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
1
vote
1
answer
107
views
Characterization on smallest element in affine Sobolev subspace
Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that
$$
\phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...
- 5,048
0
votes
1
answer
109
views
Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
- 6,377
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 6,377
7
votes
0
answers
229
views
Dimension dependence in Schauder estimates
Suppose we have a strongly elliptic PDE with Dirichlet conditions, namely: $L u = f, u_{|\partial D} = 0$ for a strongly elliptic L and a domain $D$.
Suppose also, to make life easy, $f$ is $C^{\infty}...
- 71
1
vote
0
answers
46
views
Boundary estimates for Neumann derivative of solution to Laplacian equation with Dirchlet boundary data
Let $\Omega \subset \mathbb{R}^n$ be a smooth domain. Consider the following Laplacian equation with Dirichlet boundary condition.
\begin{equation}
\begin{cases}
\Delta u=0\quad &\mbox{in $\Omega$}...
- 1,350
3
votes
1
answer
343
views
Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?
I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself.
Say we are working on the unit disc $D \subset \mathbf{R}^n$, ...
- 4,899
0
votes
0
answers
173
views
Lemma 3.10 of paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain
I am reading a paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain.
And I have a questions in the proof of lemma 3.10.
Please click the paper title for the link.
The ...
- 239
2
votes
1
answer
272
views
Even and odd solutions for the Schrödinger equation
We consider $2a$ - periodic smooth solutions for
\begin{eqnarray*}
-\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a]
\end{eqnarray*}
We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
- 6,696
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245
0
votes
2
answers
241
views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
- 39.1k
2
votes
0
answers
96
views
Evolution PDE in dual space : Generalization of a result of Gelfand
The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
- 233
29
votes
6
answers
10k
views
Square roots of the Laplace operator
In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...
- 6,853
2
votes
0
answers
148
views
Inequalities in a paper of J.Bourgain
I am reading a paper 'Periodic Nonlinear Schrodinger Equation and Invariant Measures' written by J.Bourgain. And I am wondering if I can have some help from this website.
At (2.13) of the paper, there ...
- 239
1
vote
0
answers
133
views
How can you make a PDE solution stabilize over time?
this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer.
Suppose I have a simple linear first order PDE of the form:
$$au_x+bu_y=0$$
I ...
- 251
5
votes
1
answer
279
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
- 3,584
2
votes
0
answers
122
views
PDE for the area-preserving non-parametric curve shortening flow?
In dimension $1$, it is well-known that the motion of a (unparametrized) plane curve $\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$ under the curve shortening flow can be encoded by the PDE $$\partial_t ...
- 730
1
vote
0
answers
38
views
Choice of splitting in domain decomposition algorithms
When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
- 61
9
votes
0
answers
519
views
Why is the symbol map in Atiyah–Singer paper continuous?
I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...
3
votes
1
answer
219
views
Reflection principles for linear second order elliptic pdes
Let $B_+\subset\mathbb R^n$ be the intersection of the unit ball with the upper halfspace and $u$ be a harmonic function in $B_+$ that vanishes identically in the flat part of the boundary of $B_+$.
...
- 1,848
0
votes
0
answers
77
views
On global solutions of a semilinear elliptic PDE
Consider the semilinear elliptic PDE
$$-\Delta \phi + \phi +\phi^3=f$$
for a real-valued function $\phi$ on $\mathbb{R}^n$, where $f$ is a fixed $C^2$ function. What is known about the existence and ...
- 577
2
votes
1
answer
171
views
Mean value formula for fractional heat equation
For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have
$$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$
where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
- 161
4
votes
1
answer
699
views
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
- 1,724
2
votes
1
answer
169
views
What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?
I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
- 313
8
votes
1
answer
612
views
Pfaffian systems that do not satisfy their integrability conditions
Remark: In this question I am first and foremost interested in a local problem and local solutions therefore I assume all functions are defined on open sets of real coordinate spaces and I will not ...
- 108k
2
votes
0
answers
162
views
$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?
Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 303
2
votes
1
answer
209
views
Does unique continuation also hold for derivatives of solutions?
Let $D \subset \mathbf{R}^n$ be the unit ball, and $u \in C^2(D)$ be a solution of the linear elliptic PDE
\begin{equation}
a^{ij} D_{ij} u + b^i D_i u + cu = 0 \quad \text{in $D$},
\end{equation}
...
- 39.1k
3
votes
0
answers
73
views
What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
- 111
2
votes
1
answer
232
views
Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?
I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...
- 511
2
votes
0
answers
49
views
Determining a space of differentiability
I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by
$$
P(t,s)f:...
4
votes
2
answers
543
views
Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold
Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...
CommunityBot
- 1
4
votes
1
answer
199
views
Compactly supported transverse traceless tensors
Let $(M, g)$ be a Riemanian manifold (or $\mathbb{R}^n$ if you prefer). A TT-tensor is a symmetric 2-tensor $\sigma_{ab}$ satisfying
$g^{ab} \sigma_{ab} \equiv 0$ ($\sigma$ is trace free),
$\nabla^a ...
- 21.6k
2
votes
1
answer
254
views
How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?
As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
- 63.9k
5
votes
0
answers
230
views
Concepts of Solutions to Partial Differential Equations
I already asked this question on math stackexchange (see here), but since I didn't get an answer there, I was wondering if I would be more lucky here.
I was wondering what the most used notions for ...
- 300
1
vote
0
answers
79
views
In which Sobolev spaces nonlinear wave equation is locally well-posed?
We consider nonlinear wave equation as follows:
$$\partial_t^2 \psi - \Delta \psi = \pm |\psi|^{2\sigma} \psi, \quad ( \psi(0, \cdot),\psi_t(0,\cdot))=(\psi_0,\psi_1)$$
where $\sigma \in \mathbb N, ...
- 325
4
votes
0
answers
143
views
If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
2
votes
0
answers
297
views
Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
- 246
8
votes
2
answers
2k
views
What is Young measure?
I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem:
Theorem. Assume ...
- 6,377
4
votes
0
answers
318
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
2
votes
1
answer
2k
views
Norms in Sobolev space $W^{1,\infty}$
Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ ...
- 237
4
votes
1
answer
1k
views
Where to learn about parabolic Hölder spaces and when to use them
Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...
- 313
1
vote
1
answer
202
views
Decomposing an analytic function into two functions which vanish at $0$ and $\infty?$
This question comes from exercise I-10 of The Geometry of Schemes by Joe Harris (although this question is not about schemes). It is translated to less abstract language below:
Consider the Riemann ...
- 43.2k
2
votes
1
answer
367
views
Type II singularities for 3D Ricci flow
I know that type II singularities of the Ricci flow can exist on closed 3-manifolds (e.g. on $S^3$), but on the other hand it seems to me that ODE comparison combined with Hamilton's tensor maximum ...
CommunityBot
- 1
6
votes
5
answers
3k
views
Navier-Stokes equations in Riemannian geometry
The Navier-Stokes equations can be written on a Riemannian manifold as:
$$\dot{u}+\nabla_u u+ \Delta u=(df)^* $$
$$d^* u=0$$
where $\nabla$ is the Levi-Civita connection, $u$ is a vector field, $\...
- 26.3k
0
votes
0
answers
263
views
Solving Fokker–Planck equation
Consider the Fokker–Planck equation:
$${\displaystyle {\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(...
- 134
4
votes
1
answer
313
views
Classical parabolic theory (PDEs)
I am reading the article Asymptotics of solutions to the periodic problem for a Burgers type equation by Pavel I. Naumkin andd Cristian Jesus Rojas-Milla. I'm almost done, but I don't understand this ...
- 3,065
1
vote
0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
2
votes
0
answers
153
views
unique continuation in a disk
Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy
$$ (1-r^2)^2\Delta w +w =0.$$
Prove that $w \equiv 0$.
- 4,115
5
votes
2
answers
431
views
General solution to an ultrahyperbolic PDE
$\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \...
- 63.9k
3
votes
1
answer
144
views
On the equation $[U, V] - V_x = C(x)$
While considering the zero curvature equation $U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ ...
- 108k
2
votes
0
answers
254
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 6,377