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
7
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Heat equation bounds
I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{x \in [-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.
In particular I am ...
- 143
2
votes
0
answers
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Size of the eigenfunction of Laplacian (reference request)
It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
$$||\phi||_{L^\...
- 1,692
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votes
1
answer
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From an integral equation to a differential equation
Hello,
I am wondering whether it is possible to convert the following integral equation to a partial differential equation.
Integral Equation here http://ima.epfl.ch/~lechen/images/integralEq.jpg
...
- 1,649
3
votes
2
answers
121
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Estimates on a heat process with fixed boundary data and zero initial conditions
Consider the following heat process:
For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that
$$
\partial_t p(...
- 337
3
votes
1
answer
207
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Coercivity for functional and complete orthonormal system
Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional
$$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$
I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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3
votes
1
answer
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Does replacing 'local' with 'global' in the definition of viscosity sub(super)solution change the definition?
Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this:
We say $u$ is a viscosity subsolution iff for any test function $\phi\in C^2(\...
- 383
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votes
0
answers
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Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
- 85
3
votes
0
answers
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Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...
- 93
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0
answers
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The best constant in Poincare-liked inequality in $BV$ and $BD$ space
This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
- 679
3
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2
answers
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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-...
- 539
0
votes
0
answers
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Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
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9
votes
1
answer
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Mean value property with fixed radius
Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
- 981
1
vote
1
answer
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Reference request: Spectral analysis of advection diffusion PDE
As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...
- 2,281
4
votes
1
answer
586
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Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
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1
answer
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Is there a lower bound for variance in terms of curvature?
If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (...
- 16.6k
1
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1
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Wave equation with linear coefficients
The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...
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1
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A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable
First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
$$\left[\left(\cos\phi\partial_{z}+\...
- 245
2
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0
answers
283
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Newtonian potential for continuous $f$
Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$.
Consider the Newtonian potential of $f$ (at least I think this is what it is called)
$$ v(x)=\int_{R^N} \...
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3
votes
1
answer
213
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A question about solutions to Floer's equation which are asymptotic to a stationary point
Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X_H\otimes dt)^...
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votes
1
answer
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Asymptotic decay for the wave equation
Consider the wave equation
$$
y_{tt} = \Delta y - \epsilon y_t
$$
on $\Omega\subset R^n$, with Dirichlet boundary conditions.
Where $\epsilon >0$.
Is it possible to find an explicit value $\...
- 51
1
vote
0
answers
147
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Energy inequalities for Sobolev spaces of negative integer
I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\...
- 101
2
votes
2
answers
408
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Higher order partial derivatives and global regularity.
Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...
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8
votes
1
answer
667
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How does electric potential relate to mean curvature?
Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.
Now consider the electric potential generated by this uniform mass ...
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3
votes
1
answer
215
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Checking initial data in parabolic PDE with no control on time derivative
It is possible to define a weak solution of a parabolic PDE
$$u_t - Au = f$$
$$u(0) = u_0$$
as $u \in L^2(0,T;H^1)$ such that
$$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega Au(t)\varphi(...
- 155
2
votes
1
answer
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Does the green kernel converge as a series of functions?
Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...
user50806
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1
answer
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Hopf Boundary Point Lemma
The Hopf Boundary Point Lemma
http://en.wikipedia.org/wiki/Hopf_lemma
is a result for the unit normal vector field and the normal derivative.
Is it true if one considers arbitrary directional ...
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3
votes
0
answers
105
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Can Mumford-Shah functional be adapted to lower $L^1$ space?
The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
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vote
2
answers
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Heat equation with Neumann BC
Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.
Is this true to say:
$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ ...
- 23
1
vote
0
answers
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Delta function propogation in the NLS regime
consider the $(1+1)D$ or $(2+1)D$ NLS:
$$ i\psi _t (t,{\bf x}) + \Delta \psi + |\psi|^2\psi = 0 \, ,$$
$$ \psi (t,|{\bf x}|\to \infty) =0 \, , \quad \psi(t=0,{\bf x}) = \psi_0 ({\bf x}) \, , $$
with $...
- 3,574
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0
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Help in understanding "Local well-posedness for the Maxwell-Schrodinger system"
Is there someone who knows the following paper
"Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada.
I'm trying to study it but I've some doubts. In particular I'm not ...
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votes
2
answers
812
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Estimates for Green's function
Let $n$ - dimension $\geq 3$.
Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \...
- 3
1
vote
1
answer
160
views
$L^p$ estimate for (powers of) a Laplacian with inverse square potential
I need an estimate of the form
$$ \|v\|_{L^p} \le C \|(K-\Delta- c|x|^{-2})^s v\|_{L^p} $$
where $K>0$ can be large if necessary, $c$ is positive but below the Hardy constant $(n-2)^2/4$, where $n$ ...
- 9,007
3
votes
2
answers
268
views
Maximum of the solution of a parabolic PDE
Let $u:\mathbb{R}\times [0, \infty) \rightarrow \mathbb{R}$ be defined by
$u_{xx} + u_x - u_t = u(u - 2)(u - 1)$
with $u \rightarrow 0$ as $|x| \rightarrow \infty$ and $u(x,0) = 3e^{-x^2}$. Now, let ...
- 409
2
votes
2
answers
470
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Linear coupled parabolic PDE system with Holder continuous coefficients
I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$
$$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = g$...
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1
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1
answer
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Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from $L^2$ case to $H^{-1}$ case?
I have the PDE
$$u_t(t) - \Delta f(u(t)) = 0$$
in $H^{-1}(\Omega)$ where $f$ is a nonlinear function.
Define $F(s) = \int_0^s f(s)$. Note that if $u_t(t) \in L^2(\Omega)$,
$$\frac{d}{dt}F(u(t)) = f(...
- 266
1
vote
1
answer
361
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maximum principle for a non-uniformly parabolic operator
Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator
$$
P = \partial_t - \mathrm{e}^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x} \big( G(...
- 36
2
votes
0
answers
114
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biharmonic equation with L^1 data and Navier Condition
I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
- 371
2
votes
3
answers
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Maximum principle corner
Hello,
consider a parabolic boundary value problem, for instance
$-\partial_tu+\Delta u=0$, in $\Omega$,
$\partial_\nu u=0$ on $\partial\Omega$,
in a domain $Q=(0,T)\times\Omega$, where $\Omega\...
- 225
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1
answer
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Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$? [closed]
Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging ...
- 17
0
votes
0
answers
435
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Analytic solution of Poisson's equation
Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.
Impose a von Neumann boundary condition $...
- 409
1
vote
2
answers
417
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Solving a simple PDE on a 2 dimensional manifold
On a 2 dimensional surface X is a smooth vector field with isolated zeros. When is there a vector field Y such that the Lie bracket [x,Y] is equal to fY for some given function f on the surface minus ...
- 439
4
votes
1
answer
243
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On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument
Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
- 363
4
votes
2
answers
1k
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Reference for Neumann-Laplacian
Let $\Omega\subset R^d$ be a bounded, smooth domain. Consider $A=-\Delta$ subject to homogeneous Neumann boundary conditions in $L^p$-spaces. Does anybody know a good reference book on basic results ...
- 225
3
votes
1
answer
1k
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Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 1,649
3
votes
3
answers
2k
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Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 131
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
2
votes
1
answer
358
views
Please recommend some classical books or articles on the compressible Euler equations !
Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the ...
- 35
0
votes
0
answers
515
views
If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
user24451
1
vote
2
answers
633
views
Fundamental Solutions with compact support (distributions)
Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$
We also then argue that if a fundamental solution has compact support, then it is supported ...
- 379
1
vote
0
answers
86
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Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
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