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.
4,468 questions
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An inequality for uniqueness proof of NLS
Setting
Although this detail is not relevant to my question, let me set the problem that my question arise.
We are considering an initial value problem
\begin{align*}
\begin{cases}
u\in L^\infty(I,H^{...
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2
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Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
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Density property fractional heat kernel
Let us consider $$p_t^{(n+2)}(\tilde x) , $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\...
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Does $\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le C$ imply $\Vert u_x (t,\cdot) \Vert_{L^2(\mathbb R)} \le C$ in the heat equation?
For the parabolic equation
$$u_t + f(u)_x - u_{xx} = 0$$
one has
$$\Vert u(t,\cdot) \Vert_{L^2(\mathbb R)} + 2\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le \Vert u(0,\cdot) \Vert_{L^2(\mathbb R)}.$$...
user140746
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Euler-Lagrange equation for a functional
What does it mean that the equation:
$$ \text{div}_{x,y}(y^a\nabla_{x,y}u)=0,\quad \text{in }\mathbb{R}^n\times(0,\infty),$$
is the Euler-Lagrange equation for the functional:
$$ J(u)=\int_{\mathbb{R}^...
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Two types of limits of viscosity solutions
I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to ...
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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 "...
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Asymptotics for solution of transport equation and characteristics
Consider the transport equation $$u_t(t,x) + v(t,x) \cdot \nabla u(t,x) = 0.$$
Suppose that the solution of the characteristic equation
$$\dot X(t) = v(t,X(t)) $$
decays to zero as $t \to \infty$. ...
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How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
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Scaling argument for the heat equation in a bounded domain [closed]
We want to study the long time behavior of the heat equation $u_t - u_{xx} = 0$ in the domain $[-1,1]$. Now consider the rescaling $u^{\epsilon} = u(x/\epsilon, t/\epsilon^2)$. Then
$u^\epsilon$ ...
user140746
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Existence of subsequences convergence with weak topology
Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
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finding Morse index for the following functional
not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...
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Fractional super-harmonic functions
Is this statement true. A bounded half-superharmonic function in $\mathbb R$ is a constant. That is $(-\Delta)^{1/2} u\geq 0$ implies $u\equiv 0.$
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$L^2$ bound and interpolation of Hölder norm
Consider the function
$$F(x):=\int_{\mathbb R} f(t+x)f(t-x) \ dt .$$
Clearly, we have by Cauchy-Schwarz
$$\vert F(x) \vert\le \Vert f \Vert^2_{L^2} $$
$$\vert F'(x)\vert\le 2\Vert f' \Vert_{L^2} \...
user144765
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Estimate for Laplace equation with Neumann boundary on manifold with corner
Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.
...
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Wellposedness results for the cubic Schödinger equation
Motivated by the question Relationship between the vortex filament equation and the cubic Schrödinger equation,
I'd like to ask the following:
Where can I find a reference on wellposedness ...
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Relationship between the vortex filament equation and the cubic Schrödinger equation
How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...
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Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz
Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...
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Finding Free surface elevation in semi-infinite channel
A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite ...
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Analytic approach to geodesic connectedness in Semi-Riemannian manifolds
Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
user60665
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Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
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Change of variables for double integral [closed]
Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to ...
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Accessible reference for (scattering) $\Psi DO$'s on manifolds
I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators ...
- 1,247
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Functions satisfying Neumann boundary condition
I have a question about functions satisfying a condition.
Let $D \subset \mathbb{R}^d$ be a Lipschitz domain. That is, for each $x \in \partial D$, there exists an open neighborhood $U$ of $x$ in $\...
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How do I show continuity of the mixed and weak solution to Zaremba problem?
I am interested in showing continuity/boundedness of the weak solution to the following problem pde:
\begin{align*}
0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\
0 &= \...
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Is this function positive?
Could someone tell me if my argument is correct?
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...
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Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
- 4,135
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Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
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Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
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Dispersive estimate for linear semigroup
Let's consider the propagator corresponding to the one-dimensional equation
$$
u_t=\Lambda^\alpha u_x,\; u(x,0)=f(x)
$$
where
$$
\widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi),
$$
and $-1< \...
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Critical gKdV - tutorial
I'm looking for a good introduction to the critical generelized KdV equation
$$u_t +u_{xxx}+5u^4u_x = 0 \, , $$
$$ u(t=0,x) = u_0 (x) \, , \qquad x\in \mathbb{R} \, ,$$
and its blowup solutions. There ...
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A detail in Kato's paper (Strong $L^p$-Solutions of the Navier-Stokes Equation in $\mathbb R^m$, with Applications to Weak Solutions)
A detail in Kato's paper (Strong $L^p$-Solutions of the Navier-Stokes Equation
in $\mathbb R^m$, with Applications to Weak Solution).
Here is the link: http://junon.u-3mrs.fr/monniaux/K84.pdf
In the ...
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Property Sobolev space [closed]
Let $U \subset \mathbb{R}^d$ be open, $k \in \mathbb{N}$ and $1\leq p<\infty$. Furthermore we take a function $f$ contained in the Sobolev space, $f \in W^{k,p}(U)$. Take a look at the following ...
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Estimating pointwise multiplication conjugated by a Fourier multiplier
I asked this question first on MSE but there was no activity.
Let $m(D)$ be a Fourier multiplier and $f$ a known function. I'm trying to estimate the operator
$$Tu=m^{-1}(D)(f(x)m(D)u)$$
in say $H^s$....
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Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
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Domain of the Stokes operator
Let
$\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$)
$\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$
$\mathcal H:=\overline{\mathfrak ...
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A condition for Laplacian
Let $u\in L^{2}(\mathbb{R}^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$?
Thank you in advance.
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Linearized stream function
I am trying to work through a paper Instability in Parallel Flows Revisited by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance ...
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Existence and estimates of a solution of a perturbed first order partial differential equation
My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you ...
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Is this set of function belongs to $L^\infty$?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
- 679
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The hypoellipticity of a heat-like operator
I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {...
- 5,407
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How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
$\...
- 679
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answer
528
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Backward Uniqueness for the wave equation [closed]
Does the wave equation $u_{tt} - \Delta u = 0$ have any backward uniqueness results that are similar to the ones for the heat equation (see for example Theorem 11 page 64 in Evans)? If not, are there ...
- 3
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Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?
We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:
$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$
where $U(t)= e^{it\Delta} $(free ...
- 1,051
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297
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Complex transport equation
Consider an n dimensional Riemannian manifold with boundary.
Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...
- 4,135
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261
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A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable
First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
$$\left[\left(\cos\phi\partial_{z}+\...
- 245
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Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result
Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)
Let $\Omega \...
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Nonlocal (parabolic) PDEs in the Sobolev space setting
Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev ...
- 155
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2
answers
663
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Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
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286
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weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions
I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...
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