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
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Checking initial condition of PDE is satisfied in Galerkin method
I asked this question here: https://math.stackexchange.com/questions/416885/checking-initial-condition-of-pde-is-satisfied-in-galerkin-method
But I did not receive the solution so I post it here.
...
- 21
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103
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Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
- 425
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h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
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291
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Laplacian type operator on compact Lie group
Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
- 11
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A heat kernel for Schrödinger operator with low-order terms
In "Schrödinger Operator: Heat Kernel and Its Applications", Feng computes the heat kernels associated to Schrödinger operators with at most quadratic potentials.
I am trying to see how these work ...
- 363
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440
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A problem about Joint sine and cosine fourier transform
There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
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149
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(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
- 11
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null controllability of linear wave equation
Consider the linear wave equation :
$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$
Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
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133
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nodal lines in the dirichlet problem
In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help.
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About definition of weak derivative in abstract PDE problems
I'm confused about weak derivative definition.
$u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff
$$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$
holds for all $\varphi \in C_0^...
- 11
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strong stability for the wave equation
Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
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608
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Solving a PDE involving a mixed derivative for a partial derivative
Consider a PDE of the form
\begin{equation}
\frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right)
\end{equation}
or
\begin{equation}
\frac{\partial^2u}{\partial p\...
- 21
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227
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Trace Inequality question
There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial \...
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294
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Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in L^2(0,T;...
- 113
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186
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Degeneracy and singularity of the $p$-laplace equation
In what sense is the $p$-Laplacian degenerate for $p$ greater than $2$ and singular for $p$ less than $2$?
- 129
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102
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Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product
Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
- 5,327
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315
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"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
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204
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Spectrum of Combinatorial Laplacian
The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
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Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
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127
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A critical elliptic PDE
I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
- 43
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154
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two polynomial equations
Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...
- 1,359
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171
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Compactness of solutions of elliptic equation
Consider the following nonlinear elliptic equation
$$
-\triangle u + u + u^3 = g, \quad x \in R^3.
$$
If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
- 425
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232
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Invariance of a tensor Laplacian
Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let $...
- 3,322
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503
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Relation between interpolation spaces and besov spaces
Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x u\|_{L^{\...
- 11
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When can a perturbation be treated as a regular perturbation?
I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...
- 351
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125
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base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
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elliptic system; bounds on $v$ when $u$ is small
I am interested in the following system
$-\Delta u = f(u,v) $
$-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary.
The solutions are smooth and positive. ...
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187
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Strichartz estimates over cones
I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality:
$\|...
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202
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Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
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258
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Is this Stefan-type problem an open problem?
I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
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305
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Strong minimum principle for maximal plurisubharmonic functions
Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
- 1,211
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separation of variables in differential euqtions and compact self-adjoint operators [closed]
For some partial differential equations in physics, people may separate the variables and get some eigenfunctions. And then for any solutions for that equation, people often suppose them to be a ...
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How to prove that 1 is not an eigenvalue of $T'(x)$?
Given a compact continuous operator $T$ from a Banach space $V_1$ to itself and $T$ maps a convex closed bounded set $\mathcal{B}$ into itself, how can we show that 1 is not an eigenvalue of $T'(x)$ (...
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368
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Definition of spectral gradient
Consider this differential operator
$$
\mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x}))
$$
where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow \...
- 817
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decay for spatially discrete parabolic equations with non-constant non-self-adjoint right hand side
Consider the following uniformly parabolic lattice differential equation
$ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & &...
- 471
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Geometric description of Jacobi's theorem on complete integrals of HJ eqn.
I am not sure if this question is adapted to this site, if it is not, then I will delete it.
The Hamilton--Jacobi theory is about the connection between:
the solutions of an Hamilton--Jacobi ...
- 4,306
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486
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Maximum principle for heat eq. with boundary conditions on derivatives
The Maximum principle for parabolic eq. is based on the fact that the boundary conditions are given on u.
How can this Maximum principle be used, when having boundary conditions including derivatives....
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530
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Inconsistency in definition of characteristics for a linear PDE? Folland versus Fritz John.
There seems to be a major inconsistency (perhaps due to my lack of understanding) between what Folland calls a "characteristic" and what I had previously thought was a characteristic.
For example, ...
- 2,641
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463
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Splitting wave equation for application of CPML
A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) ...
- 157
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312
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Vector field with Harmonic flow
Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle_g$...
- 356
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Analyticity of the solutions of PDE
Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*.
Is it possible a solution $f$, which is non-analytic at any point of the ...
- 4,312
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478
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Solution for a system of PDEs
I recently came across a system of PDEs
$\frac{\partial S}{\partial z}= f_1(x,y,z,w,t)$,
$\frac{\partial S}{\partial w}= f_2(x,y,z,w,t)$,
$\frac{\partial S}{\partial t}= f_3(x,y,z,w,t)$,
$S(x,y,1,1,1)...
- 459
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471
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Frobenius condition
Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...
- 4,145
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89
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Gehring Lemma in dimension 2
In Iwaniec's paper presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$.
Question. What about dimension 2: can we actually go down to $p=1$?
- 2,494
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873
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PDE - Two Dimensional Inhomogeneous?? [closed]
I'm looking at a two dimensional, second order, inhomogeneous equation which has no boundary conditions. I realize that there could be zero or infinite solutions to a problem like this, but I can't ...
0
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1
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199
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Why $-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}$ type PDE is called 'mean-field equation'?
Why $$
-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}
$$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical ...
- 809
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507
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Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
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63
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Non-existence of rapidly decaying solutions of certain elliptic semilinear equations
Consider the equation
$$ -\Delta f+mf+\lambda f^p=0$$
on $\mathbb{R}^d$, where $d>2$,$m>0$, $p>1$ is integer, and $\lambda \in \mathbb{R}$. Are there any known results regarding the non-...
- 505
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2
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197
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Concerning the decay of the ground state of certain Schrodinger operators
Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
- 219
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255
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Numerical methods for solving a hyperbolic nonlinear PDE
What type of numercial methods are there to solve PDE of the sorts of:
$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$
$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t)...
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