
answers
57
views
Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?
Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
- 107
0
votes
0
answers
88
views
Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$
Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
- 183
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0
answers
85
views
Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?
$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that.
For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
user128095
0
votes
0
answers
75
views
Partial well-posedness results on Schrödinger operators?
Set $ A_i:= -\Delta + V_i :H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \ i =1,2 $, where
\begin{equation*}
V_1 = 0, \ \ (\textrm{No interaction}) \\
V_2 = - \frac{\gamma}...
- 269
0
votes
0
answers
83
views
$ 0 $ locates in the continuous spectrum of Schrodinger operators?
This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range.
For ...
- 269