The main idea of the proof is to consider the general 2D Euler solution as a small perturbation of the 1D Burgers equation. According to Collot-Ghoul-Masmoudi [2], one of the self-similar solutions is linearly stable, and all other Burgers solutions will be drawn from that solution at the singularity point. Using the idea of Buckmaster-Shkoller-Vicol [4], we can show that the 1D Euler solution is also in C13.
Definition of a singularity and a shock
The right side of this equation is inC1, therefore we get W∈C2. By repeating the same procedure, we arrive at the conclusion W ∈C∞. The converse of the claim follows through the inverse function theorem. Since the Burgers equation has spatial translation invariance, so do steady self-similar Burgers. Since the Burgers equation has time-translation invariance, so do steady self-similar Burgers.
Riemann invariants
The main point we argue is the global well-posedness of the smooth solution and its uniqueness.
No vacuum for 1D Euler
Singularity Theorem
Set r=∂xw1, and define the new characteristics x(θ,t) satisfying dx(θdt,t)=λ1,and x(θ,0) =θ. Then the equation becomes Therefore, the C1 norm of ρ,u is bounded in [0,T]×R. Therefore, from an a priori assessment, the solution is well positioned globally. This implies that ∂xw1 blows, so the rate C1 of ρ and u explode at the same time T.
In this proof, ∂xw blows up at the singularity time when the second condition is satisfied. Therefore, we can guess that the derivative of some values corresponding to the Riemann invariants in 2D Euler will also diverge. Also, in Appendix B, we will see the regularity of the 1D Euler solution at the singularity time using the idea of Buckmaster-Shkoller-Vicol [4].
In this chapter we will see the singularity formation of 2D isentropic compressible Euler equations, introduced by Buckmaster-Shkoller-Vicol [4]. For the shock formation of multidimensional Euler equations, the various works on non-rotating flow (zero vorticity) exist with the idea that the equation turns into second-order quasilinear wave equation. Therefore, in Buckmaster-Shkoller-Vicol [4] we focus on the flow with non-zero vorticity, and not on a perturbation of plane waves.
We will consider the 1D Euler solution as a small perturbation of a Burgers solution, which will be considered a purely azimuthal 2D Euler solution.
Polar form of 2D Euler Equation
To find a relation to 1D equation, let's set a=0 for a moment so that the solution of 2D is independent of r. We can see that under this condition the equations (72b), (72c) are perfectly equivalent to 1D Euler system. To see the variables that are likely to blow up, let's turn all the variables into Riemann invariants using 1D idea.
Through our iteration in (82), we can see that all the variables {ck}and {λk} are indeed bounded. Since {λk} are the escape velocities, we now have a domain G such that all points starting in G cannot escape G. One such G is a hexagonal region bounded by the two vertical lines t=±α and the four steeply sloping lines x±β =±Kt. Our idea is by choosing small enough α, which reduces the range of Ukm,Xkm.
Like 2nK. Because the bounds become tighter as α goes to 0, the infinite series represent a convergent solution to the Cauchy problem. We can repeat the same process by proving local existence if the corresponding K is uniformly defined. But we have excluded the||w||L∞ and||z||L∞ terms from the necessary condition because these remain finite if a∈Lt1L∞x, and also the||∂θa||L∞ term.
If we set up a new vortex using our new variables, ω= ωP = 2b−∂Pθa, it satisfies∂tω+b∂θω =αaω. It is a transport equation and the vortex cannot break up, so the force term must be finite.
Self-similar variables
These new relations seem viable at first, but correspond to Burgers' self-similar solution, U(t,θ) = (−t)2i1Wi(x)withi=1, sincee−s2 = (τ(t)− t) 12. Although the domain iθ is a compact domain T, the domain of W,Z,A can be extended to R. We will prove it later. As we mentioned before, we need the initial condition (104) to share the same initial condition as Wi. Proposition 7 Changing the variables of the 1D Burgers equation using similar variables and substituting τ˙=0,ξ˙ =c,κ=c, we get (∂s−12)U+ (32x+U)Ux =0 with W as solution.
Therefore, in 2D Euler equation, τ˙=0,ξ˙ =c,κ=c is the condition equivalent to the transformation of the equation into 1D Burgers equation.
Our goal
It means that ∂θwf first inT∗. To ensure that T∗ is the smallest singularity time, we need the conditions∂θa,∂θz<∞ast→T∗. But before we show it, let's check that our solutions satisfy the conditions of positive density and nonzero vorticity all the time. a). Also, since P(θ,t)≥(α2inft|w−z|)α1, to guarantee P(θ,t)>0, we need a uniform lower bound c of P satisfying|w−z | ≥c>0. Here, we will define the following initial data conditions for ease of future calculations. From here on, we will denote W =W1, where W1 was a solution of Burgers' consistent self-similar equation.
We deliberately make the equation in the form of the last term to use the second maximum principle in Appendix A. x0)23+f, which is the kernel notation in the second maximum principle. Since the second term is non-negative at x∗(s) and the first term is equal to dsν(φ(x,s),s) for the characteristicφ satisfying ∂ φ∂s =U(x,s), we obtain. For this we need to check: the final initial data, the final velocity characteristics and the final term of the force.
Now we will check whether all terms corresponding to force and speed terms are finite or not. To do this, let's see if both conditions can be satisfied at the same time. Since we want to see W as a solution of Burgers equation at γ=3, we hope to extend the domain to R.
Let's rearrange the inequalities we needed for our purpose and prove it using the bootstrap method.
Assumptions on the initial datum
Bootstrap assumptions
Closing the assumptions
However, since finer bounds are needed for W than the previous assumptions, we will focus on a toy problem to get an idea of the calculation. Now, to evaluate the damping term of (207c), split the middle terms into two, so that one contains only the functions we already know, such as W, and the other has the remainder, because the middle terms can be new depreciation term. 208a) We do not know the exact value of the second term, but we want a situation where the first term has a positive lower bound large enough to cover the second term, so that it can have a positive lower bound of the total amortization term. For this, we will define a new variable V(x,s) =g(x,s)W˜x with some functionsg(x,s). Here, g will be determined later in the process of establishing a positive lower bound. can use the relations ˜Wx=Vg,W˜xx=Vx−. The first term becomes a damping term of V, and the second obviously does not.
We will find a limit of V, but because F =0, the limit will be constant by the maximum principle. At this point, we see that we cannot use the first maximum principle because the lower bound goes to 0 as x decreases to 0. Since|F|=0 and K=0, their conditions satisfy the second maximum principle. 1+x2)12. Therefore, to have a constant bound on ˜V, additional initial guesses on the partial derivatives of W are required.
To satisfy the next condition ||V˜(x,s)||L∞(|x| Therefore, any open set that satisfies the rest second conditions and the third one, which is transformed to satisfy the first condition, will satisfy all the initial conditions of W. Bootstrap method Also, there exists a minimum time s∗ and its corresponding x∗ such that |f(x∗,s∗)|= 3m4 because f is a continuous function (if not, the transport equation is not well defined.). When z=0 in (76), we proved that w∈C13 is at the singularity time when the initial date W is in a special small neighborhood of W in C4. So we are interested in the solution (w,z) (252) that satisfies w6=0,z6=0. They are not solutions of (252), but we can show the decay rate of Win in the same way as 2D Euler. Like 2D Euler, we can change the variables using self-similar variables and modulations. According to Theorem 2, when the derivatives of one Riemann invariant are negative for some point, the 1D Euler solution explodes at the same singularity time T , and before such T the solution is well defined. To show the regularity of w at the singularity time, we changed the equation of w to a new variableν, and applied the second maximum principle to find upper bound ofν. Finally, by the relation between W and w, we obtained w∈C13. In that idea, we have never used (76b) and (76c), so we do not need to modify the proof at all, for the proof of 1D Euler solution. This substitution does not disturb the limit of power term, therefore we can apply the second maximum principle as in chapter 4.3. The proof of 2D uses Proposition 7, to say that w and z would not affect the divergence of e∂θz. For this, we will make the adequate property corresponding to Proposition 7. This was because the proof of 2D Euler solution almost ignores the modulations and derivatives of Z and A due to their sufficiently small limits. In this paper, we have studied that the initial data for solutions of 1D Burgers, 1D Euler, 2D Euler whose profile at the time of singularity is inC13 are in an open set iC4. For this, we first introduced the 1D Burgers self-similar solution W, and saw that 1D Euler, 2D Euler equations can be expressed as a perturbed form of 1D Burgers, in Appendix B and Chapter 4. And then we explained that the general solution of these equations has the same decay rate as W, so we proved that the difference between Wx and the spatial derivative of the general solution is bounded. First, we changed the 2D isentropic compressible Euler to polar form and tried to find a solution independent of r to use the property solution of the 1D Euler equation. The new a,b,P equation coincided with the 1D Euler equation at a=0 and we found that the a=0 condition is equivalent to γ=3. At this point, since one of the Riemannian invariants of the 1D Euler solution has a diverging derivative at the singularity point, we could guess that the same would happen on the 2D Euler generala6=0 conditions. In particular, the fact that the regularity of w at the singularity point becomes C13 like it is a new achievement in [4]. Wang, “The Cauchy problem for the Euler equations for compressible fluids,” in Handbook of Mathematical Fluid Dynamics. Finally, I would like to thank my family, who always encouraged me to continue my studies, and Hyunsoo Kang, who always supported me for seven years.Maximum principles
Perturbed form of 1D Burgers
Closing bootstrap assumption
