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.

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14 views

Uniqueness for a transport-diffusion equation with low integrable drift

Consider the equation $$ \frac{\partial f}{\partial t} + u \cdot \nabla f - \Delta f = 0 $$ in $(0,T) \times \mathbb R^N$, with initial condition $$ f \vert_{t=0} = f_0 $$ for some given $f_0 \colon \...
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0answers
46 views

Relation between the $L^2$ norm of the Poisson bracket of $f$ and $g$ and their $H^1$ norms

Let $f,g\in H^1(\Omega)$ where $\Omega$ is a sufficiently nice bounded domain in $\mathbb{R}^2$. If $\{\cdot,\cdot\}:H^1(\Omega)\times H^1(\Omega)\to L^2(\Omega)$ is the Poisson bracket, is there some ...
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3answers
300 views

about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
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1answer
187 views
+50

A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$. What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
2
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1answer
96 views

Gronwall estimate with a Fourier transform

Suppose I have the following equality $$\hat{u}_\epsilon(t,k) = \alpha(t,k) + \int_0^t\int_{\mathbb{R}^n} e^{ik\cdot(x +\epsilon \phi(s,x))}u_\epsilon(s,x)dxds$$ Where $\alpha(t,k) \geq 0$ and $\alpha(...
3
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0answers
134 views

Nonconstant smooth maps $f:\mathbb{R}^4-\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$

Does there exist a nonconstant smooth map $f:\mathbb{R}^4\setminus\{0\}\to L\left(\mathbb{R}^4,\mathbb{R}^5\right)$ such that each $f(x):\mathbb{R}^4\to\mathbb{R}^5$ is an isometric linear map and $x\...
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66 views

Does my functional satisfy the Palais Smale condition?

Consider the functional $$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$ where $u:\mathbb{R}^2 ...
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67 views

Linearization Navier-Stokes

I am considering the classical form of the stationary Navier-Stokes equation given by $$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...
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0answers
63 views

Behavior of the principal eigenfunction of fractional Laplacian

How does the first eigenfunction $\phi_{1}$ behave near the boundary of $\Omega$ where $$(-\Delta)^s\phi_{1}=\lambda_{1}\phi_{1},\text{ in } \Omega; \phi_{1} =0 \text{ in } \Omega^c$$ in a n-...
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14 views

Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic

I first asked this question on math.stackexchange as I first thought it was somewhat innocent. I would like if one can solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where $...
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1answer
99 views

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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0answers
46 views

transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$

Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$. Applying the transformation $\Phi (\psi)=e^{...
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0answers
29 views

Boundary regularity of rectifiable multiplicity 1 hypercurrents

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
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0answers
73 views

Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem

I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein. Theorem A: Assume that $m: (0, \infty)\to \...
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2answers
113 views

Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...
2
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0answers
30 views

On maximum principle of spectral fractional Laplacian

Suppose $(-\Delta)^s u=g \geq 0$ in $\Omega$ and $u=0$ in $ \partial \Omega.$ Also suppose $u$ is $C^{2}$ non-negative and $(-\Delta)^s u=0$ in $\Omega \setminus B$ and $u\leq a$ on $\partial B $ ...
2
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0answers
51 views

Itō formula for the solution of a SPDE in the distributional sense

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
1
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1answer
101 views

first order derivative of the parabolic equation

Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE $$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...
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0answers
34 views

Modelling fluid flows with mean curvature flow

A while ago I was wondering if the displacement of fluid described in this blog post could be modelled with mean curvature flow or some other flow, but when I asked someone in Engineering they replied ...
3
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0answers
40 views

Harnack inequality for $F(D^{2}u, Du,x)=f$

I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$, ...
5
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1answer
278 views

When is a distribution having a finite support actually zero?

Let $D$ be a differential operator with smooth coefficients in $\mathbb{R}^n$. Suppose $E$ is a bounded open set in $\mathbb{R}^n$. Suppose $u$ is a function that is smooth up to the boundary of $E$. ...
3
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0answers
46 views

Complex Monge Ampere equal with zero right hand side

Let $(X, \omega)$ be a compact Kaehler manifold, with $K_{X}$ numerically effective. Suppose that $[\zeta] = c_1(K_X)$ and $\int_{X}\zeta^n = 0$. I am interested in solving \begin{equation} (\zeta + i ...
3
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1answer
91 views

Existence of a special function

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$. Is there any smooth function $...
2
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0answers
85 views

Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...
5
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1answer
172 views

Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation) $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1] $$ with periodic or ...
3
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1answer
117 views

Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
2
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0answers
48 views

Modified variational formulation of heat equation

The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to $$ u_t = \Delta u, $$ subject to certain boundary conditions and can alternatively be described, in variational form, as ...
2
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0answers
72 views

Can Schauder's fixed point theorem apply to a metric space?

I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
5
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0answers
98 views

Sobolev extension from a discrete set of points

Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define $$...
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0answers
71 views

Change of one boundary condition in a problem

This is a problem in modeling in hydraulic fracturing field. It's quite long so hopefully someone can patiently read and help me. The equation numbers are match those of the reference paper by ...
5
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0answers
83 views

Relations between $\mathcal D$-modules and Exterior Differential Systems?

As a followup of this answer, I wonder relations between $\mathcal D$-modules and Exterior Differential Systems, both of which could be used to describe systems of PDEs. If I understand correctly, $\...
30
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2answers
1k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
15
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4answers
4k views

unique continuation principle

I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on ...
6
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0answers
321 views

Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test $$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
3
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1answer
78 views

How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by $$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$ where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
10
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3answers
817 views

Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
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0answers
22 views

Some scattering properties of cubic NLKG (how can we check the mixed-norm is bounded)?

Let $n=3$ and $u$ be the solution to Klein-Gordon equation \begin{equation} \begin{cases}\dot{u}-\Delta u +u=u^3 \\ u(0)=u_0, \partial_t u(0)=u_1, \end{cases} \end{equation} where $(u_0,u_1) \in H^...
3
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0answers
106 views

Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of Regular Lagrangian flow Filippov solution Caratheodory solution of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
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0answers
66 views

Reference for Rellich Kondrachov theorem on bounded domains and spaces with finite measure

I read at the top of the page $580$ of this paper that the imbedding $W^{1,2}(\Gamma,d\mu) \hookrightarrow L^2(\Gamma,d\mu)$ is compact for a bounded domain $\Gamma \subset \mathbb{R}^n$ and a measure ...
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1answer
143 views

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 ...
1
vote
1answer
71 views

A completion equality in product of Sobolev spaces

Let $\Omega$ be a nice bounded domain of $\mathbb{R}^n$. For $s\ge 0$ we define $$H_{0}^{s}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{H^{s}(\Omega)}.$$ My question: is the following equality true? $$...
11
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1answer
1k views

A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
1
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0answers
46 views

Proof of uniqueness of solution of the Poisson's equation for given boundary conditions [closed]

I asked this question last week on Mathematics Stack Exchange, but I didn't get any feedback. It is not a research level question, however, I post it here, perhaps somebody could help me: I would ...
1
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0answers
43 views

propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$ $$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$ with given initial condition $u_0(x)$. Assume that: $u_0$ is rotation invariant, ...
2
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0answers
44 views

Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain $$ \begin{cases} \rho_t + q_x = 0 \\ q_t + (q^2/\rho + \rho)_x = - q \\ u|_{t=0} = u_0 \\ u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t) \end{cases} $$ in which ...
2
votes
1answer
143 views

Numerical analysis of parabolic obstacle problem

I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$ \begin{align*} \langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\...
2
votes
1answer
70 views

Explicit solution of a free boundary problem for heat equation

Consider the free boundary problem $$ \min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\ u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\ u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...
1
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0answers
84 views

Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$

Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
0
votes
0answers
53 views

Inf sup condition for discrete Stokes problem

I am considering the discrete inf-sup condition for the discrete Stokes problem $ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$ This means ...
2
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0answers
41 views

$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

Consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial condition ...

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