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
21
votes
3
answers
9k
views
A Hölder continuous function which does not belong to any Sobolev space
I'm seeking a function which is Hölder continuous but does not belong to any Sobolev space.
Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \...
- 60.5k
4
votes
1
answer
159
views
diffusions corresponding to estimators
I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
CommunityBot
- 1
26
votes
5
answers
5k
views
Book Recommendation - PDE's for geometricians / topologists
I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
- 27.5k
5
votes
1
answer
418
views
Robin-Laplacian in unbounded domains
Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If $\Omega$...
CommunityBot
- 1
5
votes
1
answer
932
views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-...
- 34.7k
3
votes
1
answer
365
views
Two equivalent definitions of weak solution to parabolic PDE; don't understand proof
(Crossposted from MSE due to no replies)
I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the ...
- 6,210
6
votes
1
answer
181
views
Ergodic Mean for Schrodinger flow
Let us consider the linear Schrödinger equation in $\mathbb{R}^N$
$$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$
with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its solution....
- 61
3
votes
2
answers
365
views
analysis question related to $L^p$ type inequalities
Dear mathoverflowers.
Just wondering if the following inequality is true. For all $ p >1$ there is some $C$ such that
$ | |x+1|^p-|y+1|^p -p(x-y)| \le C ( |x|+|y| + |x|^{p-1} + |y|^{p-1} ) |x-...
CommunityBot
- 1
5
votes
0
answers
428
views
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: \...
- 3,245
1
vote
1
answer
400
views
Nonlinear parabolic PDEs existence with Galerkin method?
Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...
- 4,729
1
vote
1
answer
396
views
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 \...
CommunityBot
- 1
6
votes
2
answers
2k
views
Eigenvalues of Laplacian
What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be
$$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...
- 6,210
11
votes
2
answers
2k
views
the spectrum of the Laplacian and Dirac operator on $S^3$
A paper on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$:
The eigenvalues of the vector Laplacian on divergenceless vector
fields is $(\ell + 1)^2$ with ...
CommunityBot
- 1
6
votes
1
answer
361
views
Local boundedness of weak solutions of heat equations...?
(Please, what is going on?) The following claim is from a paper which was apparently reviewed by László Erdös, Zhongwei Shen, and Bernard Heffler. Someone tell me it's true. Surely it's true. The ...
- 251
1
vote
2
answers
106
views
Bound deg 3 partial differential operator on Laplace eigenfunction?
I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
- 34.7k
40
votes
2
answers
4k
views
Recent fundamental new directions in PDEs
My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
- 2,106
5
votes
1
answer
421
views
A moving boundary in rock mechanics
I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...
CommunityBot
- 1
0
votes
1
answer
291
views
Application of Toms- Stein restriction theorem for Strichartz estimates
The initial value problem for one dimensional Shrödinger equation is
$$iu_{t}+u_{xx}=0,$$
$$u(x, 0)= f(x),$$
where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued ...
CommunityBot
- 1
1
vote
0
answers
142
views
Elliptic problem on half space; infinite boundary values; Liouville theorem
In a the study of a boundary value problem the following problem is arising:
$-\Delta v(x)= e^{v(x)}$ in $ R^N_+$
$v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$.
...
- 539
15
votes
2
answers
785
views
Is there a spectral theory approach to non-explicit Plancherel-type theorems?
Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...
- 23k
4
votes
1
answer
261
views
Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
- 614
0
votes
1
answer
180
views
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 ...
- 39.1k
1
vote
1
answer
527
views
Reference request: Spectral analysis of advection diffusion PDE
As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...
- 2,243
15
votes
1
answer
5k
views
Basis for the space of Harmonic homogeneous polynomial in N variables.
Hello,
Does someone know an explicit basis of the space of harmonic homogeneous polynomial in N variables.
When $N=3$, if I'm not mistaking Legendre polynomial allow to write an explicit basis.
Is ...
- 8,597
3
votes
1
answer
703
views
unique continuation property for overdetermined elliptic PDE
On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is injective....
- 31
0
votes
1
answer
236
views
analytic solution to elliptic PDE in R^n
I am looking for (minimal) conditions, which guarantee that the problem
Lu = 0 in R^n,
where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...
CommunityBot
- 1
42
votes
5
answers
6k
views
Why is symplectic geometry so important in modern PDE ?
First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds ...
- 2,239
2
votes
3
answers
3k
views
Jacobi method on first order partial differential equations
Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance
CommunityBot
- 1
3
votes
1
answer
785
views
on an inequality of Brezis-Lieb
In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
- 25.8k
6
votes
2
answers
1k
views
How to solve the linearized Navier-Stokes equations in L^P?
Let $\Omega\subset \mathbb{R}^3$ be an open set with smooth boundary $\partial \Omega$.
Consider the following linearized Navier-Stokes equations in $Q_T=\Omega\times (0,T)$ for an arbitrarily fixed $...
- 171
1
vote
2
answers
1k
views
Reference request: learn measure theory for PDEs
I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...
CommunityBot
- 1
2
votes
0
answers
170
views
Elliptic equations with divergence-free drift terms
Given
$\
\mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ bounded, $div$$(\mathbf{u})=0$, $\...
- 171
1
vote
2
answers
213
views
Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data
I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial ...
CommunityBot
- 1
1
vote
1
answer
335
views
In which way is this a linearization of the Gross-Pitaevskii-Equation?
In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation
$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - \widetilde{v}...
CommunityBot
- 1
1
vote
1
answer
567
views
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 $P:=\mathbb{R}^k$ ("...
CommunityBot
- 1
1
vote
0
answers
103
views
Kernel of perturbation of biharmonic operator
Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form:
$$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$
with $P_{ij}(x)...
- 149
2
votes
2
answers
462
views
Regularity of parabolic equation; divergence free drift
I am curious about weak solutions of the parabolic problem
$ u_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is ...
- 171
1
vote
0
answers
222
views
local existence for a singular quasilinear parabolic equation
I'm considering the following type of PDE:
$u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$
with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, $...
- 11
5
votes
0
answers
153
views
Critical elliptic equation; kernel of linearized operator
I am interested in the critical equation
$- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin (...
- 539
5
votes
0
answers
198
views
Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs
Question:
Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$:
$$
\frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t),
$$
with smooth initial data $...
- 301
51
votes
1
answer
6k
views
Unconditional nonexistence for the heat equation with rapidly growing data?
Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [...
- 1,655
0
votes
1
answer
89
views
approximating functions pointwise [closed]
If we have in certain norm
1). $g_j(x) \rightarrow h(x), j\rightarrow\infty$ and
2). $ f_{ij}(x)\rightarrow g_j(x), i\rightarrow \infty, \forall j$ ,
then we can choose a subsequence $\{f_{ij_{(i)...
- 129
7
votes
3
answers
1k
views
PDEs involving measures; where to begin?
If I want to learn about existence of weak solutions to PDEs of the form
$$u_t + Au = f$$
or
$$Au = f$$
where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for ...
- 1,587
3
votes
0
answers
179
views
How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded
With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider
$$u' + Au = f$$
in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
- 41
2
votes
0
answers
159
views
Helmhotz decomposition and Regularity in Stokes equation
It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely
decomposed as
\begin{eqnarray*}
\
f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)}
\
\end{eqnarray*} with $f_{0}\in L_{...
- 171
4
votes
0
answers
343
views
Tangential boundary regularity for optimal transport maps
I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II:
Assume $u$ is a convex ...
- 4,899
2
votes
1
answer
461
views
Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?
I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
CommunityBot
- 1
2
votes
2
answers
324
views
worst regularity of f ensuring u is locally Lipschitz for $\Delta u = f$
Assume $M$ is a Riemannian manifold, and $\Omega $ is a bounded domain. Consider the Poisson equation:
$$\Delta u = f \qquad \text{with }u \in {W^{1,2}}$$
What is the worst regularity of $f$ which ...
- 914
3
votes
1
answer
518
views
A closed extension of the Laplace operator with respect to the supremum norm
Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
- 3,968
2
votes
0
answers
219
views
A microlocal representation for quantum operator dynamics
In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
- 21