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
2
votes
0
answers
127
views
PDE Parameter-Dependent Center Manifolds
In the book Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems by Mariana Haragus, parameter-dependent center manifolds are discussed. Here it is assumed ...
- 21
1
vote
0
answers
61
views
Convergence of PDE/PsiDE - expansion of pseudo-differential operators
I have am working with a nonlinear pseudo-differential evolution equation of the form
$$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$
where $\mathcal{N}$ is a nonlinear operator and $\mathcal{...
- 155
6
votes
1
answer
216
views
On a conjecture of Lions for the wave equation
In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :
Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider
$\phi'' - \triangle \phi = F$
$\...
- 63
1
vote
2
answers
2k
views
The integrability of fundamental solution of laplace equation follows from integrability of f ?
Hi, I am really struggling with this question.
The question is :
Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta ...
- 13
1
vote
0
answers
70
views
Normal form of Principal type $\Psi$DO's
Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I ...
- 4,115
1
vote
1
answer
333
views
Weak convergence of a sequence
I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : $\int_{[...
- 185
1
vote
1
answer
136
views
Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$
Is there anything known about uniqueness of classical solutions to
$$
\partial_t u -u\Delta u=0\quad u(0,\cdot)=1
$$
on smooth domains $[0,T]\times D$ without boundary conditions? I know that $u(0,\...
- 435
1
vote
1
answer
367
views
weak*closure of {f:||f||=1} in dual.
What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology.
So what is the weak* closure of this set. Thanks.
- 1,269
1
vote
1
answer
237
views
Interpolation and embeddings for parabolic function spaces
I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 632
1
vote
0
answers
166
views
Examples for differential operators first order
Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function ...
- 223
1
vote
1
answer
159
views
Nodal sets under the heat flow
Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $...
3
votes
0
answers
134
views
Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$
Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...
- 601
3
votes
1
answer
367
views
BMO spaces on the torus
I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as
$$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$
where $u_Q$ is ...
- 31
1
vote
1
answer
3k
views
How to show this Holder bound?
Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \...
- 67
5
votes
3
answers
2k
views
Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way
I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on $\...
- 7,197
4
votes
1
answer
418
views
Frobenius theorem with lesser regularity
Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius ...
- 1,211
2
votes
0
answers
246
views
A integral equation with Discrete to result by inverse problem
Problem
I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
- 4,280
3
votes
1
answer
363
views
First integrals of a 3D incompressible flow
Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
- 41
0
votes
0
answers
92
views
Sum of quasi-m-accretive operators
Kato's existence theory (see link) gives conditions under which a PDE has a (unique) solution. The equation of evolution he considers is
$$ \frac{dv}{dt} + A(v)v = f(v),$$
where $A(y) \in L(Y,X)$ for $...
- 101
2
votes
1
answer
840
views
Lebesgue Riemann Theorem.
Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that
Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x)...
- 1,594
2
votes
2
answers
507
views
Positivity of Second-Order Elliptic Differential Operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-...
- 1,701
4
votes
1
answer
227
views
Besov Characterization of Strichartz Estimate.
On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to
$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial_tu(0,x)=g(x)\end{...
- 43
2
votes
0
answers
113
views
Continuous inclusions Sobolev theorem, question [closed]
How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
- 369
2
votes
1
answer
1k
views
Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces
Consider a smooth metric measure space in which the integral of a gradient is meaningful. For example in the sense of upper gradients of Heinonen, or on a riemannian manifold with the associated ...
- 469
3
votes
1
answer
229
views
Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?
Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...
- 131
3
votes
0
answers
169
views
Why a cone/parabolic set for the nontangential maximal function?
Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...
- 81
2
votes
0
answers
110
views
Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
- 271
3
votes
1
answer
375
views
Exponential decay for the gradient of a solution
Dear all,
I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$:
$$
\sqrt{-\Delta+m^2} u +u= f(u),
$$
where I can assume that $m \neq ...
- 31
1
vote
0
answers
431
views
Lyapunov Schmidt; basic example
I am attempting to understand the Lyapunov-Schmidt method with a simple example but I am running into trouble. Here is the example I am considering. Suppose $ v>0$ satisfies $ -\Delta v - v=0 $ ...
- 539
2
votes
0
answers
77
views
Solving a system of Laplace equations
Let $u_0$ and $u_1$ to be smooth functions defined on $\Omega\subset\mathbb{R}^n$, consider the following system of equations
$$\triangle u_1 = C_1(\partial_{ij}u_0),$$
$$\triangle u_0 = C_0u_1,$$
...
- 21
3
votes
1
answer
659
views
Short time existence on Hyperbolic Ricci flow in non-compact case
We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...
user21574
1
vote
1
answer
1k
views
Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
- 1,644
2
votes
1
answer
2k
views
A question about the proofs of the Sobolev embedding theorem.
I have seen two proofs of the Sobolev embedding theorem. One uses the Hardy-Littlewood-Sobolev inequality
$
\displaystyle \|f\|_{L^q({\bf R}^d)} \leq C_{p,q,d} \|f\|_{W^{1,p}({\bf R}^d)}.
$
One ...
- 23
1
vote
1
answer
462
views
Strichartz estimates of damped wave equation
If $w(t,x)$ is a solution of wave equation
$$
w_{tt}-\triangle w = 0, w(0)=w_0, w_t(0)=w_1,
$$
then $w$ satisfies the following Strichartz esitmates
$$
\|w\|_{L^q_tL^r_x} \lesssim \|w_0\|_{H^1} + \|...
- 425
1
vote
1
answer
360
views
Existence of the solution of a linear parabolic pde
Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in L^q(0,T;...
- 243
2
votes
2
answers
413
views
Replacing large-dimensional ODE systems with one PDE [closed]
Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
6
votes
0
answers
191
views
behaviour of first eigenfunction near the boundary
Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses.
My interest is in how $ \...
- 1,385
1
vote
0
answers
191
views
$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value
Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
- 21
0
votes
1
answer
809
views
Wave equation v.s.Schrödinger equation
The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$
From the above that a wave operator can be ...
- 1,644
1
vote
0
answers
44
views
Uniqueness of positive solutions to the n-vortex type equation
The $n$-vortex equation, in the context of optical vortex solitons, is of the form
\begin{equation}
-(ru_{r})_r+\dfrac{n^2}{r}u+\beta ru= f(u^2)ru,\quad r\in (0,\infty),\\
u(0)=0=u(\infty),
\end{...
-2
votes
1
answer
690
views
A Poincare inequality for the Laplace-Beltrami operator [closed]
Suppose $w \in C^2 (S^{n-1}), \Lambda$ is Laplace-Beltrami operator on the sphere $S^{n-1}$, How can I prove follow Poincare inequality :
$\int_{S^{n-1}} w\Lambda w d\sigma \leq (1-n) \int_{S^{n-1}} |...
- 21
1
vote
0
answers
227
views
Showing existence of positive weak solution of a PDE by CoV
Given the following PDE
$$
\begin{cases}
-\Delta u+\alpha=u^q &x\in\Omega\\
u=0 &x\in\partial\Omega
\end{cases}
$$
where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, $1<...
- 679
1
vote
1
answer
2k
views
Basic questions about parabolic Holder space
Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...
- 67
1
vote
2
answers
942
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 1,471
0
votes
1
answer
156
views
Prove a function, defined by integration of a harmonic function, is log-convex [closed]
Let $u$ be a harmonic function and we define
$$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$
The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function ...
- 679
1
vote
1
answer
121
views
A coarea formula when proving maximum principles for strong solutions in Chapter9 in Gilbarg-Trudinger's book
strong text
In GT's book(1998 Edition) Chapter9 P223,
Let $g$ be a nonnegative, locally integrable function in $\mathbb{R}^n$ and $u\in C^2(\Omega)\bigcap C^0(\bar\Omega)$.
How to prove
$\int_{Du(\...
- 11
1
vote
0
answers
101
views
Suggestion for books in Pertubation theory with an emphasis on the theory
As the title suggest I am looking for another good coverage of the theory of Pertubation theory.
Currently I am working through Murodock's book: Pertubations: Theory and Methods.
But I am rest assure ...
- 1,594
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
2
votes
0
answers
142
views
elliptic regularity for Neumann BVP on square
I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(...
- 1,385
2
votes
1
answer
333
views
Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$
Consider the elliptic eigenvalue problem
$$
\begin{cases}
\int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\
\qquad \qquad \qquad \quad u&...
- 632